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\title{The $α \to 1$ Limit of the Sharp Quantum \renyi Divergence}
\author{Bjarne Bergh}
\author{Robert Salzmann}
\author{Nilanjana Datta}
\affil{DAMTP, University of Cambridge, United Kingdom}


\begin{document}
	\maketitle
	\begin{abstract}
		Fawzi and Fawzi \cite{fawzi_defining_2020} recently defined the sharp \renyi divergence, $D_\alpha^\#$, for $\alpha \in (1, \infty)$, as an additional quantum \renyi divergence with nice mathematical properties and applications in quantum channel discrimination and quantum communication. One of their open questions was the limit $α \to 1$ of this divergence. By finding a new expression of the sharp divergence in terms of a minimization of the geometric \renyi divergence, we show that this limit is equal to the Belavkin-Staszewski relative entropy. Analogous minimizations of arbitrary generalized divergences lead to a new family of generalized divergences that we call {\em{kringel divergences}}, and for which we prove various properties including the data-processing inequality. 
	\end{abstract}
	\section{Geometric and sharp R\'enyi divergences} 
	Let $\HS$ be a complex finite-dimensional Hilbert space, and $\BO$ the set of linear operators on $\HS$. We write $\pos$ for the set of positive semi-definite operators on $\HS$ and $\posdef$ for the set of positive definite operators. Let $\DM$ denote the set of density matrices, i.e.~the set of positive semi-definite operators with trace 1. For $A, B \in \pos$ we further write $A \ll B$ if $\supp{A} \subseteq \supp{B}$.
	
	
	For $A, B \in \posdef$ and $α \in \reals$ the \emph{weighted matrix geometric mean} is defined as
	
	\begin{equation}\label{def:matrix-geom-mean}
	    A \#_α B \coloneqq A^{\frac{1}{2}}\left(A^{-\frac{1}{2}}B A^{-\frac{1}{2}}\right)^α A^{\frac{1}{2}} \,.
	\end{equation}
	For $α \ge 0$ this definition can be easily extended to more general $A, B \in \pos$, $B \ll A$, by restricting the Hilbert space to the support of $A$ (this also corresponds to directly using pseudo-inverses in (\ref{def:matrix-geom-mean})). For $α \in [0,1]$ the weighted matrix geometric mean satisfies many desirable properties of a matrix mean \cite{kubo_means_1979}. 
	
	The geometric \renyi divergence (also called the maximal \renyi divergence), first introduced by Matsumoto~\cite{matsumoto_new_2013}, can be defined in terms of the weighted matrix geometric mean. For $ρ, σ \in \pos$, $ρ \ll σ$ and $α \in (0, \infty)$ define the \emph{geometric trace function} as 
	\begin{equation}
	    \label{def:geometric-trace-fn}
		\qrd[\widehat{Q}][α]{ρ}{σ} \coloneqq \tr (\sigma \#_{\alpha} \rho) = \tr(σ^{\frac{1}{2}}(σ^{-\frac{1}{2}}ρ σ^{-\frac{1}{2}})^α σ^{\frac{1}{2}})\,.
	\end{equation}
	This can be extended to general $ρ, σ \in \pos$ by setting
	\begin{equation}
	   \qrd[\widehat{Q}][α]{ρ}{σ} \coloneqq \lim_{ε \to 0} \qrd[\widehat{Q}][α]{ρ}{σ + ε \IdentityMatrix} \, .
	\end{equation}
	For $α > 1$ this limit is equal to $+ \infty$ if $ρ \centernot\ll σ$, whereas for $α \in (0, 1]$ the limit is always finite and explicit expressions for it can be found in~\cite{matsumoto_new_2013, hiai_different_2017, katariya_geometric_2020}.
	For $α \in (0, 1) \cup (1, \infty)$ one then defines the \emph{geometric \renyi divergence} as \cite{matsumoto_new_2013, petz_contraction_1998,  fang_geometric_2019, katariya_geometric_2020}
	\begin{equation}\label{def:geometric-div}
		\qrd[\widehat{D}][α]{ρ}{σ} \coloneqq \frac{1}{α - 1} \log \qrd[\widehat{Q}][α]{ρ}{σ}\, .
	\end{equation}
	
	It reduces to the classical $α$-\renyi divergence for commuting states and satisfies the data-processing inequality for $α \in (0, 1) \cup (1, 2]$~\cite{matsumoto_new_2013}. Further, it is known~\cite{matsumoto_new_2013, tomamichel_quantum_2016, katariya_geometric_2020} that for $ρ \in \DM$ and $σ \in \pos$
		\begin{equation}
		  \lim\limits_{α \to 1} 	\qrd[\widehat{D}][α]{ρ}{σ}  = \widehat{D}(\rho||\sigma) \coloneqq  \tr(ρ \log(ρ^{\frac{1}{2}}σ^{-1}ρ^{\frac{1}{2}})),
		\end{equation}
	the Belavkin-Staszewski relative entropy \cite{belavkin_c-algebraic_1982}. 
	
	In~\cite{fawzi_defining_2020}, Fawzi and Fazwi defined the sharp trace function for $α \in (1, \infty)$:
	\begin{equation}\label{def:sharp-trace-fn}
		\qrd[Q^\#][α]{ρ}{σ} = \min_{A \geq 0}\Set{ \tr A | ρ \leq σ\#_{\frac{1}{α}} A }.
	\end{equation}
    They defined the sharp R\'enyi divergence of order $\alpha$ in terms of it as follows:
	\begin{equation}\label{def:sharp-div}
		\qrd[D^\#][α]{ρ}{σ} \coloneqq \frac{1}{α - 1} \log(\qrd[Q^\#][α]{ρ}{σ}),
	\end{equation}
	and proved that it satisfies the data-processing inequality, and that it also reduces to the classical $α$-\renyi divergence for commuting states. Further, they showed that this divergence has several desirable computational
properties such as an efficient semidefinite programming representation for states and quantum channels (i.e.~linear, completely positive trace-preserving maps), and a crucial chain rule property
which can be exploited to obtain important information-theoretic results concerning quantum channel discrimination and quantum channel capacities.\smallskip

A natural question to ask is: {\em{What is $\lim_{\alpha \to 1} \qrd[D^\#][α]{ρ}{σ}$?}} 
\smallskip



This was left as an open question in~\cite{fawzi_defining_2020}.
In this paper we answer this question by proving that this limit is given by the Belavkin-Staszewski relative entropy (see Theorem~\ref{α1:theorem-sharp-bs-limit} below).

\smallskip
To address the limit of $α \to 1$ we want to make use of the following alternative characterization of the sharp trace function and sharp R\'enyi divergence.
    \begin{proposition} \label{prop:key}  
		For $α \in (1,\infty)$ and $ρ, σ \in \pos$
		\begin{align}\label{sharp-min-geometric}
		 \qrd[Q^\#][α]{ρ}{σ} &= \min_{A \geq ρ} \qrd[\widehat{Q}][α]{A}{σ}, \\
		 \qrd[D^\#][α]{ρ}{σ} &= \min_{A \geq ρ} \qrd[\widehat{D}][α]{A}{σ}.
		 \end{align}
	\end{proposition}
	\begin{remark}
	Note that the $A$ in the above expressions are in general unnormalized states and we use the definitions of the geometric trace function and the geometric R\'enyi divergence as in (\ref{def:geometric-trace-fn}) and (\ref{def:geometric-div}) without additional normalization factors.
	\end{remark}
	For what is going to follow, we prove a slightly stronger version of the above statement, given by the following lemma.
	
	\begin{lemma}\label{α1:compact-restriction}
		%For $1 < a <\infty$ and $α \in (1, a]$ as well as $ρ, σ \in \pos$,
		For $α \in (1, a]$, with $a \in (1, \infty)$, and $ρ, σ \in \pos$,
		\begin{equation}
			\qrd[Q^\#][α]{ρ}{σ} = \min_{A \geq ρ} \qrd[\widehat{Q}][α]{A}{σ},
		\end{equation}
		and the minimization can be restricted to $A \in K$ where $K$ is a compact subset of $\Set{A \in \pos| A \ll σ}$ depending only on $ρ, σ, a$, but not on $α$.
	\end{lemma}
	\begin{proof}
		Recall the definition of $\qrd[Q^\#][α]{ρ}{σ}$ for $\alpha >1$~\cite{fawzi_defining_2020}:
		\begin{equation}
			\qrd[Q^\#][α]{ρ}{σ} = \min_{A \geq 0}\Set{\tr A | ρ \leq σ^{\frac{1}{2}} (σ^{-\frac{1}{2}} A σ^{-\frac{1}{2}})^{\frac{1}{α}} σ^{\frac{1}{2}}}\,.
		\end{equation}
	    Fawzi and Fawzi~\cite{fawzi_defining_2020} showed that the minimization can be further restricted to $0 \leq A \leq c^{α - 1} \tr(ρ) \Pi_σ$, where $c = \mnorm{σ^{-\frac{1}{2}} ρ σ^{-\frac{1}{2}}}$ is the spectral norm, and $\Pi_σ$ is the projection onto the support of $σ$. It is easy to see that if $ρ \centernot\ll σ$ the minimization is infeasible, and so $\qrd[Q^\#][α]{ρ}{σ} = +\infty$. Moreover, if $ρ \centernot\ll σ$ and $A \ge ρ$ then also $A \centernot\ll σ$, hence $\min_{A \ge ρ}\qrd[\widehat{Q}][α]{A}{σ} = +\infty$ and the statement holds. Thus, we can assume $ρ \ll σ$.
		Taking $C \coloneqq \tr (ρ)\sup{_{α \in [1, a]}} c^{α - 1}$, which is always finite, we get:
		\begin{align}
			\qrd[Q^\#][α]{ρ}{σ} &= \min_{A \geq 0}\Set{\tr A | ρ \leq σ^{\frac{1}{2}} (σ^{-\frac{1}{2}} A σ^{-\frac{1}{2}})^{\frac{1}{α}} σ^{\frac{1}{2}}, \; A \leq C \Pi_σ} \\
			&=\min_{A \geq 0}\Set{\tr(σ^{\frac{1}{2}} A σ^{\frac{1}{2}}) | ρ \leq σ^{\frac{1}{2}} A^{\frac{1}{α}} σ^{\frac{1}{2}}, \; A \leq C σ^{-\frac{1}{2}}\Pi_σ σ^{-\frac{1}{2}} \leq C \mnorm{σ^{-1}} \Pi_σ }\\
			&= \min_{A \geq 0} \Set{\tr(σ^{\frac{1}{2}} A^α σ^{\frac{1}{2}}) | ρ \leq σ^{\frac{1}{2}} A σ^{\frac{1}{2}}, \; A^α \leq C \mnorm{σ^{-1}} \Pi_σ }
		\end{align}
		where we redefined $A$ multiple times, and understand $\mnorm{σ^{-1}}$ as the operator norm of the pseudo-inverse.
		While for general matrices $A, B \geq 0$, the statements $A^α \leq B$ and $A \leq B^\frac{1}{α}$ are not equivalent, they are equivalent in the case in which $B$ is a constant times a projector onto a subspace that includes the support of $A$. 
		This is because in this case $B$ is diagonal in the same basis in which $A$ (and hence also $A^\alpha$) is diagonal. 
		Hence, the operator inequality $A^α \leq B$ turns into a condition only on the eigenvalues of $A$, which is then equivalent to $A \leq B^\frac{1}{α}$. Thus, we finally get:
		\begin{align}
			\qrd[Q^\#][α]{ρ}{σ} &= \min_{A \geq 0} \Set{\tr(σ^{\frac{1}{2}} A^α σ^{\frac{1}{2}}) | ρ \leq σ^{\frac{1}{2}} A σ^{\frac{1}{2}}, \; A \leq (C \mnorm{σ^{-1}})^\frac{1}{α} \Pi_σ } \\
			&= \min_{A \geq 0}   \Set{\qrd[\widehat{Q}][α]{A}{σ} | ρ \leq A , \; A \leq  (C \mnorm{σ^{-1}})^\frac{1}{α} σ^{\frac{1}{2}} \Pi_σ σ^{\frac{1}{2}} \leq (C \mnorm{σ^{-1}})^\frac{1}{α}\mnorm{σ}\Pi_σ } 
		\end{align}
		With $\tilde{C} := \sup_{α \in [1, a]} (C \mnorm{σ^{-1}})^\frac{1}{α}\mnorm{σ} $, which exists, and $K := \set{A\in \pos | ρ \leq A \leq \tilde{C} \Pi_σ}$, which is compact, we have for all $α \in (1,b]$:
		\begin{equation}
			\qrd[Q^\#][α]{ρ}{σ} = \min_{A \in K} \qrd[\widehat{Q}][α]{A}{σ}\, .
		\end{equation}
	\end{proof}	
	
	Since the logarithm on $\reals$ is monotone, Lemma~\ref{α1:compact-restriction} implies that also $\qrd[D^\#][α]{ρ}{σ} = \min_{A \geq ρ} \qrd[\widehat{D}][α]{A}{σ}$ for all $α > 1$. This completes the proof of Proposition~\ref{prop:key}. Note, that for $α > 2$ the geometric divergence does not satisfy the data-processing inequality, while the sharp divergence does. 
	
	
	\section{The $α \to 1$ limit}
	We show in Theorem~\ref{α1:theorem-sharp-bs-limit} at the end of this section that the limit $α \to 1$ of the sharp divergence is the Belavkin-Staszewski relative entropy. 
	The key step in the proof is to show that $\dv{α} \qrd[Q^\#][α]{ρ}{σ} |_{α = 1}$ exists and is equal to $\dv{α} \qrd[\widehat{Q}][α]{ρ}{σ} |_{α = 1}$.
	One way to establish this is to use the following theorem~\autocite[Theorem~2.2.1]{demyanov_theory_1971}:
	\begin{theorem}[Dem'yanov and Malozemov (1971)]\label{α1:demyanov-theorem}
		Let $U \subset \reals^n$ be open, $K \subset \reals^m$ compact, and $f \colon U\times K \to \reals$ continuous and also that $\nabla_u f(u,k)$ is (jointly) continuous. Then, the function $g(u) = \min_{k \in K} f(u,k)$ has for every $u \in U$ a one-sided directional derivative along every $v \in \reals^n$, which can be computed as
		\begin{equation}
			\lim\limits_{t \searrow 0} \frac{g(u + v t) - g(u)}{t} = \min_{k \in R(u)} \langle \nabla_u f(u, k), v \rangle,
		\end{equation}
	where $R(u) = \set{k \in K | f(u,k) = g(u)}$.\footnote{In \cite{demyanov_theory_1971} the theorem is phrased with a maximum instead of a minimum, but it is easy to see that this is equivalent upon setting $f \mapsto -f$.}
	\end{theorem}
The following lemmas establish the properties of $\widehat{Q}_α$ which are necessary in order to apply Theorem~\ref{α1:demyanov-theorem}.


    \begin{lemma}\label{lemma:matrix-continuity}
        Let $\Omega$ be a subset of $\reals^n$ and $h\colon \Omega \times [0, \infty) \to [0, \infty)$ be a jointly continuous function. Then, the corresponding matrix function $h \colon \Omega \times \pos \to \pos$ is jointly continuous on $\Omega \times \pos$. 
    \end{lemma}
    \begin{proof}
        For a fixed $α \in \Omega$, the continuity of the matrix function $h(α, ρ)$  in $ρ$ follows from the continuity of $h(α, x)$ in $x$, for $x \in [0, \infty)$, by \cite[Theorem 6.2.37]{horn_topics_1991}. To see joint continuity, let $ρ_n \to ρ$ be a converging sequence in $\pos$ and $α_n \to α$ be a converging sequence in $\Omega$, as $n \to \infty$. Then, taking operator norms,
		\begin{align}\label{α1:continuity_triangle}
			\mnorm{h(α_n, ρ_n) - h(α, ρ)} \leq \mnorm{h(α_n, ρ_n) - h(α, ρ_n)} + \mnorm{h(α, ρ_n) - h(α, ρ)} \,.
		\end{align}
	    The second term on the right hand side goes to zero as $n \to \infty$ by the continuity of $h(α, ρ)$ in $ρ$. For the first term, note that $h(α_n, ρ_n)$ and $h(α, ρ_n)$ commute and the operator norm can be evaluated as
		\begin{equation}\label{α1:continuity_eigenvalues}
			\max_{\lambda \in \text{spec}(ρ_n)} |h(α_n, λ) - h(α, λ)| \,.
		\end{equation}
		Since the limit $α_n \to α$ exists, $α_n$ is eventually bounded, so there exists a compact subset $K$ of $\Omega$, such that $α_n \in K$ for all large enough $n$. 
		Analogously, the limit $ρ_n \to ρ$ exists, and so the spectrum of $ρ_n$ is eventually bounded, and hence there exists a compact subset $K'$ of $[0, \infty)$ such that $\text{spec}(ρ_n) \subset K'$ for all large enough $n$. Since $K\times K'$ is compact, $h$ is in fact uniformly continuous on $K \times K'$. Hence, the expression in (\ref{α1:continuity_eigenvalues}) goes to zero as $n \to \infty$ uniformly in all elements of the maximum and also uniformly in $ρ_n$. This shows that 
		\begin{equation}
		    \lim_{n \to \infty} \mnorm{h(α_n, ρ_n) - h(α, ρ)} = 0
		\end{equation}
		which completes the proof. 
	\end{proof}



	\begin{lemma}\label{α1:posdef-continuity}
	For $σ \in \pos$ fixed, the functions $f_ρ(α) = f(α, ρ) = \qrd[\widehat{Q}][α]{ρ}{σ}$ and $f'_ρ(α) = f'(α, ρ) = \dv{α} \qrd[\widehat{Q}][α]{ρ}{σ}$ are jointly continuous on $(0, \infty) \times \Set{A \in \pos| A \ll σ}$.
	\end{lemma}
	\begin{proof}
	    We have:
		\begin{align}
			\qrd[\widehat{Q}][α]{ρ}{σ} &= \tr(σ^{\frac{1}{2}}\qty(σ^{-\frac{1}{2}}ρ σ^{-\frac{1}{2}})^α σ^{\frac{1}{2}}) \\
			\dv{α}\qrd[\widehat{Q}][α]{ρ}{σ} &= \tr(σ^{\frac{1}{2}}\log(σ^{-\frac{1}{2}}ρ σ^{-\frac{1}{2}})\qty(σ^{-\frac{1}{2}}ρ σ^{-\frac{1}{2}})^α σ^{\frac{1}{2}})\, .
		\end{align}
		It is easy to check that the real functions $(α, x) \mapsto x^α$ and $(α, x) \mapsto x^α \log(x)$ are jointly continuous on $(0, \infty) \times [0, \infty)$. Also, for $A \in \Set{A \in \pos| A \ll σ}$ the expression $σ^{-\frac{1}{2}} A σ^{-\frac{1}{2}}$ is continuous in $A$ (remember that the inverses are all pseudo-inverses).  Hence, the continuity follows from Lemma \ref{lemma:matrix-continuity}, the continuity of the matrix product and the continuity of the trace.
	\end{proof}
	\begin{theorem}\label{α1:theorem-sharp-bs-limit}
		For $ρ\in \DM$ and $σ \in \pos$:
		\begin{equation}\label{α1:equation_theorem_limit}
			\lim\limits_{α \to 1} \qrd[D^\#][α]{ρ}{σ} = \widehat{D}(\rho||\sigma) 
		\end{equation}
	the Belavkin-Staszewski relative entropy. 
	\end{theorem}
	\begin{proof}
		As the sharp R\'enyi divergence, $D_\alpha^{\#}$, is only defined for $α > 1$, all there is to prove is a limit from above. Let us fix $ρ \in \DM$ and $σ \in \pos$.
		We can restrict our proof to the case $ρ \ll σ$, since otherwise $\qrd[D^\#][α]{ρ}{σ} = \widehat{D}_\alpha(\rho||\sigma) = \infty$ for all $α > 1$ and the statement clearly holds. 
		By Lemma \ref{α1:compact-restriction} there exists a compact set $K \subseteq \Set{A\in \pos| A \ll σ}$\footnote{$K$ depends on $ρ$ and $σ$, but since they are fixed in the entire proof we do not make this dependence explicit.} such that for $α \in (1, 2]$,
		\begin{equation}
		    \qrd[Q^\#][α]{ρ}{σ} = \min_{A \in K} \qrd[\widehat{Q}][α]{A}{σ} \, .
		\end{equation}
		For $α \in (0, \infty)$,\footnote{To apply Theorem \ref{α1:demyanov-theorem} we require an open interval in $α$ which includes 1.} we define
		\begin{align}
			f(α, A) &\coloneqq \qrd[\widehat{Q}][α]{A}{σ} \\
			g(α) &\coloneqq \min_{A \in K} f(α, A) = \min_{A \in K} \qrd[\widehat{Q}][α]{A}{σ} \,.
		\end{align}
		By Lemma \ref{α1:posdef-continuity}, $f(α, A)$ is jointly continuous on $(0, \infty) \times K$ and has a jointly continuous derivative in $α$. Hence by Theorem \ref{α1:demyanov-theorem}, $g(α)$ has the the following one-sided derivatives:
		\begin{align}
			\lim_{α \searrow 1} \frac{g(α) - g(1)}{α - 1} &= \min_{A \in R(α)} \dv{α} f(α, A) \label{α1:theorem-limit:limit-above} \\
			\lim_{α \nearrow 1} \frac{g(α) - g(1)}{α - 1} &= -\min_{A \in R(α)} \left(-\dv{α} f(α, A)\right)\label{α1:theorem-limit:limit-below}\, ,
		\end{align}
		where $R(α) := \set{A \in K | f(α, A) = \min_{B \in K} f(α, B)}$. Recall that the sharp \renyi divergence is only defined for $α > 1$ and so all we really need here is the limit from above (\ref{α1:theorem-limit:limit-above}). However, establishing that this is also equal to the limit from below makes things slightly simpler, as we are then able to directly use the chain rule later in (\ref{eq:limit-theorem:chain-rule}).
		
		For $α = 1$ the set $R(α)$ only contains $ρ$, since
		\begin{align}
		    \min_{A \geq ρ} \qrd[\widehat{Q}][1]{A}{σ} = \min_{A \geq ρ} \Tr(A) = \Tr(ρ),
		\end{align}
		and $A \geq \rho $, together with $\Tr(A) = \Tr(ρ)$, imply that $A = ρ$. 
		Hence, the two one-sided derivatives are equal and
		\begin{align}
		\label{eq:deriequal}
            \dv{α} g(α)\bigg|_{α = 1} = \dv{α} f(α, ρ) \bigg|_{α = 1} = \dv{α} \qrd[\widehat{Q}][α]{ρ}{σ}\bigg|_{α = 1}\,.
		\end{align}
		We further have $g(α) = \qrd[Q^\#][α]{ρ}{σ}$ for $α \in (1, 2]$, and $g(1) = \tr(ρ) = 1$, so
		\begin{align}
			\lim\limits_{α \searrow 1} \qrd[D^\#][α]{ρ}{σ} &= \lim\limits_{α \searrow 1} \frac{\log \qrd[Q^\#][α]{ρ}{σ}}{α - 1} = \lim\limits_{α \searrow 1} \frac{\log g(α) - \log g(1)}{α - 1} \nonumber\\
			&= \dv{α} \log g(α) \bigg|_{α = 1}  = \frac{\dv{α} g(α) \big|_{α = 1}}{g(1)} = \dv{α} g(α) \bigg|_{α = 1} \,. \label{eq:limit-theorem:chain-rule}
		\end{align}
	As the same argument also gives \begin{align}\lim\limits_{α \to 1} \qrd[\widehat{D}][α]{ρ}{σ} = \dv{α} \widehat Q_\alpha(\rho\|\sigma) \bigg|_{α = 1}, \end{align} we get by using  \eqref{eq:deriequal}
	\begin{equation}\label{α1:limit-theorem:final_eq}
		\lim\limits_{α \to 1} \qrd[D^\#][α]{ρ}{σ} = \lim\limits_{α \to 1} \grd{ρ}{σ} = \widehat{D}(\rho||\sigma)\, .
	\end{equation}
	%the last equality was shown i.e. in~\cite{tomamichel_quantum_2016}.
	\end{proof}
\section{Kringel divergences and their properties}

 A key ingredient of our main result, Theorem~\ref{α1:theorem-sharp-bs-limit}, was Proposition~\ref{prop:key}, which allowed us to express
the sharp divergence as a minimization of the geometric R\'enyi divergence. Analogous minimizations of arbitrary generalized divergences lead to an interesting new family of generalized divergences which we call {\em{kringel divergences}}\footnote{Since the symbols $D$, $\widetilde{D}_\alpha$, $\widehat{D}_\alpha$, $D^\flat_α$ and $D^\#_\alpha$ all refer to existing divergences, and generalized divergences are usually denoted as $\bD$, we use the symbol $\bD^\circ$ to denote this new family; hence the name
{\em{kringel divergences}} (kringel = circle in German).
  }. We introduce them in this section and prove some of their properties, including the data-processing inequality.
\medskip

	For a function $\bD: \pos\times\pos \to \R\cup\{-\infty,\infty\}$  we define for $\rho,\sigma\in\pos$
	\begin{align}
    \bD^\circ(\rho\|\sigma) = \inf_{A\ge\rho}\bD(A\|\sigma).
    \end{align}
    Moreover, we say  $\bD: \pos\times\pos \to \R\cup\{-\infty,\infty\}$  is a {\em{generalized divergence}} if it satisfies the data-processing inequality, i.e.~for any $\rho,\sigma\in\pos$ and any quantum channel $\cN$, we have
	\begin{align}
	\label{eq:DPI}
	\bD(\cN(\rho)\|\cN(\sigma)) \le \bD(\rho\|\sigma).
	\end{align}
	%
	\begin{remark}
	Note that the above definition of the generalized divergence is an extension of the standard definition (see e.g.~in~\cite{khatri_principles_2020}). In the latter, the generalized divergence is considered as a function $\bD: \mathcal{D}(\mathcal{H}) \times\pos \to \R\cup\{\infty\}$, where $\mathcal{D}(\mathcal{H})$ denotes the set of density matrices (quantum states) on $\mathcal{H}$.
	\end{remark}
	\smallskip
	
	\noindent
	The following lemma shows that if $\bD$ is a generalized divergence then so is $\bD^\circ$. In this case we call $\bD^\circ$ the \emph{kringel divergence} of $\bD$.
 
\begin{lemma}[Data-processing inequality] \label{lem:DPI}
	Let $\bD$ be a generalized divergence, $\rho,\sigma\in\pos$ and $\cN$ be a quantum channel. Then
	\begin{align}
	\bD^\circ(\cN(\rho)\|\cN(\sigma)) \le \bD^\circ(\rho\|\sigma).
	\end{align}
\end{lemma}
\begin{proof}
	We have
	\begin{align}
	\nn\bD^\circ(\cN(\rho)\|\cN(\sigma)) &= \inf_{A\ge\cN(\rho)}\bD(A\|\cN(\sigma)) \le \inf_{\cN(A)\ge\cN(\rho)} \bD(\cN(A)\|\cN(\sigma))\\&\le\inf_{\cN(A)\ge\cN(\rho)} \bD(A\|\sigma)\le \inf_{A\ge\rho}\bD(A\|\sigma) = \bD^\circ(A\|\sigma).
	\end{align}
	Here, for the first inequality we have used the fact that the minimum increases when we only optimise over operators of the form $\cN(A)$ and for the second inequality we have used the data-processing inequality for $\bD$. For the third inequality we have used the fact that $A\ge\rho$ implies $\cN(A)\ge\cN(\rho)$, which follows by positivity and linearity of $\cN$.
\end{proof}
\begin{remark}
By Proposition~\ref{prop:key} we know that for $\alpha\in(1,\infty)$ we have
\begin{align}
\widehat D^\circ_\alpha = D^\#_\alpha,
\end{align}
where the left hand side denotes the kringel divergence corresponding to the geometric \renyi divergence \eqref{def:geometric-div}. Note that even though the geometric \renyi divergence satisfies the data-processing inequality only for $\alpha\in(0,1)\cup(1,2]$, by \cite[Proposition 3.2]{fawzi_defining_2020-1} its corresponding kringel divergence actually satisfies the data-processing inequality for all $\alpha\in(0,1)\cup(1,\infty).$ 
\end{remark}
\begin{lemma}
	\label{lem:subadd}
	If $\bD$ is subadditive then so is $\bD^\circ$, i.e. for all $\rho_1,\rho_2,\sigma_1,\sigma_2\in\pos$ we have
	\begin{align}
	\bD^\circ(\rho_1\otimes\rho_2\|\sigma_1\otimes\sigma_2) \le \bD^\circ(\rho_1\|\sigma_1) + \bD^\circ(\rho_2\|\sigma_2).
	\end{align}
\end{lemma}
\begin{proof}
	We have
	\begin{align}
	\nn \bD^\circ(\rho_1\otimes\rho_2\| \sigma_1\otimes\sigma_2) &= \inf_{A\ge\rho_1\otimes\rho_2}\bD(A\| \sigma_1\otimes\sigma_2) \le \inf_{A_1\otimes A_2\ge\rho_1\otimes\rho_2}\bD(A_1\otimes A_2\| \sigma_1\otimes\sigma_2) \\ &\le\nn  \inf_{A_1\otimes A_2\ge\rho_1\otimes\rho_2} \Big(\bD(A_1\|\sigma_1) + \bD(A_2\|\sigma_2)\Big)  \\&= \inf_{A_1\ge \rho_1} \bD(A_1\|\sigma_1)  +\inf_{A_2\ge \rho_2} \bD(A_2\|\sigma_2)\nn\\& = \bD^\circ(\rho_1\|\sigma_1) + \bD^\circ(\rho_2\|\sigma_2).
	\end{align}
    The third line above follows from the fact that $A_1\ge \rho_1$ and $A_2\ge\rho_2$ implies that $A_1\otimes A_2\ge\rho_1\otimes\rho_2$ because $A_1\otimes A_2 - \rho_1\otimes\rho_2 = \left(A_1-\rho_1\right)\otimes A_2 + \rho_1\otimes\left(A_2 - \rho_2\right)$.
\end{proof}

\subsection{Kringel divergences for $\alpha$-\renyi divergences}
We say $\bD_\alpha: \pos\times\pos \to \R\cup\{-\infty,\infty\}$ is a \emph{quantum generalization of the $\alpha$-\renyi divergence} if it reduces to the corresponding classical \renyi divergence if both entries commute. That is, if $\rho,\sigma\in\pos$ are commuting operators then
\begin{align}
\bD_\alpha(\rho\|\sigma) = \frac{1}{\alpha-1}\log\Big(\sum_i p_i^{\alpha} q_i^{1-\alpha}\Big),
\end{align}
where the $\{p_i\}_i$ and $\{q_i\}_i$ are eigenvalues of $\rho$ and $\sigma$ respectively with respect to a simultaneous eigenbasis.
The following lemma states that, for $\alpha>1$, if $\bD_\alpha$ is a quantum generalization of the \renyi relative entropy, then $\bD^\circ_\alpha$ too reduces to the classical \renyi divergence in the commuting case, and hence is itself a quantum generalization of the \renyi relative entropy.
\begin{lemma}
\label{lem:Reny}
Let $\alpha>1$ and $\bD_\alpha$ be a quantum generalization of the $\alpha$-\renyi divergence satisfying the data-processing inequality \eqref{eq:DPI}. Then also $\bD^\circ_{\alpha}$ is a quantum generalization of the $\alpha$-\renyi divergence.
\end{lemma}
\begin{proof}
Let $\rho, \sigma \in \pos $ such that $[\rho, \sigma]=0$. Then there exists a simultaneous eigenbasis $\{\ket{i}\}_i$ of $\rho$ and $\sigma$ such that 
\begin{align}
\rho = \sum_i p_i\kb{i},\quad\quad \sigma = \sum_{i}q_i\kb{i}.
\end{align} Clearly,
\begin{align}
\bD_\alpha^\circ(\rho\|\sigma) \le \bD_\alpha(\rho\|\sigma) = \frac{1}{\alpha-1}\log\Big(\sum_i p_i^{\alpha} q_i^{1-\alpha}\Big).
\end{align}
For the reversed inequality let $\cP$ be the pinching map defined as $\cP(\cdot) := \sum_{i}\kb{i}\cdot\kb{i}$. Hence, denoting for $A\ge \rho$ the diagonal entries by $a_i=\bra{i}A\ket{i}\ge p_i$, we see
\begin{align}
\nn \bD_\alpha^\circ(\rho\|\sigma) &= \inf_{A\ge\rho}\bD_\alpha(A\|\sigma) \ge\inf_{A\ge\rho}\bD_\alpha(\cP(A)\|\cP(\sigma)) 
\\&=\inf_{A\ge\rho} \frac{1}{\alpha-1}\log\Big(\sum_{i} a_i^{\alpha}q_i^{1-\alpha} \Big)=  \frac{1}{\alpha-1}\log\Big(\sum_{i} p_i^{\alpha}q_i^{1-\alpha}\Big),
\end{align}
where we have used data-processing inequality for $\bD_\alpha$ in the first inequality.
\end{proof}


\begin{remark}
Note that for $\alpha\in(0,1)$ and $\bD_\alpha$ being a quantum generalization of the $\alpha$-\renyi divergence, one easily sees that
\begin{align}
\bD_{\alpha}^\circ(\rho\|\sigma) = -\infty,
\end{align} 
for all $\rho,\sigma\in\pos$ with $\sigma\neq0$,
which is why we excluded this range of $\alpha$ in Lemma~\ref{lem:Reny} above and Proposition~\ref{prop:KringelMingel} below.
\end{remark}
Important quantum generalizations of the $\alpha$-R\'enyi divergence include the geometric- and sharp R\'enyi divergences, as well as the {\em{Petz R\'enyi divergence}} ($ D_\alpha$) \cite{petz_quasi-entropies_1986} and the {\em{sandwiched \renyi divergence}} ($\widetilde D_\alpha$) \cite{muller-lennert_quantum_2013, wilde_strong_2014}. The latter two are defined, respectively, as follows: for $\rho, \sigma \in \pos$
\begin{align}
D_\alpha (\rho||\sigma) &:= \frac{1}{\alpha -1} \log \tr \left(\rho^\alpha \sigma^{1-\alpha}\right), \nonumber\\
 \widetilde D_\alpha (\rho||\sigma) &:= \frac{1}{\alpha -1} \log \tr \left(\sigma^{\frac{1-\alpha}{2 \alpha}}\rho \sigma^{\frac{1-\alpha}{2 \alpha}}\right)^\alpha
\end{align}
if $\alpha\in(0,1)$, or $\alpha\in(1,\infty)$ and $\rho \ll\sigma$. If $\alpha\in(1,\infty)$ and $\rho\centernot\ll\sigma$, then $D_\alpha(\rho\|\sigma)=\widetilde D_\alpha(\rho\|\sigma)=\infty$. Both the Petz- and the sandwiched \renyi divergences are additive. Moreover, $D_\alpha$ for $\alpha\in(0,1)\cup(1,2]$, and $\widetilde D_\alpha$ for $\alpha\in[1/2,1)\cup (1,\infty)$, satisfy the data-processing inequality \eqref{eq:DPI}, as was shown
in \cite{petz_quasi-entropies_1986,tomamichel_fullyquantum_2009} and \cite{frank_monotonicity_2013}, respectively. Therefore, for both of these divergences the following proposition applies in the corresponding ranges of $\alpha$.
\begin{proposition}
    \label{prop:KringelMingel}
	Let $\alpha>1$ and $\bD_\alpha$ be a quantum generalization of the $\alpha$-\renyi divergence which is subadditive and satisfies the data-processing inequality~\eqref{eq:DPI}. Then
	\begin{align}
	\label{eq:bounds}
	\widetilde D_\alpha \le \bD^\circ_\alpha \le D^\#_\alpha.
	\end{align}
	In particular, this gives for any $\rho,\sigma\in\pos$
	\begin{align}
	\label{eq:reg}
	\lim_{n\to\infty} \frac{1}{n}\bD^\circ_\alpha(\rho^{\otimes n}\|\sigma^{\otimes n}) = 	\widetilde D_\alpha(\rho\|\sigma).
	\end{align}
	Moreover, in the case of the sandwiched divergence 
	\begin{align}
	\label{eq:sandwichKringel}
	\widetilde D^\circ_\alpha  = 	\widetilde D_\alpha. 
	\end{align}
\end{proposition}
\begin{proof}
As $\bD_\alpha$ satisfies the data-processing inequality, we know that $\bD_\alpha \le \widehat D_\alpha$; this follows from the argument~\cite[Section 4.2.3]{tomamichel_quantum_2016} (also see \cite{matsumoto_new_2013} where the argument originally appeared)\footnote{Note that in
in~\cite{tomamichel_quantum_2016, matsumoto_new_2013} the argument is presented for the case in which $\rho$ and $\sigma$ are states. However, it can be easily seen that, by a slight modification, the argument also works for the case $\rho,\sigma\in\pos$.}. This gives
\begin{align}
\bD^\circ_\alpha(\rho\|\sigma) = \min_{A\ge\rho}\bD_\alpha(A\|\sigma) \le \min_{A\ge\rho}\widehat D_\alpha(A\|\rho)  = \widehat D^\circ_\alpha(\rho\|\sigma) =  D^\#_\alpha(\rho\|\sigma).
\end{align}
Moreover, as $\alpha>1$ and additionally $\bD_\alpha$ is subadditive, Lemmas~\ref{lem:DPI}, \ref{lem:subadd} and \ref{lem:Reny} give that also $\bD^\circ_\alpha$ is a subadditive quantum generalization of the $\alpha$-\renyi divergence satisfying the data-processing inequality. Therefore, by the argument in~\cite[Section 4.2.2]{tomamichel_quantum_2016} (note that actually only subadditivity instead of additivity is used there) we have
\begin{align}
\widetilde D_\alpha \le \bD^\circ_\alpha,
\end{align}
which gives \eqref{eq:bounds}. Since, trivially, $\widetilde D^\circ_\alpha  \le	\widetilde D_\alpha$, \eqref{eq:sandwichKringel} follows immediately.

Lastly, using $\alpha>1$ again and~\cite[Proposition 3.4]{fawzi_defining_2020} gives for any $\rho,\sigma\in\pos$
\begin{align}
\lim_{n\to\infty} \frac{1}{n}D^\#_\alpha(\rho^{\otimes n}\|\sigma^{\otimes n}) = 	\widetilde D_\alpha(\rho\|\sigma),
\end{align}
which together with the additivity of the sandwiched R\'enyi divergence and \eqref{eq:bounds} gives \eqref{eq:reg}.

\end{proof}

\bigskip
\noindent
\textbf{Acknowledgements.}
The authors would like to thank the Institute for Pure and Applied Mathematics (IPAM) at UCLA for hosting a stimulating workshop on {\em{Entropy Inequalities, Quantum Information and Quantum Physics}} (February 8-11, 2021). The main question addressed in this paper was also posed as an open question at the workshop. The authors would also like to thank Eric Carlen and Mark~M.~Wilde for helpful comments on earlier versions of this paper. Bjarne Bergh is supported by the UK Engineering and Physical Sciences Research Council (EPSRC), grant number EP/V52024X/1. Robert Salzmann is supported by the Cambridge Commonwealth, European and International Trust. 

\medskip
\noindent
\textbf{Data availability.}
Data sharing is not applicable to this article as no new data were created or analyzed in this study.
\printbibliography
\end{document}
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bibliography.bib

@article{ando_concavity_1979,
  title = {Concavity of Certain Maps on Positive Definite Matrices and Applications to {{Hadamard}} Products},
  author = {Ando, T.},
  date = {1979-08-01},
  journaltitle = {Linear Algebra and its Applications},
  shortjournal = {Linear Algebra and its Applications},
  volume = {26},
  pages = {203--241},
  doi = {10.1016/0024-3795(79)90179-4},
  url = {http://www.sciencedirect.com/science/article/pii/0024379579901794},
  urldate = {2020-12-03},
  abstract = {If f is a positive function on (0, ∞) which is monotone of order n for every n in the sense of Löwner and if Φ1 and Φ2 are concave maps among positive definite matrices, then the following map involving tensor products: (A,B)↦f[Φ1(A)−1⊗Φ2(B)]·(Φ1(A)⊗I) is proved to be concave. If Φ1 is affine, it is proved without use of positivity that the map (A,B)↦f[Φ1(A)⊗Φ2(B)−1]·(Φ1(A)⊗I) is convex. These yield the concavity of the map (A,B)↦A1−p⊗Bp (0},
  langid = {english}
}

@article{belavkin_c-algebraic_1982,
  title = {C\textsuperscript{*}-algebraic generalization of relative entropy and entropy},
  author = {Belavkin, V. P. and Staszewski, P.},
  date = {1982},
  journaltitle = {Annales de l'I.H.P. Physique théorique},
  volume = {37},
  pages = {51--58},
  url = {http://www.numdam.org/item/AIHPA_1982__37_1_51_0/},
  urldate = {2021-01-24},
  langid = {french},
  number = {1}
}

@online{berta_quantum_2019,
  title = {Quantum {{Brascamp}}-{{Lieb Dualities}}},
  author = {Berta, Mario and Sutter, David and Walter, Michael},
  date = {2019-09-30},
  url = {http://arxiv.org/abs/1909.02383},
  urldate = {2020-11-24},
  abstract = {Brascamp-Lieb inequalities are entropy inequalities which have a dual formulation as generalized Young inequalities. In this work, we introduce a fully quantum version of this duality, relating quantum relative entropy inequalities to matrix exponential inequalities of Young type. We demonstrate this novel duality by means of examples from quantum information theory - including entropic uncertainty relations, strong data-processing inequalities, super-additivity inequalities, and many more. As an application we find novel uncertainty relations for Gaussian quantum operations that can be interpreted as quantum duals of the well-known family of geometric Brascamp-Lieb inequalities.},
  archivePrefix = {arXiv},
  eprint = {1909.02383},
  eprinttype = {arxiv},
  keywords = {Mathematical Physics,Quantum Physics},
  primaryClass = {math-ph, physics:quant-ph}
}

@article{berta_variational_2017,
  title = {On {{Variational Expressions}} for {{Quantum Relative Entropies}}},
  author = {Berta, Mario and Fawzi, Omar and Tomamichel, Marco},
  date = {2017-12},
  journaltitle = {Letters in Mathematical Physics},
  shortjournal = {Lett Math Phys},
  volume = {107},
  pages = {2239--2265},
  doi = {10.1007/s11005-017-0990-7},
  url = {http://arxiv.org/abs/1512.02615},
  urldate = {2021-02-09},
  abstract = {Distance measures between quantum states like the trace distance and the fidelity can naturally be defined by optimizing a classical distance measure over all measurement statistics that can be obtained from the respective quantum states. In contrast, Petz showed that the measured relative entropy, defined as a maximization of the Kullback-Leibler divergence over projective measurement statistics, is strictly smaller than Umegaki's quantum relative entropy whenever the states do not commute. We extend this result in two ways. First, we show that Petz' conclusion remains true if we allow general positive operator valued measures. Second, we extend the result to Renyi relative entropies and show that for non-commuting states the sandwiched Renyi relative entropy is strictly larger than the measured Renyi relative entropy for \$\textbackslash alpha \textbackslash in (\textbackslash frac12, \textbackslash infty)\$, and strictly smaller for \$\textbackslash alpha \textbackslash in [0,\textbackslash frac12)\$. The latter statement provides counterexamples for the data-processing inequality of the sandwiched Renyi relative entropy for \$\textbackslash alpha {$<$} \textbackslash frac12\$. Our main tool is a new variational expression for the measured Renyi relative entropy, which we further exploit to show that certain lower bounds on quantum conditional mutual information are superadditive.},
  archivePrefix = {arXiv},
  eprint = {1512.02615},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematical Physics,Quantum Physics},
  number = {12}
}

@book{bhatia_matrix_1997,
  title = {Matrix Analysis},
  author = {Bhatia, Rajendra},
  date = {1997},
  publisher = {{Springer}},
  location = {{New York}},
  isbn = {978-0-387-94846-1},
  keywords = {Matrices},
  langid = {english},
  number = {169},
  pagetotal = {347},
  series = {Graduate Texts in Mathematics}
}

@article{bluhm_strengthened_2020,
  title = {A Strengthened Data Processing Inequality for the {{Belavkin}}-{{Staszewski}} Relative Entropy},
  author = {Bluhm, Andreas and Capel, Angela},
  date = {2020-03},
  journaltitle = {Reviews in Mathematical Physics},
  shortjournal = {Rev. Math. Phys.},
  volume = {32},
  pages = {2050005},
  doi = {10.1142/S0129055X20500051},
  url = {http://arxiv.org/abs/1904.10768},
  urldate = {2020-10-13},
  abstract = {In this work, we provide a strengthening of the data processing inequality for the relative entropy introduced by Belavkin and Staszewski (BS-entropy). This extends previous results by Carlen and Vershynina for the relative entropy and other standard \$f\$-divergences. To this end, we provide two new equivalent conditions for the equality case of the data processing inequality for the BS-entropy. Subsequently, we extend our result to a larger class of maximal \$f\$-divergences. Here, we first focus on quantum channels which are conditional expectations onto subalgebras and use the Stinespring dilation to lift our results to arbitrary quantum channels.},
  archivePrefix = {arXiv},
  eprint = {1904.10768},
  eprinttype = {arxiv},
  keywords = {Mathematical Physics,Quantum Physics},
  number = {02},
  version = {2}
}

@online{boche_effectiveness_2021,
  title = {On the {{Effectiveness}} of {{Fekete}}'s {{Lemma}}},
  author = {Boche, Holger and Böck, Yannik and Deppe, Christian},
  date = {2021-01-01},
  url = {http://arxiv.org/abs/2010.09896},
  urldate = {2021-02-09},
  abstract = {Fekete's lemma is a well known combinatorial result pertaining to number sequences and shows the existence of limits of superadditive sequences. In this paper we analyze Fekete's lemma with respect to effective convergence and computability. We show that Fekete's lemma exhibits no constructive derivation. That is, a form of the axiom of choice is needed for the proof. We characterize when the limit value in Fekete's lemma can be computed and when it is not possible. We also investigate the speed of convergence. Given certain conditions, we can show effective convergence. In general, there is no effective convergence when applying Fekete's lemma.},
  archivePrefix = {arXiv},
  eprint = {2010.09896},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory},
  primaryClass = {cs, math}
}

@book{boyd_convex_2004,
  title = {Convex Optimization},
  author = {Boyd, Stephen P. and Vandenberghe, Lieven},
  date = {2004},
  publisher = {{Cambridge University Press}},
  location = {{Cambridge, UK ; New York}},
  isbn = {978-0-521-83378-3},
  keywords = {Convex functions,Mathematical optimization},
  langid = {english},
  pagetotal = {716}
}

@online{brown_computing_2020,
  title = {Computing Conditional Entropies for Quantum Correlations},
  author = {Brown, Peter and Fawzi, Hamza and Fawzi, Omar},
  date = {2020-08-21},
  url = {http://arxiv.org/abs/2007.12575},
  urldate = {2020-11-30},
  abstract = {The rates of quantum cryptographic protocols are usually expressed in terms of a conditional entropy minimized over a certain set of quantum states. In particular, in the device-independent setting, the minimization is over all the quantum states jointly held by the adversary and the parties that are consistent with the statistics that are seen by the parties. Here, we introduce a method to approximate such entropic quantities. Applied to the setting of device-independent randomness generation and quantum key distribution, we obtain improvements on protocol rates in various settings. In particular, we find new upper bounds on the minimal global detection efficiency required to perform device-independent quantum key distribution without additional preprocessing. Furthermore, we show that our construction can be readily combined with the entropy accumulation theorem in order to establish full finite-key security proofs for these protocols. In order to achieve this we introduce the family of iterated mean quantum R\textbackslash 'enyi divergences with parameters \$\textbackslash alpha\_k = 1+\textbackslash frac\{1\}\{2\^\{k\}-1\}\$ for positive integers \$k\$. We then show that the corresponding conditional entropies admit a particularly nice form which, in the context of device-independent optimization, can be relaxed to a semidefinite programming problem using the Navascu\textbackslash 'es-Pironio-Ac\textbackslash 'in hierarchy.},
  archivePrefix = {arXiv},
  eprint = {2007.12575},
  eprinttype = {arxiv},
  keywords = {Quantum Physics},
  primaryClass = {quant-ph}
}

@online{carlen_recovery_2019,
  title = {Recovery Map Stability for the {{Data Processing Inequality}}},
  author = {Carlen, Eric A. and Vershynina, Anna},
  date = {2019-05-30},
  url = {http://arxiv.org/abs/1710.02409},
  urldate = {2021-02-08},
  abstract = {The Data Processing Inequality (DPI) says that the Umegaki relative entropy \$S(\textbackslash rho||\textbackslash sigma) := \{\textbackslash rm Tr\}[\textbackslash rho(\textbackslash log \textbackslash rho - \textbackslash log \textbackslash sigma)]\$ is non-increasing under the action of completely positive trace preserving (CPTP) maps. Let \$\{\textbackslash mathcal M\}\$ be a finite dimensional von Neumann algebra and \$\{\textbackslash mathcal N\}\$ a von Neumann subalgebra if it. Let \$\{\textbackslash mathcal E\}\_\textbackslash tau\$ be the tracial conditional expectation from \$\{\textbackslash mathcal M\}\$ onto \$\{\textbackslash mathcal N\}\$. For density matrices \$\textbackslash rho\$ and \$\textbackslash sigma\$ in \$\{\textbackslash mathcal N\}\$, let \$\textbackslash rho\_\{\textbackslash mathcal N\} := \{\textbackslash mathcal E\}\_\textbackslash tau \textbackslash rho\$ and \$\textbackslash sigma\_\{\textbackslash mathcal N\} := \{\textbackslash mathcal E\}\_\textbackslash tau \textbackslash sigma\$. Since \$\{\textbackslash mathcal E\}\_\textbackslash tau\$ is CPTP, the DPI says that \$S(\textbackslash rho||\textbackslash sigma) \textbackslash geq S(\textbackslash rho\_\{\textbackslash mathcal N\}||\textbackslash sigma\_\{\textbackslash mathcal N\})\$, and the general case is readily deduced from this. A theorem of Petz says that there is equality if and only if \$\textbackslash sigma = \{\textbackslash mathcal R\}\_\textbackslash rho(\textbackslash sigma\_\{\textbackslash mathcal N\} )\$, where \$\{\textbackslash mathcal R\}\_\textbackslash rho\$ is the Petz recovery map, which is dual to the Accardi-Cecchini coarse graining operator \$\{\textbackslash mathcal A\}\_\textbackslash rho\$ from \$\{\textbackslash mathcal M\} \$ to \$\{\textbackslash mathcal N\} \$. In it simplest form, our bound is \$\$S(\textbackslash rho||\textbackslash sigma) - S(\textbackslash rho\_\{\textbackslash mathcal N\} ||\textbackslash sigma\_\{\textbackslash mathcal N\} ) \textbackslash geq \textbackslash left(\textbackslash frac\{1\}\{8\textbackslash pi\}\textbackslash right)\^\{4\} \textbackslash |\textbackslash Delta\_\{\textbackslash sigma,\textbackslash rho\}\textbackslash |\^\{-2\} \textbackslash | \{\textbackslash mathcal R\}\_\{\textbackslash rho\_\{\textbackslash mathcal N\}\} -\textbackslash sigma\textbackslash |\_1\^4 \$\$ where \$\textbackslash Delta\_\{\textbackslash sigma,\textbackslash rho\}\$ is the relative modular operator. We also prove related results for various quasi-relative entropies. Explicitly describing the solutions set of the Petz equation \$\textbackslash sigma = \{\textbackslash mathcal R\}\_\textbackslash rho(\textbackslash sigma\_\{\textbackslash mathcal N\} )\$ amounts to determining the set of fixed points of the Accardi-Cecchini coarse graining map. Building on previous work, we provide a throughly detailed description of the set of solutions of the Petz equation, and obtain all of our results in a simple self, contained manner.},
  archivePrefix = {arXiv},
  eprint = {1710.02409},
  eprinttype = {arxiv},
  keywords = {Mathematics - Operator Algebras,Quantum Physics},
  primaryClass = {quant-ph}
}

@online{carlen_trace_2018,
  title = {Some Trace Inequalities for Exponential and Logarithmic Functions},
  author = {Carlen, Eric A. and Lieb, Elliott H.},
  date = {2018-04-30},
  url = {http://arxiv.org/abs/1709.05450},
  urldate = {2021-02-09},
  abstract = {Consider a function \$F(X,Y)\$ of pairs of positive matrices with values in the positive matrices such that whenever \$X\$ and \$Y\$ commute \$F(X,Y)= X\^pY\^q.\$ Our first main result gives conditions on \$F\$ such that \$\{\textbackslash rm Tr\}[ X \textbackslash log (F(Z,Y))] \textbackslash leq \{\textbackslash rm Tr\}[X(p\textbackslash log X + q \textbackslash log Y)]\$ for all \$X,Y,Z\$ such that \$\{\textbackslash rm Tr\} Z = \{\textbackslash rm Tr\} X\$. (Note that \$Z\$ is absent from the right side of the inequality.) We give several examples of functions \$F\$ to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables \$X,Y,Z\$ instead of just \$X,Y\$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy \$D(X||Y) = \{\textbackslash rm Tr\} [X(\textbackslash log X-\textbackslash log Y])\$, and two others, the Donald relative entropy \$D\_D(X||Y)\$, and the Belavkin-Stasewski relative entropy \$D\_\{BS\}(X||Y)\$. They are known to satisfy \$D\_D(X||Y) \textbackslash leq D(X||Y)\textbackslash leq D\_\{BS\}(X||Y)\$. We prove that the Donald relative entropy provides the sharp upper bound, independent of \$Z\$, on \$\{\textbackslash rm Tr\}[ X \textbackslash log (F(Z,Y))]\$ in a number of cases in which \$(Z,Y)\$ is homogeneous of degree \$1\$ in \$Z\$ and \$-1\$ in \$Y\$. We also investigate the Legendre transforms in \$X\$ of \$D\_D(X||Y)\$ and \$D\_\{BS\}(X||Y)\$, and show how our results for these lead to new refinements of the Golden-Thompson inequality.},
  archivePrefix = {arXiv},
  eprint = {1709.05450},
  eprinttype = {arxiv},
  keywords = {Mathematical Physics},
  primaryClass = {math-ph}
}

@online{carlen_trace_nodate,
  title = {Trace Inequalities and Quantum Entropy: {{An}} Introductory Course | {{Request PDF}}},
  shorttitle = {Trace Inequalities and Quantum Entropy},
  author = {Carlen, Eric},
  doi = {10.1090/conm/529/10428},
  url = {https://www.researchgate.net/publication/265820700_Trace_inequalities_and_quantum_entropy_An_introductory_course},
  urldate = {2020-09-14},
  abstract = {Request PDF | Trace inequalities and quantum entropy: An introductory course | We give an elementary introduction to the subject of trace inequalities and related topics in analysis, with a special focus on results that are... | Find, read and cite all the research you need on ResearchGate},
  langid = {english},
  organization = {{ResearchGate}}
}

@article{clarke_generalized_1975,
  title = {Generalized Gradients and Applications},
  author = {Clarke, Frank H.},
  date = {1975},
  journaltitle = {Transactions of the American Mathematical Society},
  shortjournal = {Trans. Amer. Math. Soc.},
  volume = {205},
  pages = {247--262},
  doi = {10.1090/S0002-9947-1975-0367131-6},
  url = {https://www.ams.org/tran/1975-205-00/S0002-9947-1975-0367131-6/},
  urldate = {2021-01-21},
  abstract = {A theory of generalized gradients for a general class of functions is developed, as well as a corresponding theory of normals to arbitrary closed sets. It is shown how these concepts subsume the usual gradients and normals of smooth functions and manifolds, and the subdifferentials and normals of convex analysis. A theorem is proved concerning the differentiability properties of a function of the form . This result unifies and extends some theorems of Danskin and others. The results are then applied to obtain a characterization of flow-invariant sets which yields theorems of Bony and Brezis as corollaries.},
  keywords = {directional derivatives,flow-invariant sets,generalized gradients,Lipschitz,max functions,Nondifferentiable functions,tangent cones},
  langid = {english}
}

@article{cooney_strong_2016,
  title = {Strong {{Converse Exponents}} for a {{Quantum Channel Discrimination Problem}} and {{Quantum}}-{{Feedback}}-{{Assisted Communication}}},
  author = {Cooney, Tom and Mosonyi, Milán and Wilde, Mark M.},
  date = {2016-06-01},
  journaltitle = {Communications in Mathematical Physics},
  shortjournal = {Commun. Math. Phys.},
  volume = {344},
  pages = {797--829},
  doi = {10.1007/s00220-016-2645-4},
  url = {https://doi.org/10.1007/s00220-016-2645-4},
  urldate = {2021-02-11},
  abstract = {This paper studies the difficulty of discriminating between an arbitrary quantum channel and a “replacer" channel that discards its input and replaces it with a fixed state. The results obtained here generalize those known in the theory of quantum hypothesis testing for binary state discrimination. We show that, in this particular setting, the most general adaptive discrimination strategies provide no asymptotic advantage over non-adaptive tensor-power strategies. This conclusion follows by proving a quantum Stein’s lemma for this channel discrimination setting, showing that a constant bound on the Type I error leads to the Type II error decreasing to zero exponentially quickly at a rate determined by the maximum relative entropy registered between the channels. The strong converse part of the lemma states that any attempt to make the Type II error decay to zero at a rate faster than the channel relative entropy implies that the Type I error necessarily converges to one. We then refine this latter result by identifying the optimal strong converse exponent for this task. As a consequence of these results, we can establish a strong converse theorem for the quantum-feedback-assisted capacity of a channel, sharpening a result due to Bowen. Furthermore, our channel discrimination result demonstrates the asymptotic optimality of a non-adaptive tensor-power strategy in the setting of quantum illumination, as was used in prior work on the topic. The sandwiched Rényi relative entropy is a key tool in our analysis. Finally, by combining our results with recent results of Hayashi and Tomamichel, we find a novel operational interpretation of the mutual information of a quantum channel \$\$\{\textbackslash mathcal\{N\}\}\$\$Nas the optimal Type II error exponent when discriminating between a large number of independent instances of \$\$\{\textbackslash mathcal\{N\}\}\$\$Nand an arbitrary “worst-case” replacer channel chosen from the set of all replacer channels.},
  langid = {english},
  number = {3}
}

@online{courtade_euclidean_2019,
  title = {Euclidean {{Forward}}-{{Reverse Brascamp}}-{{Lieb Inequalities}}: {{Finiteness}}, {{Structure}} and {{Extremals}}},
  shorttitle = {Euclidean {{Forward}}-{{Reverse Brascamp}}-{{Lieb Inequalities}}},
  author = {Courtade, Thomas A. and Liu, Jingbo},
  date = {2019-08-29},
  url = {http://arxiv.org/abs/1907.12723},
  urldate = {2020-11-24},
  abstract = {A new proof is given for the fact that centered gaussian functions saturate the Euclidean forward-reverse Brascamp-Lieb inequalities, extending the Brascamp-Lieb and Barthe theorems. A duality principle for best constants is also developed, which generalizes the fact that the best constants in the Brascamp-Lieb and Barthe inequalities are equal. Finally, as the title hints, the main results concerning finiteness, structure and gaussian-extremizability for the Brascamp-Lieb inequality due to Bennett, Carbery, Christ and Tao are generalized to the setting of the forward-reverse Brascamp-Lieb inequality.},
  archivePrefix = {arXiv},
  eprint = {1907.12723},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematics - Classical Analysis and ODEs,Mathematics - Functional Analysis},
  primaryClass = {cs, math}
}

@article{demyanov_theory_1971,
  title = {On the {{Theory}} of {{Non}}-{{Linear Minimax Problems}}},
  author = {Dem'yanov, V. F. and Malozemov, V. N.},
  date = {1971-06-30},
  journaltitle = {Russian Mathematical Surveys},
  shortjournal = {Russ. Math. Surv.},
  volume = {26},
  pages = {57},
  publisher = {{IOP Publishing}},
  doi = {10.1070/RM1971v026n03ABEH003834},
  url = {https://iopscience.iop.org/article/10.1070/RM1971v026n03ABEH003834/meta},
  urldate = {2021-01-23},
  langid = {english},
  number = {3}
}

@article{fang_chain_2020,
  title = {A Chain Rule for the Quantum Relative Entropy},
  author = {Fang, Kun and Fawzi, Omar and Renner, Renato and Sutter, David},
  date = {2020-03-10},
  journaltitle = {Physical Review Letters},
  shortjournal = {Phys. Rev. Lett.},
  volume = {124},
  pages = {100501},
  doi = {10.1103/PhysRevLett.124.100501},
  url = {http://arxiv.org/abs/1909.05826},
  urldate = {2021-01-31},
  abstract = {The chain rule for the classical relative entropy ensures that the relative entropy between probability distributions on multipartite systems can be decomposed into a sum of relative entropies of suitably chosen conditional distributions on the individual systems. Here, we prove a similar chain rule inequality for the quantum relative entropy in terms of channel relative entropies. The new chain rule allows us to solve an open problem in the context of asymptotic quantum channel discrimination: surprisingly, adaptive protocols cannot improve the error rate for asymmetric channel discrimination compared to non-adaptive strategies. In addition, we give examples of quantum channels showing that the channel relative entropy is not additive under the tensor product.},
  archivePrefix = {arXiv},
  eprint = {1909.05826},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Quantum Physics},
  number = {10}
}

@online{fang_geometric_2019,
  title = {Geometric {{Rényi Divergence}} and Its {{Applications}} in {{Quantum Channel Capacities}}},
  author = {Fang, Kun and Fawzi, Hamza},
  date = {2019-09-12},
  url = {http://arxiv.org/abs/1909.05758},
  urldate = {2020-09-14},
  abstract = {We present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties. For example we prove a chain rule inequality that immediately implies the "amortization collapse" for the geometric Rényi divergence, addressing an open question by Berta et al. [arXiv:1808.01498, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.},
  archivePrefix = {arXiv},
  eprint = {1909.05758},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematical Physics,Quantum Physics},
  primaryClass = {math-ph, physics:quant-ph}
}

@online{fawzi_defining_2020,
  ids = {fawzi_defining_2020-1},
  title = {Defining Quantum Divergences via Convex Optimization},
  author = {Fawzi, Hamza and Fawzi, Omar},
  date = {2020-07-24},
  url = {http://arxiv.org/abs/2007.12576},
  urldate = {2020-11-19},
  abstract = {We introduce a new quantum R\textbackslash 'enyi divergence \$D\^\{\textbackslash\#\}\_\{\textbackslash alpha\}\$ for \$\textbackslash alpha \textbackslash in (1,\textbackslash infty)\$ defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum R\textbackslash 'enyi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched \$\textbackslash alpha\$-R\textbackslash 'enyi divergence between quantum channels for \$\textbackslash alpha {$>$} 1\$. Second it allows us to prove a chain rule property for the sandwiched \$\textbackslash alpha\$-R\textbackslash 'enyi divergence for \$\textbackslash alpha {$>$} 1\$ which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.},
  archivePrefix = {arXiv},
  eprint = {2007.12576},
  eprinttype = {arxiv},
  keywords = {Mathematics - Optimization and Control,Quantum Physics},
  primaryClass = {quant-ph},
  version = {1}
}

@article{fawzi_liebs_2017,
  ids = {fawzi_liebs_2020},
  title = {Lieb's Concavity Theorem, Matrix Geometric Means, and Semidefinite Optimization},
  author = {Fawzi, Hamza and Saunderson, James},
  date = {2017-01-15},
  journaltitle = {Linear Algebra and its Applications},
  shortjournal = {Linear Algebra and its Applications},
  volume = {513},
  pages = {240--263},
  doi = {10.1016/j.laa.2016.10.012},
  url = {http://www.sciencedirect.com/science/article/pii/S0024379516304852},
  urldate = {2020-11-30},
  abstract = {A famous result of Lieb establishes that the map (A,B)↦tr[K⁎A1−tKBt] is jointly concave in the pair (A,B) of positive definite matrices, where K is a fixed matrix and t∈[0,1]. In this paper we show that Lieb's function admits an explicit semidefinite programming formulation for any rational t∈[0,1]. Our construction makes use of a semidefinite formulation of weighted matrix geometric means. We provide an implementation of our constructions in Matlab.},
  archivePrefix = {arXiv},
  eprint = {1512.03401},
  eprinttype = {arxiv},
  keywords = {Lieb's concavity theorem,Linear matrix inequalities,Mathematics - Optimization and Control,Matrix convexity,Matrix geometric means,Quantum Physics,Semidefinite optimization},
  langid = {english}
}

@article{frank_monotonicity_2013,
  title = {Monotonicity of a Relative {{Renyi}} Entropy},
  author = {Frank, Rupert L. and Lieb, Elliott H.},
  date = {2013-12},
  journaltitle = {Journal of Mathematical Physics},
  shortjournal = {Journal of Mathematical Physics},
  volume = {54},
  pages = {122201},
  doi = {10.1063/1.4838835},
  url = {http://arxiv.org/abs/1306.5358},
  urldate = {2020-12-07},
  abstract = {We show that a recent definition of relative R\textbackslash 'enyi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of M\textbackslash "uller-Lennert et al.},
  archivePrefix = {arXiv},
  eprint = {1306.5358},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematical Physics,Mathematics - Functional Analysis,Quantum Physics},
  number = {12}
}

@online{hayashi_discrimination_2008,
  title = {Discrimination of Two Channels by Adaptive Methods and Its Application to Quantum System},
  author = {Hayashi, Masahito},
  date = {2008-04-04},
  doi = {10.1109/TIT.2009.2023726},
  url = {http://arxiv.org/abs/0804.0686},
  urldate = {2021-02-04},
  abstract = {The optimal exponential error rate for adaptive discrimination of two channels is discussed. In this problem, adaptive choice of input signal is allowed. This problem is discussed in various settings. It is proved that adaptive choice does not improve the exponential error rate in these settings. These results are applied to quantum state discrimination.},
  archivePrefix = {arXiv},
  eprint = {0804.0686},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematics - Statistics Theory,Quantum Physics},
  primaryClass = {quant-ph, stat}
}

@online{hayashi_error_2006,
  title = {Error {{Exponent}} in {{Asymmetric Quantum Hypothesis Testing}} and {{Its Application}} to {{Classical}}-{{Quantum Channel}} Coding},
  author = {Hayashi, Masahito},
  date = {2006-11-11},
  doi = {10.1103/PhysRevA.76.062301},
  url = {http://arxiv.org/abs/quant-ph/0611013},
  urldate = {2021-02-04},
  abstract = {In the simple quantum hypothesis testing problem, upper bound with asymmetric setting is shown by using a quite useful inequality by Audenaert et al, quant-ph/0610027, which was originally invented for symmetric setting. Using this upper bound, we obtain the Hoeffding bound, which are identical with the classical counter part if the hypotheses, composed of two density operators, are mutually commutative. Our upper bound improves the bound by Ogawa-Hayashi, and also provides a simpler proof of the direct part of the quantum Stein's lemma. Further, using this bound, we obtain a better exponential upper bound of the average error probability of classical-quantum channel coding.},
  archivePrefix = {arXiv},
  eprint = {quant-ph/0611013},
  eprinttype = {arxiv},
  keywords = {Quantum Physics}
}

@book{hayashi_quantum_2017,
  title = {Quantum {{Information Theory}}: {{Mathematical Foundation}}},
  shorttitle = {Quantum {{Information Theory}}},
  author = {Hayashi, Masahito},
  date = {2017},
  publisher = {{Springer Berlin Heidelberg}},
  location = {{Berlin, Heidelberg}},
  doi = {10.1007/978-3-662-49725-8},
  url = {http://link.springer.com/10.1007/978-3-662-49725-8},
  urldate = {2020-09-14},
  isbn = {978-3-662-49723-4 978-3-662-49725-8},
  langid = {english},
  series = {Graduate {{Texts}} in {{Physics}}}
}

@article{hiai_different_2017,
  title = {Different Quantum F-Divergences and the Reversibility of Quantum Operations},
  author = {Hiai, Fumio and Mosonyi, Milan},
  date = {2017-08},
  journaltitle = {Reviews in Mathematical Physics},
  shortjournal = {Rev. Math. Phys.},
  volume = {29},
  pages = {1750023},
  doi = {10.1142/S0129055X17500234},
  url = {http://arxiv.org/abs/1604.03089},
  urldate = {2020-10-15},
  abstract = {The concept of classical \$f\$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\textbackslash 'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\textbackslash 'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard \$f\$-divergences), Matsumoto's maximal \$f\$-divergences, measured \$f\$-divergences, and sandwiched and \$\textbackslash alpha\$-\$z\$-R\textbackslash 'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum \$f\$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum \$f\$-divergence by a quantum operation. In particular, we compare the standard and the maximal \$f\$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum \$f\$-divergences are strictly different for non-commuting operators unless \$f\$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard \$f\$-divergences. We also study the monotonicity of the \$\textbackslash alpha\$-\$z\$-R\textbackslash 'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\textbackslash 'enyi divergence invariant, and determine domains of the parameters \$\textbackslash alpha,z\$ where monotonicity holds, and where the preservation of the \$\textbackslash alpha\$-\$z\$-R\textbackslash 'enyi divergence implies the reversibility of the quantum operation.},
  archivePrefix = {arXiv},
  eprint = {1604.03089},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematical Physics,Quantum Physics},
  number = {07}
}

@article{hiai_quantum_2011,
  title = {Quantum F-Divergences and Error Correction},
  author = {Hiai, F. and Mosonyi, M. and Petz, D. and Beny, C.},
  date = {2011-08},
  journaltitle = {Reviews in Mathematical Physics},
  shortjournal = {Rev. Math. Phys.},
  volume = {23},
  pages = {691--747},
  doi = {10.1142/S0129055X11004412},
  url = {http://arxiv.org/abs/1008.2529},
  urldate = {2020-12-08},
  abstract = {Quantum f-divergences are a quantum generalization of the classical notion of f-divergences, and are a special case of Petz' quasi-entropies. Many well known distinguishability measures of quantum states are given by, or derived from, f-divergences; special examples include the quantum relative entropy, the Renyi relative entropies, and the Chernoff and Hoeffding measures. Here we show that the quantum f-divergences are monotonic under the dual of Schwarz maps whenever the defining function is operator convex. This extends and unifies all previously known monotonicity results. We also analyze the case where the monotonicity inequality holds with equality, and extend Petz' reversibility theorem for a large class of f-divergences and other distinguishability measures. We apply our findings to the problem of quantum error correction, and show that if a stochastic map preserves the pairwise distinguishability on a set of states, as measured by a suitable f-divergence, then its action can be reversed on that set by another stochastic map that can be constructed from the original one in a canonical way. We also provide an integral representation for operator convex functions on the positive half-line, which is the main ingredient in extending previously known results on the monotonicity inequality and the case of equality. We also consider some special cases where the convexity of f is sufficient for the monotonicity, and obtain the inverse Holder inequality for operators as an application. The presentation is completely self-contained and requires only standard knowledge of matrix analysis.},
  archivePrefix = {arXiv},
  eprint = {1008.2529},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematical Physics,Quantum Physics},
  number = {07}
}

@book{horn_matrix_2012,
  title = {Matrix Analysis},
  author = {Horn, Roger A. and Johnson, Charles R.},
  date = {2012},
  edition = {2nd ed},
  publisher = {{Cambridge University Press}},
  location = {{Cambridge ; New York}},
  abstract = {"The thoroughly revised and updated second edition of this acclaimed text has several new and expanded sections and more than 1,100 exercises"--},
  isbn = {978-0-521-83940-2},
  keywords = {MATHEMATICS / Algebra / Abstract,Matrices},
  langid = {english},
  pagetotal = {643}
}

@book{horn_topics_1991,
  title = {Topics in {{Matrix Analysis}}},
  author = {Horn, Roger A. and Johnson, Charles R.},
  date = {1991},
  publisher = {{Cambridge University Press}},
  location = {{Cambridge}},
  doi = {10.1017/CBO9780511840371},
  url = {https://www.cambridge.org/core/books/topics-in-matrix-analysis/B988495A235F1C3406EA484A2C477B03},
  urldate = {2021-02-10},
  abstract = {Building on the foundations of its predecessor volume, Matrix Analysis, this book treats in detail several topics in matrix theory not included in the previous volume, but with important applications and of special mathematical interest. As with the previous volume, the authors assume a background knowledge of elementary linear algebra and rudimentary analytical concepts. Many examples and exercises of varying difficulty are included.},
  isbn = {978-0-521-46713-1}
}

@online{katariya_geometric_2020,
  title = {Geometric Distinguishability Measures Limit Quantum Channel Estimation and Discrimination},
  author = {Katariya, Vishal and Wilde, Mark M.},
  date = {2020-04-22},
  url = {http://arxiv.org/abs/2004.10708},
  urldate = {2021-02-15},
  abstract = {Quantum channel estimation and discrimination are fundamentally related information processing tasks of interest in quantum information science. In this paper, we analyze these tasks by employing the right logarithmic derivative Fisher information and the geometric R\textbackslash 'enyi relative entropy, respectively, and we also identify connections between these distinguishability measures. A key result of our paper is that a chain-rule property holds for the right logarithmic derivative Fisher information and the geometric R\textbackslash 'enyi relative entropy for the interval \$\textbackslash alpha\textbackslash in(0,1) \$ of the R\textbackslash 'enyi parameter \$\textbackslash alpha\$. In channel estimation, these results imply a condition for the unattainability of Heisenberg scaling, while in channel discrimination, they lead to improved bounds on error rates in the Chernoff and Hoeffding error exponent settings. More generally, we introduce the amortized quantum Fisher information as a conceptual framework for analyzing general sequential protocols that estimate a parameter encoded in a quantum channel, and we use this framework, beyond the aforementioned application, to show that Heisenberg scaling is not possible when a parameter is encoded in a classical-quantum channel. We then identify a number of other conceptual and technical connections between the tasks of estimation and discrimination and the distinguishability measures involved in analyzing each. As part of this work, we present a detailed overview of the geometric R\textbackslash 'enyi relative entropy of quantum states and channels, as well as its properties, which may be of independent interest.},
  archivePrefix = {arXiv},
  eprint = {2004.10708},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Mathematical Physics,Mathematics - Statistics Theory,Quantum Physics},
  primaryClass = {math-ph, physics:quant-ph, stat}
}

@book{kato_perturbation_1995,
  title = {Perturbation Theory for Linear Operators},
  author = {Katō, Tosio},
  date = {1995},
  publisher = {{Springer}},
  location = {{Berlin}},
  isbn = {978-3-540-58661-6},
  keywords = {Linear operators,Perturbation (Mathematics)},
  langid = {english},
  pagetotal = {619},
  series = {Classics in Mathematics}
}

@online{khatri_principles_2020,
  title = {Principles of {{Quantum Communication Theory}}: {{A Modern Approach}}},
  shorttitle = {Principles of {{Quantum Communication Theory}}},
  author = {Khatri, Sumeet and Wilde, Mark M.},
  date = {2020-11-09},
  url = {http://arxiv.org/abs/2011.04672},
  urldate = {2021-02-11},
  abstract = {This is a preliminary version of a book in progress on the theory of quantum communication. We adopt an information-theoretic perspective throughout and give a comprehensive account of fundamental results in quantum communication theory from the past decade (and earlier), with an emphasis on the modern one-shot-to-asymptotic approach that underlies much of today's state-of-the-art research in this field. In Part I, we cover mathematical preliminaries and provide a detailed study of quantum mechanics from an information-theoretic perspective. We also provide an extensive and thorough review of the quantum entropy zoo, and we devote an entire chapter to the study of entanglement measures. Equipped with these essential tools, in Part II we study classical communication (with and without entanglement assistance), entanglement distillation, quantum communication, secret key distillation, and private communication. In Part III, we cover the latest developments in feedback-assisted communication tasks, such as quantum and classical feedback-assisted communication, LOCC-assisted quantum communication, and secret key agreement.},
  archivePrefix = {arXiv},
  eprint = {2011.04672},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Condensed Matter - Statistical Mechanics,High Energy Physics - Theory,Mathematical Physics,Quantum Physics},
  primaryClass = {cond-mat, physics:hep-th, physics:math-ph, physics:quant-ph}
}

@article{kubo_means_1979,
  title = {Means of Positive Linear Operators.},
  author = {Kubo, Fumio and Ando, Tsuyoshi},
  date = {1979},
  journaltitle = {Mathematische Annalen},
  volume = {246},
  pages = {205--224},
  url = {https://eudml.org/doc/163339},
  urldate = {2020-12-03},
  langid = {und}
}

@article{leditzky_approaches_2018,
  title = {Approaches for Approximate Additivity of the {{Holevo}} Information of Quantum Channels},
  author = {Leditzky, Felix and Kaur, Eneet and Datta, Nilanjana and Wilde, Mark M.},
  date = {2018-01-25},
  journaltitle = {Physical Review A},
  shortjournal = {Phys. Rev. A},
  volume = {97},
  pages = {012332},
  doi = {10.1103/PhysRevA.97.012332},
  url = {http://arxiv.org/abs/1709.01111},
  urldate = {2021-01-28},
  abstract = {We study quantum channels that are close to another channel with weakly additive Holevo information and derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above that measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise, and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.},
  archivePrefix = {arXiv},
  eprint = {1709.01111},
  eprinttype = {arxiv},
  keywords = {Quantum Physics},
  number = {1}
}

@article{leditzky_data_2017,
  title = {Data Processing for the Sandwiched {{Renyi}} Divergence: A Condition for Equality},
  shorttitle = {Data Processing for the Sandwiched {{R}}\textbackslash 'enyi Divergence},
  author = {Leditzky, Felix and Rouzé, Cambyse and Datta, Nilanjana},
  date = {2017-01},
  journaltitle = {Letters in Mathematical Physics},
  shortjournal = {Lett Math Phys},
  volume = {107},
  pages = {61--80},
  doi = {10.1007/s11005-016-0896-9},
  url = {http://arxiv.org/abs/1604.02119},
  urldate = {2020-11-22},
  abstract = {The \$\textbackslash alpha\$-sandwiched R\textbackslash 'enyi divergence satisfies the data processing inequality, i.e. monotonicity under quantum operations, for \$\textbackslash alpha\textbackslash geq 1/2\$. In this article, we derive a necessary and sufficient algebraic condition for equality in the data processing inequality for the \$\textbackslash alpha\$-sandwiched R\textbackslash 'enyi divergence for all \$\textbackslash alpha\textbackslash geq 1/2\$. For the range \$\textbackslash alpha\textbackslash in [1/2,1)\$, our result provides the only condition for equality obtained thus far. To prove our result, we first consider the special case of partial trace, and derive a condition for equality based on the original proof of the data processing inequality by Frank and Lieb [J. Math. Phys. 54.12 (2013), p. 122201] using a strict convexity/concavity argument. We then generalize to arbitrary quantum operations via the Stinespring Representation Theorem. As applications of our condition for equality in the data processing inequality, we deduce conditions for equality in various entropic inequalities. We formulate a R\textbackslash 'enyi version of the Araki-Lieb inequality and analyze the case of equality, generalizing a result by Carlen and Lieb [Lett. Math. Phys. 101.1 (2012), pp. 1-11] about equality in the original Araki-Lieb inequality. Furthermore, we prove a general lower bound on a R\textbackslash 'enyi version of the entanglement of formation, and observe that it is attained by states saturating the R\textbackslash 'enyi version of the Araki-Lieb inequality. Finally, we prove that the known upper bound on the entanglement fidelity in terms of the usual fidelity is saturated only by pure states.},
  archivePrefix = {arXiv},
  eprint = {1604.02119},
  eprinttype = {arxiv},
  keywords = {Mathematical Physics,Quantum Physics},
  number = {1}
}

@online{liu_resource_2019,
  title = {Resource Theories of Quantum Channels and the Universal Role of Resource Erasure},
  author = {Liu, Zi-Wen and Winter, Andreas},
  date = {2019-04-08},
  url = {http://arxiv.org/abs/1904.04201},
  urldate = {2021-01-28},
  abstract = {We initiate the systematic study of resource theories of quantum channels, i.e. of the dynamics that quantum systems undergo by completely positive maps, in abstracto: Resources are in principle all maps from one quantum system to another, but some maps are deemed free. The free maps are supposed to satisfy certain axioms, among them closure under tensor products, under composition and freeness of the identity map (the latter two say that the free maps form a monoid). The free maps act on the resources simply by tensor product and composition. This generalizes the much-studied resource theories of quantum states, and abolishes the distinction between resources (states) and the free maps, which act on the former, leaving only maps, divided into resource-full and resource-free ones. We discuss the axiomatic framework of quantifying channel resources, and show two general methods of constructing resource monotones of channels. Furthermore, we show that under mild regularity conditions, each resource theory of quantum channels has a distinguished monotone, the robustness (and its smoothed version), generalizing the analogous concept in resource theories of states. We give an operational interpretation of the log-robustness as the amount of heat dissipation (randomness) required for resource erasure by random reversible free maps, valid in broad classes of resource theories of quantum channels. Technically, this is based on an abstract version of the recent convex-split lemma, extended from its original domain of quantum states to ordered vector spaces with sufficiently well-behaved base norms (which includes the case of quantum channels with diamond norm or variants thereof). Finally, we remark on several key issues concerning the asymptotic theory.},
  archivePrefix = {arXiv},
  eprint = {1904.04201},
  eprinttype = {arxiv},
  keywords = {Quantum Physics},
  primaryClass = {quant-ph}
}

@online{matsumoto_condition_2014,
  title = {On the Condition of Conversion of Classical Probability Distribution Families into Quantum Families},
  author = {Matsumoto, Keiji},
  date = {2014-12-11},
  url = {http://arxiv.org/abs/1412.3680},
  urldate = {2021-02-15},
  abstract = {The purpose of the paper is to study the condition for a probability distribution family to a quantum state family. This is an (relatively) easy example of quantum version of "comparison of statistical experiments", which had turned out to supply deep insight into the foundation of classical and quantum statistics \textbackslash cite\{Strasser\}\textbackslash cite\{Torgersen\}. It turns out use of maximal quantum \$f\$-divergence is useful in characterizing the classical-quantum transformability.},
  archivePrefix = {arXiv},
  eprint = {1412.3680},
  eprinttype = {arxiv},
  keywords = {Quantum Physics},
  primaryClass = {quant-ph}
}

@online{matsumoto_new_2013,
  title = {A New Quantum Version of F-Divergence},
  author = {Matsumoto, Keiji},
  date = {2013-11-19},
  url = {https://arxiv.org/abs/1311.4722v4},
  urldate = {2021-01-24},
  abstract = {This paper proposes and studies new quantum version of \$f\$-divergences, a class of convex functionals of a pair of probability distributions including Kullback-Leibler divergence, Rnyi-type relative entropy and so on. There are several quantum versions so far, including the one by Petz. We introduce another quantum version (\$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$, below), defined as the solution to an optimization problem, or the minimum classical \$f\$- divergence necessary to generate a given pair of quantum states. It turns out to be the largest quantum \$f\$-divergence. The closed formula of \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$ is given either if \$f\$ is operator convex, or if one of the state is a pure state. Also, concise representation of \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$ as a pointwise supremum of linear functionals is given and used for the clarification of various properties of the quality.   Using the closed formula of \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$, we show: Suppose \$f\$ is operator convex. Then the\textbackslash{} maximum \$f\textbackslash,\$- divergence of the probability distributions of a measurement under the state \$ρ\$ and \$σ\$ is strictly less than \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\textbackslash left( ρ\textbackslash Vertσ\textbackslash right) \$. This statement may seem intuitively trivial, but when \$f\$ is not operator convex, this is not always true. A counter example is \$f\textbackslash left( λ\textbackslash right) =\textbackslash left\textbackslash vert 1-λ\textbackslash right\textbackslash vert \$, which corresponds to total variation distance.   We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.},
  archivePrefix = {arXiv},
  eprint = {1311.4722},
  eprinttype = {arxiv},
  langid = {english},
  primaryClass = {quant-ph}
}

@online{matsumoto_quantum_2014,
  title = {Quantum {{Fidelities}}, {{Their Duals}}, {{And Convex Analysis}}},
  author = {Matsumoto, Keiji},
  date = {2014-08-14},
  url = {http://arxiv.org/abs/1408.3462},
  urldate = {2021-02-15},
  abstract = {We study tree kinds of quantum fidelity. Usual Uhlmann's fidelity, minus of f-divergence when \$f(x)=-\textbackslash sqrt\{x\}\$, and the one introduced by the author via reverse test. All of them are quantum extensions of classical fidelity, where the first one is the largest and the third one is the smallest. We characterize them in terms of convex optimization, and introduce their 'dual' quantity, or the polar of the minus of the fidelity. They turned out to be monotone increasing by unital completely positive maps, concave, and linked to its classical version via optimization about classical-to-quantum maps and quantum-to-classical maps.},
  archivePrefix = {arXiv},
  eprint = {1408.3462},
  eprinttype = {arxiv},
  keywords = {Quantum Physics},
  primaryClass = {quant-ph}
}

@article{moakher_differential_2005,
  title = {A {{Differential Geometric Approach}} to the {{Geometric Mean}} of {{Symmetric Positive}}-{{Definite Matrices}}},
  author = {Moakher, Maher},
  date = {2005-01},
  journaltitle = {SIAM Journal on Matrix Analysis and Applications},
  shortjournal = {SIAM J. Matrix Anal. \& Appl.},
  volume = {26},
  pages = {735--747},
  doi = {10.1137/S0895479803436937},
  url = {http://epubs.siam.org/doi/10.1137/S0895479803436937},
  urldate = {2020-12-16},
  abstract = {In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization of this mean.},
  langid = {english},
  number = {3}
}

@article{mosonyi_quantum_2011,
  title = {On the {{Quantum Rényi Relative Entropies}} and {{Related Capacity Formulas}}},
  author = {Mosonyi, M. and Hiai, F.},
  date = {2011-04},
  journaltitle = {IEEE Transactions on Information Theory},
  volume = {57},
  pages = {2474--2487},
  doi = {10.1109/TIT.2011.2110050},
  abstract = {Following Csiszár's approach in classical information theory, it is shown that the quantum α-relative entropies with parameter α ∈ (0,1) can be represented as generalized cutoff rates, and hence a direct operational interpretation of the quantum α-relative entropies are provided. It is also shown that various generalizations of the Holevo capacity, defined in terms of the α-relative entropies, coincide for the parameter range α ∈ (0,2], and an upper bound on the one-shot ε-capacity of a classical-quantum channel in terms of these capacities is given.},
  eventtitle = {{{IEEE Transactions}} on {{Information Theory}}},
  keywords = {Channel capacity,classical information theory,classical-quantum channel,Correlation,Csiszár's approach,direct operational interpretation,entropy,Entropy,Error probability,generalized cutoff rates,Hilbert space,Hoeffding distances,Holevo capacity,Joints,one-shot capacities,one-shot ε-capacity,quantum channels,quantum Rényi relative entropy,quantum theory,quantum α-relative entropy,related capacity formulas,Relativistic quantum mechanics,Rényi relative entropies,α-capacities},
  number = {4}
}

@article{muller-lennert_quantum_2013,
  title = {On Quantum {{Rényi}} Entropies: {{A}} New Generalization and Some Properties},
  shorttitle = {On Quantum {{Rényi}} Entropies},
  author = {Müller-Lennert, Martin and Dupuis, Frédéric and Szehr, Oleg and Fehr, Serge and Tomamichel, Marco},
  date = {2013-12-01},
  journaltitle = {Journal of Mathematical Physics},
  shortjournal = {Journal of Mathematical Physics},
  volume = {54},
  pages = {122203},
  publisher = {{American Institute of Physics}},
  doi = {10.1063/1.4838856},
  url = {https://aip.scitation.org/doi/full/10.1063/1.4838856},
  urldate = {2021-02-11},
  abstract = {The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.},
  number = {12}
}

@online{noauthor_linear_nodate,
  title = {Linear Algebra - {{Inverse}} of the Anti-Commutator},
  url = {https://math.stackexchange.com/questions/2362983/inverse-of-the-anti-commutator},
  urldate = {2020-12-16},
  organization = {{Mathematics Stack Exchange}}
}

@online{noauthor_linear_nodate-1,
  title = {Linear Algebra - {{Is}} There Any Inverse-Commutator for Matrices},
  url = {https://math.stackexchange.com/questions/1307098/is-there-any-inverse-commutator-for-matrices},
  urldate = {2020-12-16},
  organization = {{Mathematics Stack Exchange}}
}

@online{noauthor_real_nodate,
  title = {Real Analysis - {{Derivatives}} of Infimum},
  url = {https://math.stackexchange.com/questions/29867/derivatives-of-infimum},
  urldate = {2020-12-16},
  organization = {{Mathematics Stack Exchange}}
}

@online{noauthor_semidefinite_nodate,
  title = {Semidefinite {{Optimization}} and {{Convex Algebraic Geometry}} | {{Optimization}}, {{OR}} and Risk Analysis},
  url = {https://www.cambridge.org/us/academic/subjects/mathematics/optimization-or-and-risk-analysis/semidefinite-optimization-and-convex-algebraic-geometry, https://www.cambridge.org/us/academic/subjects/mathematics/optimization-or-and-risk-analysis},
  urldate = {2020-12-14},
  langid = {english},
  organization = {{Cambridge University Press}}
}

@article{petz_contraction_1998,
  title = {Contraction of {{Generalized Relative Entropy Under Stochastic Mappings}} on {{Matrices}}},
  author = {Petz, Dénes and Ruskai, Mary Beth},
  date = {1998-01-01},
  journaltitle = {Infinite Dimensional Analysis, Quantum Probability and Related Topics},
  shortjournal = {Infin. Dimens. Anal. Quantum. Probab. Relat. Top.},
  volume = {01},
  pages = {83--89},
  publisher = {{World Scientific Publishing Co.}},
  doi = {10.1142/S0219025798000077},
  url = {https://www.worldscientific.com/doi/abs/10.1142/S0219025798000077},
  urldate = {2021-02-15},
  abstract = {The contraction coefficient of stochastic mappings between full matrix algebras is introduced with respect to a generalized relative entropy labeled by an operator convex function g. It is proved that the coefficient is actually independent of g, in particular, it can be most conveniently computed by means of the square function.},
  number = {01}
}

@article{petz_monotonicity_2003,
  title = {Monotonicity of Quantum Relative Entropy Revisited},
  author = {Petz, D.},
  date = {2003-03},
  journaltitle = {Reviews in Mathematical Physics},
  shortjournal = {Rev. Math. Phys.},
  volume = {15},
  pages = {79--91},
  doi = {10.1142/S0129055X03001576},
  url = {http://arxiv.org/abs/quant-ph/0209053},
  urldate = {2020-11-23},
  abstract = {Monotonicity under coarse-graining is a crucial property of the quantum relative entropy. The aim of this paper is to investigate the condition of equality in the monotonicity theorem and in its consequences such as the strong sub-additivity of the von Neumann entropy, the Golden-Thompson trace inequality and the monotonicity of the Holevo quantity.The relation to quantum Markovian states is briefly indicated.},
  archivePrefix = {arXiv},
  eprint = {quant-ph/0209053},
  eprinttype = {arxiv},
  keywords = {Quantum Physics},
  number = {01}
}

@article{petz_quasi-entropies_1986,
  title = {Quasi-Entropies for Finite Quantum Systems},
  author = {Petz, Dénes},
  date = {1986-02-01},
  journaltitle = {Reports on Mathematical Physics},
  shortjournal = {Reports on Mathematical Physics},
  volume = {23},
  pages = {57--65},
  doi = {10.1016/0034-4877(86)90067-4},
  url = {https://www.sciencedirect.com/science/article/pii/0034487786900674},
  urldate = {2021-02-11},
  abstract = {Convexity properties of entropy-like functionals on states of a finite dimensional algebra are discussed. The treatment covers both the quantum mechanical and the classical cases. The purpose is to generalize Lieb's convexity theorem and the monotonicity of the relative entropy using the Jensen inequality of operator convex functions. From the quasi-entropies defined here the quantum version of Rényi's α- entropies can be deduced.},
  langid = {english},
  number = {1}
}

@book{rockafellar_convex_1997,
  title = {Convex {{Analysis}}},
  shorttitle = {Convex {{Analysis}}},
  author = {Rockafellar, Ralph Tyrell},
  date = {1997},
  publisher = {{Princeton University Press}},
  location = {{Princeton, N.J.}},
  url = {https://ezp.lib.cam.ac.uk/login?url=https://search.ebscohost.com/login.aspx?direct=true&db=nlebk&AN=969057&site=ehost-live&scope=site},
  urldate = {2021-01-21},
  abstract = {Available for the first time in paperback, R. Tyrrell Rockafellar's classic study presents readers with a coherent branch of nonlinear mathematical analysis that is especially suited to the study of optimization problems. Rockafellar's theory differs from classical analysis in that differentiability assumptions are replaced by convexity assumptions. The topics treated in this volume include: systems of inequalities, the minimum or maximum of a convex function over a convex set, Lagrange multipliers, minimax theorems and duality, as well as basic results about the structure of convex sets and the continuity and differentiability of convex functions and saddle- functions. This book has firmly established a new and vital area not only for pure mathematics but also for applications to economics and engineering. A sound knowledge of linear algebra and introductory real analysis should provide readers with sufficient background for this book. There is also a guide for the reader who may be using the book as an introduction, indicating which parts are essential and which may be skipped on a first reading.},
  isbn = {978-0-691-01586-6},
  keywords = {Convex domains,Convex functions,Mathematical analysis,MATHEMATICS / Optimization},
  langid = {english},
  series = {Princeton {{Paperbacks}}}
}

@incollection{simon_2_2019,
  title = {2\,×\,2 {{Matrices}}: {{The Donoghue}} and {{Hansen}}–{{Tomiyama Theorems}}},
  shorttitle = {2\,×\,2 {{Matrices}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {145--151},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_14},
  url = {https://doi.org/10.1007/978-3-030-22422-6_14},
  urldate = {2020-11-28},
  abstract = {Loewner’s theorem provides a simple characterization of \textbackslash (\{\textbackslash mathcal M\}\_\textbackslash infty (a,b)\textbackslash ) but it is not so simple to describe which functions are in a general \textbackslash (\{\textbackslash mathcal M\}\_n(a,b)\textbackslash ).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_ameurs_2019,
  title = {Ameur’s {{Proof Using Quadratic Interpolation}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {313--320},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_30},
  url = {https://doi.org/10.1007/978-3-030-22422-6_30},
  urldate = {2020-11-28},
  abstract = {This chapter relies on the theory of quadratic interpolation as presented in Chapter 15.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_convexity_2019,
  title = {Convexity, {{III}}: {{Hansen}}–{{Jensen}}–{{Pedersen}} ({{HJP}}) {{Inequality}}},
  shorttitle = {Convexity, {{III}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {117--122},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_11},
  url = {https://doi.org/10.1007/978-3-030-22422-6_11},
  urldate = {2020-11-28},
  abstract = {Jensen’s inequality in its original form says that if f is a scalar convex function (on an open convex set, A, of a vector space, V) and if \textbackslash (\textbackslash\{x\_j\textbackslash\}\_\{j=1\}\^n \textbackslash subset A; \textbackslash, 0 \textbackslash le \textbackslash theta \_j \textbackslash le 1, \textbackslash, j=1,\textbackslash dots ,n\textbackslash ) with \textbackslash (\textbackslash sum \_\{j=1\}\^n \textbackslash theta \_j = 1\textbackslash ), then \textbackslash (f(\textbackslash sum \_\{j=1\}\^n \textbackslash theta \_j x\_j) \textbackslash le \textbackslash sum \_\{j=1\}\^n \textbackslash theta \_j f(x\_j)\textbackslash ).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_convexity_2019-1,
  title = {Convexity, {{II}}: {{Concavity}} and {{Monotonicity}}},
  shorttitle = {Convexity, {{II}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {107--116},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_10},
  url = {https://doi.org/10.1007/978-3-030-22422-6_10},
  urldate = {2020-11-28},
  abstract = {This chapter will first provide a remarkable equivalence between matrix concavity and matrix monotonicity for positive functions not even hinted at in the scalar case. Then we’ll discuss a connection between matrix convexity and Loewner matrices .},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_convexity_2019-2,
  title = {Convexity, {{V}}: {{Strongly Operator Convex Functions}}},
  shorttitle = {Convexity, {{V}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {135--144},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_13},
  url = {https://doi.org/10.1007/978-3-030-22422-6_13},
  urldate = {2020-11-28},
  abstract = {Let f be a real-valued function on \textbackslash ((a,b) \textbackslash subset \{\textbackslash mathbb \{R\}\}\textbackslash ).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_convexity_2019-3,
  title = {Convexity, {{I}}: {{The Theorem}} of {{Bendat}}–{{Kraus}}–{{Sherman}}–{{Uchiyama}}},
  shorttitle = {Convexity, {{I}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {95--105},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_9},
  url = {https://doi.org/10.1007/978-3-030-22422-6_9},
  urldate = {2020-11-28},
  abstract = {Recall that.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_convexity_2019-4,
  title = {Convexity, {{IV}}: {{Bhatia}}–{{Hiai}}–{{Sano}} ({{BHS}}) {{Theorem}}},
  shorttitle = {Convexity, {{IV}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {123--134},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_12},
  url = {https://doi.org/10.1007/978-3-030-22422-6_12},
  urldate = {2020-11-28},
  abstract = {In Chapter 5, given a C1 function, f, on \textbackslash ((a,b) \textbackslash subset \{\textbackslash mathbb \{R\}\}\textbackslash ), we defined the Loewner matrix by.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_divided_2019,
  title = {Divided {{Differences}} and {{Polynomial Approximation}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {235--245},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_22},
  url = {https://doi.org/10.1007/978-3-030-22422-6_22},
  urldate = {2020-11-28},
  abstract = {In this chapter, as warmup for rational approximation, we discuss polynomial approximation. Since rational approximation has f = P∕Q ⇔ Qf = P, finding P and Q can be viewed as a kind of polynomial approximation.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_divided_2019-1,
  title = {Divided {{Differences}} and {{Multipoint Rational Interpolation}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {247--261},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_23},
  url = {https://doi.org/10.1007/978-3-030-22422-6_23},
  urldate = {2020-11-28},
  abstract = {This chapter is the most involved and most central of the chapters on rational approximation.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_generalities_2019,
  title = {Some {{Generalities}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {11--16},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_2},
  url = {https://doi.org/10.1007/978-3-030-22422-6_2},
  urldate = {2020-11-28},
  abstract = {In this chapter, we will discuss several issues that will help us prove the easy half of Loewner’s theorem in the next chapter and then, we will extend Theorem 1.6 to infinite dimensions.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_hardy_2019,
  title = {Hardy {{Spaces}} and the {{Rosenblum}}–{{Rovnyak Proof}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {339--344},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_33},
  url = {https://doi.org/10.1007/978-3-030-22422-6_33},
  urldate = {2020-11-28},
  abstract = {In this chapter, we present a proof of the hard part of Loewner’s theorem due to Rosenblum and Rovnyak based on their general theorem that appeared as Theorem 19.8.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_heinavaaras_2019,
  title = {Heinävaara’s {{Integral Formula}} and the {{Dobsch}}–{{Donoghue Theorem}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {73--83},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_6},
  url = {https://doi.org/10.1007/978-3-030-22422-6_6},
  urldate = {2020-11-28},
  abstract = {Recall (see ( 5.38)) that given a C2n−1 function, f, on (a, b), we defined its Dobsch matrix at x0 ∈ (a, b) by \$\$\textbackslash displaystyle B\_n(x\_0;f)\_\{k\textbackslash ell \} = \textbackslash frac \{f\^\{(k+\textbackslash ell -1)\}(x\_0)\}\{(k+\textbackslash ell -1)!\} \textbackslash qquad 1\textbackslash le k,\textbackslash ell \textbackslash le n. \$\$},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_heinavaaras_2019-1,
  title = {Heinävaara’s {{Second Proof}} of the {{Dobsch}}–{{Donoghue Theorem}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {89--93},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_8},
  url = {https://doi.org/10.1007/978-3-030-22422-6_8},
  urldate = {2020-11-28},
  abstract = {In this chapter, we provide a really short, simple, and elegant proof due to Heinävaara of Theorem 6.1. It will be a direct consequence of the mean value theorem for divided differences. Below, \textbackslash (\{\textbackslash mathcal P\}\_n\textbackslash ) will denote the real polynomials of degree at most n, a real vector space of dimension n + 1.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_herglotz_2019,
  title = {The {{Herglotz Representation Theorems}} and the {{Easy Direction}} of {{Loewner}}’s {{Theorem}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {17--31},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_3},
  url = {https://doi.org/10.1007/978-3-030-22422-6_3},
  urldate = {2020-11-28},
  abstract = {In this chapter, we will show (b)⇔(c) in Theorems 1.3 and 1.6 and also (b)⇒(a) and the equivalent part of Theorem 1.8. We’ll have a different proof of (c)⇒(a) in Chapter 16. We want to study functions f on \textbackslash (\{\textbackslash mathbb \{C\}\}\_+\textbackslash ) with \textbackslash ( \textbackslash operatorname \{\textbackslash mathrm \{Im\}\} f(x){$>$}0\textbackslash ), but begin with functions f on \textbackslash (\{\textbackslash mathbb \{D\}\} =\textbackslash\{z\textbackslash in \{\textbackslash mathbb \{C\}\}\textbackslash mid \textbackslash lvert z\textbackslash rvert {$<$}1\textbackslash\}\textbackslash ) with \textbackslash ( \textbackslash operatorname \{\textbackslash mathrm \{Re\}\} f {$>$}0\textbackslash ) and use conformal mapping.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_hilbert_2019,
  title = {Hilbert {{Space Methods}} and the {{Korányi Proof}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {283--287},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_27},
  url = {https://doi.org/10.1007/978-3-030-22422-6_27},
  urldate = {2020-11-28},
  abstract = {The moral of Chapter 17 and many other situations is that given a positive matrix, use it to form an inner product.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_introduction_2019,
  title = {Introduction: {{The Statement}} of {{Loewner}}’s {{Theorem}}},
  shorttitle = {Introduction},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {3--9},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_1},
  url = {https://doi.org/10.1007/978-3-030-22422-6_1},
  urldate = {2020-11-28},
  abstract = {Most of this monograph will consider finite matrices, but since we will be interested in positivity defined in terms of a (Euclidean) inner product, we make definitions in a general context.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_kreinmilman_2019,
  title = {The {{Krein}}–{{Milman Theorem}} and {{Hansen}}’s {{Variant}} of the {{Hansen}}–{{Pedersen Proof}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {289--299},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_28},
  url = {https://doi.org/10.1007/978-3-030-22422-6_28},
  urldate = {2020-11-28},
  abstract = {In this chapter, we will present a proof of Loewner’s theorem due to Hansen–Pedersen that relies on the Krein–Milman theorem; we follow a variant of Hansen [133].},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_lieb_2019,
  title = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{II}}: {{Effros}}’ {{Proof}}},
  shorttitle = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{II}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {393--394},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_39},
  url = {https://doi.org/10.1007/978-3-030-22422-6_39},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll prove Lieb Concavity as a special case of the following theorem (we explain the name in the Notes).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_lieb_2019-1,
  title = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{III}}: {{Ando}}’s {{Proof}}},
  shorttitle = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{III}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {395--396},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_40},
  url = {https://doi.org/10.1007/978-3-030-22422-6_40},
  urldate = {2020-11-28},
  abstract = {In Chapter 37, we showed that if f is a matrix monotone function on (0, ∞), then A, B↦A1∕2f(A−1∕2BA−1∕2)A1∕2 is jointly concave on pairs of positive operators (see Corollary 37.2).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_lieb_2019-2,
  title = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{IV}}: {{Aujla}}–{{Hansen}}–{{Uhlmann Proof}}},
  shorttitle = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{IV}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {397--398},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_41},
  url = {https://doi.org/10.1007/978-3-030-22422-6_41},
  urldate = {2020-11-28},
  abstract = {In this chapter we’ll prove a generalization of Lieb Concavity (which was already in his original paper).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_lieb_2019-3,
  title = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{I}}: {{Basics}}},
  shorttitle = {Lieb {{Concavity}} and {{Lieb}}–{{Ruskai Strong Subadditivity Theorems}}, {{I}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {385--391},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_38},
  url = {https://doi.org/10.1007/978-3-030-22422-6_38},
  urldate = {2020-11-28},
  abstract = {The next four chapters concern some important matrix inequalities originally of significance in quantum statistical mechanics and, more recently, in quantum information theory.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_loewner_2019,
  title = {Loewner {{Matrices}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {43--71},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_5},
  url = {https://doi.org/10.1007/978-3-030-22422-6_5},
  urldate = {2020-11-28},
  abstract = {In this chapter, we will reduce matrix monotonicity to the positivity of certain matrices and determinants. Seven of the eleven proofs we’ll give of the hard part of Loewner’s theorem rely on this reduction.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_loewners_2019,
  title = {Loewner’s {{Theorem}} via {{Rational Interpolation}}: {{Loewner}}’s {{Proof}}},
  shorttitle = {Loewner’s {{Theorem}} via {{Rational Interpolation}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {267--272},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_25},
  url = {https://doi.org/10.1007/978-3-030-22422-6_25},
  urldate = {2020-11-28},
  abstract = {Our main goal in this chapter is to prove the following.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_loewners_2019-1,
  title = {Loewner’s {{Theorem}} for {{General Open Sets}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {359--369},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_35},
  url = {https://doi.org/10.1007/978-3-030-22422-6_35},
  urldate = {2020-11-28},
  abstract = {In this chapter, we discuss versions of Loewner’s theorem that replace (a, b) by a more general open subset of \textbackslash (\{\textbackslash mathbb \{R\}\}\textbackslash ). Let \textbackslash (\textbackslash Omega \textbackslash subset \{\textbackslash mathbb \{R\}\}\textbackslash ) be an open subset of \textbackslash (\{\textbackslash mathbb \{R\}\}\textbackslash ).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_mellin_2019,
  title = {Mellin {{Transforms}}: {{Boutet}} de {{Monvel}}’s {{Proof}}},
  shorttitle = {Mellin {{Transforms}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {345--357},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_34},
  url = {https://doi.org/10.1007/978-3-030-22422-6_34},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll see that Mellin transforms are a natural approach to proving the hard part of Loewner’s theorem. We’ve already seen several reformulations of the integral representation for matrix monotone functions on (0, ∞).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_moment_2019,
  title = {The {{Moment Problem}} and the {{Bendat}}–{{Sherman Proof}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {273--282},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_26},
  url = {https://doi.org/10.1007/978-3-030-22422-6_26},
  urldate = {2020-11-28},
  abstract = {The two proofs so far have involved approximation by rational functions, and that will also be true of the proofs in Chapters 31 and 32.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_monotonicity_2019,
  title = {Monotonicity of the {{Square Root}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {33--41},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_4},
  url = {https://doi.org/10.1007/978-3-030-22422-6_4},
  urldate = {2020-11-28},
  abstract = {As explained in the preface, most applications of Loewner’s theorem involve the easy half of the theorem. This chapter is an aside involving the two most significant and common matrix monotone functions: fractional powers and the log.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_multipoint_2019,
  title = {Multipoint {{Continued Fractions}}: {{A New Proof}}},
  shorttitle = {Multipoint {{Continued Fractions}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {333--337},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_32},
  url = {https://doi.org/10.1007/978-3-030-22422-6_32},
  urldate = {2020-11-28},
  abstract = {This chapter has a variant of the proof of the last one that we regard as a new proof of the hard part of Loewner’s theorem. The proof of the last section was fairly straightforward but the details of the proof of convergence to the right limit were involved—one might even say tedious.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_n1nmathcal_2019,
  title = {ℳn+1≠{{ℳn}}\$\$\{\textbackslash mathcal \vphantom\}{{M}}\vphantom\{\}\_\{n+1\} \textbackslash ne \{\textbackslash mathcal \vphantom\}{{M}}\vphantom\{\}\_n\$\$},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {85--88},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_7},
  url = {https://doi.org/10.1007/978-3-030-22422-6_7},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll present explicit examples that show that \textbackslash (\{\textbackslash mathcal M\}\_\{n+1\}(a,b)\textbackslash ) is strictly smaller than \textbackslash (\{\textbackslash mathcal M\}\_\{n\}(a,b)\textbackslash ) and that show that 2n − 3 in Theorem 5.2 is optimal.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_one-point_2019,
  title = {One-{{Point Continued Fractions}}: {{The Wigner}}–von {{Neumann Proof}}},
  shorttitle = {One-{{Point Continued Fractions}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {321--332},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_31},
  url = {https://doi.org/10.1007/978-3-030-22422-6_31},
  urldate = {2020-11-28},
  abstract = {In this chapter and the next, we provide proofs of the hard part of Loewner’s theorem that rely on continued fractions and a key lemma on the Loewner matrix under stripping off the front of the fraction.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_operator_2019,
  ids = {simon_operator_2019-1},
  title = {Operator {{Means}}, {{II}}: {{Kubo}}–{{Ando Theorem}}},
  shorttitle = {Operator {{Means}}, {{II}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {379--384},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_37},
  url = {https://doi.org/10.1007/978-3-030-22422-6_37},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll see a natural 1–1 correspondence between operators means A, B↦AσB and non-negative matrix monotone functions, f on (0, ∞), normalized by f(1) = 1.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_operator_2019-2,
  title = {Operator {{Means}}, {{I}}: {{Basics}} and {{Examples}}},
  shorttitle = {Operator {{Means}}, {{I}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {373--378},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_36},
  url = {https://doi.org/10.1007/978-3-030-22422-6_36},
  urldate = {2020-11-28},
  abstract = {If a, b {$>$} 0, there are various “means” that have turned out to be useful, in particular, the arithmetic mean, \textbackslash (\textbackslash tfrac \{1\}\{2\}(a+b)\textbackslash ), the harmonic mean, 2(a−1 + b−1)−1 and the geometric mean, \textbackslash (\textbackslash sqrt \{ab\}\textbackslash ).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_pick_2019,
  title = {Pick {{Interpolation}}, {{V}}: {{Rational Interpolation Proof}}},
  shorttitle = {Pick {{Interpolation}}, {{V}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {263--266},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_24},
  url = {https://doi.org/10.1007/978-3-030-22422-6_24},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll use the theory of rational interpolation of the previous chapter to prove the harder direction of Pick’s theorem for Herglotz functions, Theorem 16.1.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_pick_2019-1,
  title = {Pick {{Interpolation}}, {{III}}: {{Continued Fraction Proof}}},
  shorttitle = {Pick {{Interpolation}}, {{III}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {183--188},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_18},
  url = {https://doi.org/10.1007/978-3-030-22422-6_18},
  urldate = {2020-11-28},
  abstract = {In this chapter, we will prove the following which provides the missing part (given Chapter 16) of Theorem 16.4 and thereby completes (again) the proofs of Theorems 16.1, 16.3, and 16.4.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_pick_2019-2,
  title = {Pick {{Interpolation}}, {{II}}: {{Hilbert Space Proof}}},
  shorttitle = {Pick {{Interpolation}}, {{II}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {175--181},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_17},
  url = {https://doi.org/10.1007/978-3-030-22422-6_17},
  urldate = {2020-11-28},
  abstract = {Our goal here is to prove the following part of Theorem 16.1 which, by the results of the last chapter, completes the proofs of Theorems 16.1, 16.3, and 16.4.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_pick_2019-3,
  title = {Pick {{Interpolation}}, {{I}}: {{The Basics}}},
  shorttitle = {Pick {{Interpolation}}, {{I}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {165--173},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_16},
  url = {https://doi.org/10.1007/978-3-030-22422-6_16},
  urldate = {2020-11-28},
  abstract = {In 1916, Pick asked and answered the following question: Given indexed collections \{zj\}j ∈ I in \textbackslash (\{\textbackslash mathbb \{D\}\}\textbackslash ) and \{wj\}j ∈ I in \textbackslash (\textbackslash\{w\textbackslash mid \textbackslash operatorname \{\textbackslash mathrm \{Re\}\} w{$>$}0\textbackslash\}\textbackslash ) with the zj distinct, when does there exist an analytic function F on \textbackslash (\{\textbackslash mathbb \{D\}\}\textbackslash ) with \textbackslash ( \textbackslash operatorname \{\textbackslash mathrm \{Re\}\} F {$>$}0\textbackslash ) so that for all j ∈ I.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_pick_2019-4,
  title = {Pick {{Interpolation}}, {{IV}}: {{Commutant Lifting Proof}}},
  shorttitle = {Pick {{Interpolation}}, {{IV}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {189--214},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_19},
  url = {https://doi.org/10.1007/978-3-030-22422-6_19},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll present a proof of Pick’s theorem, Theorem 16.4, due to Sarason [294], that relies on the Hardy spaces H2 and H∞.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_positive_2019,
  title = {Positive {{Functions}} and {{Sparr}}’s {{Proof}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {301--312},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_29},
  url = {https://doi.org/10.1007/978-3-030-22422-6_29},
  urldate = {2020-11-28},
  abstract = {The key to many proofs of Loewner’s theorem is finding a measure.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_proof_2019,
  title = {A {{Proof}} of {{Loewner}}’s {{Theorem}} as a {{Degenerate Limit}} of {{Pick}}’s {{Theorem}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {215--220},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_20},
  url = {https://doi.org/10.1007/978-3-030-22422-6_20},
  urldate = {2020-11-28},
  abstract = {In this chapter, we will give our first proof of the hard part of Loewner’s theorem. Despite the fact that the approach is very natural, this proof seems to be new (although, as I will explain in the Notes, it can be regarded as a kind of variant of Korányi’s proof of Loewner’s theorem and also Loewner’s).},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_quadratic_2019,
  title = {Quadratic {{Interpolation}}: {{The Foiaş}}–{{Lions Theorem}}},
  shorttitle = {Quadratic {{Interpolation}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {153--159},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_15},
  url = {https://doi.org/10.1007/978-3-030-22422-6_15},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll begin by considering a mathematically interesting problem that seems unconnected to the subject of matrix monotone functions.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_rational_2019,
  title = {Rational {{Approximation}} and {{Orthogonal Polynomials}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {221--233},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_21},
  url = {https://doi.org/10.1007/978-3-030-22422-6_21},
  urldate = {2020-11-28},
  abstract = {A key element of many proofs of Loewner’s theorem is the approximation of f by rational Herglotz functions.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_unitarily_2019,
  title = {Unitarily {{Invariant Norm Inequalities}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {409--414},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_43},
  url = {https://doi.org/10.1007/978-3-030-22422-6_43},
  urldate = {2020-11-28},
  abstract = {In this chapter, we’ll prove two beautiful results of Ando.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@incollection{simon_unitarily_2019-1,
  title = {Unitarily {{Invariant Norms}} and {{Rearrangement}}},
  booktitle = {Loewner's {{Theorem}} on {{Monotone Matrix Functions}}},
  author = {Simon, Barry},
  editor = {Simon, Barry},
  date = {2019},
  pages = {399--407},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-030-22422-6_42},
  url = {https://doi.org/10.1007/978-3-030-22422-6_42},
  urldate = {2020-11-28},
  abstract = {In the next chapter, we will discuss some operator norm inequalities for matrix monotone functions and also some functions which are functional inverses of matrix monotone functions.},
  isbn = {978-3-030-22422-6},
  langid = {english},
  series = {Grundlehren Der Mathematischen {{Wissenschaften}}}
}

@book{steele_probability_1997,
  title = {Probability Theory and Combinatorial Optimization},
  author = {Steele, J. Michael},
  date = {1997},
  publisher = {{Society for Industrial and Applied Mathematics SIAM Market Street, Floor 6, Philadelphia, PA 19104}},
  location = {{Philadelphia, Pa.}},
  abstract = {This monograph provides an introduction to the state of the art of the probability theory that is most directly applicable to combinatorial optimization. The questions that receive the most attention are those that deal with discrete optimization problems for points in Euclidean space, such as the minimum spanning tree, the traveling-salesman tour, and minimal-length matchings. Still, there are several nongeometric optimization problems that receive full treatment, and these include the problems of the longest common subsequence and the longest increasing subsequence. The philosophy that guides the exposition is that analysis of concrete problems is the most effective way to explain even the most general methods or abstract principles. There are three fundamental probabilistic themes that are examined through our concrete investigations. First, there is a systematic exploitation of martingales. The second theme that is explored is the systematic use of subadditivity of several flavors, ranging from the naïve subadditivity of real sequences to the subtler subadditivity of stochastic processes. The third and deepest theme developed here concerns the application of Talagrand's isoperimetric theory of concentration inequalities.},
  editora = {{Society for Industrial and Applied Mathematics}},
  editoratype = {collaborator},
  isbn = {978-1-61197-002-9},
  keywords = {Probabilities; Combinatorial optimization; Probability; Combinatorial optimization; Objective function; Stochastic models; Subadditivity; Talagrand's theory},
  langid = {english},
  number = {69},
  pagetotal = {155},
  series = {{{CBMS}}-{{NSF}} Regional Conference Series in Applied Mathematics}
}

@article{sun_strong_2003,
  title = {Strong {{Semismoothness}} of {{Eigenvalues}} of {{Symmetric Matrices}} and {{Its Application}} to {{Inverse Eigenvalue Problems}}},
  author = {Sun, Defeng and Sun, Jie},
  date = {2003},
  journaltitle = {SIAM Journal on Numerical Analysis},
  volume = {40},
  pages = {2352--2367},
  publisher = {{Society for Industrial and Applied Mathematics}},
  abstract = {It is well known that the eigenvalues of a real symmetric matrix are not everywhere differentiable. A classical result of Ky Fan states that each eigenvalue of a symmetric matrix is the difference of two convex functions, which implies that the eigenvalues are semismooth functions. Based on a recent result of the authors, it is further proved in this paper that the eigenvalues of a symmetric matrix are strongly semismooth everywhere. As an application, it is demonstrated how this result can be used to analyze the quadratic convergence of Newton's method for solving inverse eigenvalue problems (IEPs) and generalized IEPs with multiple eigenvalues.},
  eprint = {4100999},
  eprinttype = {jstor},
  number = {6}
}

@thesis{tomamichel_framework_2012,
  ids = {tomamichel_framework_2013},
  title = {A {{Framework}} for {{Non}}-{{Asymptotic Quantum Information Theory}}},
  author = {Tomamichel, Marco},
  date = {2012-03-09},
  institution = {{ETH Zürich}},
  url = {https://arxiv.org/abs/1203.2142v2},
  urldate = {2020-07-29},
  abstract = {This thesis consolidates, improves and extends the smooth entropy framework for non-asymptotic information theory and cryptography. We investigate the conditional min- and max-entropy for quantum states, generalizations of classical R\textbackslash 'enyi entropies. We introduce the purified distance, a novel metric for unnormalized quantum states and use it to define smooth entropies as optimizations of the min- and max-entropies over a ball of close states. We explore various properties of these entropies, including data-processing inequalities, chain rules and their classical limits. The most important property is an entropic formulation of the asymptotic equipartition property, which implies that the smooth entropies converge to the von Neumann entropy in the limit of many independent copies. The smooth entropies also satisfy duality and entropic uncertainty relations that provide limits on the power of two different observers to predict the outcome of a measurement on a quantum system. Finally, we discuss three example applications of the smooth entropy framework. We show a strong converse statement for source coding with quantum side information, characterize randomness extraction against quantum side information and prove information theoretic security of quantum key distribution using an intuitive argument based on the entropic uncertainty relation.},
  archivePrefix = {arXiv},
  eprint = {1203.2142},
  eprinttype = {arxiv},
  keywords = {Mathematical Physics,Quantum Physics},
  langid = {english}
}

@article{tomamichel_fully_2009,
  ids = {tomamichel_fullyquantum_2009},
  title = {A {{Fully Quantum Asymptotic Equipartition Property}}},
  author = {Tomamichel, Marco and Colbeck, Roger and Renner, Renato},
  date = {2009-12},
  journaltitle = {IEEE Transactions on Information Theory},
  shortjournal = {IEEE Trans. Inform. Theory},
  volume = {55},
  pages = {5840--5847},
  doi = {10.1109/TIT.2009.2032797},
  url = {http://arxiv.org/abs/0811.1221},
  urldate = {2021-02-15},
  abstract = {The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, we prove a fully quantum generalization of this property, where both the output of the experiment and side information are quantum. We give an explicit bound on the convergence, which is independent of the dimensionality of the side information. This naturally leads to a family of Renyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.},
  archivePrefix = {arXiv},
  eprint = {0811.1221},
  eprinttype = {arxiv},
  keywords = {Quantum Physics},
  number = {12}
}

@book{tomamichel_quantum_2016,
  title = {Quantum {{Information Processing}} with {{Finite Resources}}},
  author = {Tomamichel, Marco},
  date = {2016},
  volume = {5},
  publisher = {{Springer International Publishing}},
  location = {{Cham}},
  doi = {10.1007/978-3-319-21891-5},
  url = {http://link.springer.com/10.1007/978-3-319-21891-5},
  urldate = {2020-09-14},
  isbn = {978-3-319-21891-5},
  langid = {english},
  series = {{{SpringerBriefs}} in {{Mathematical Physics}}}
}

@article{tomamichel_strong_2017,
  title = {Strong Converse Rates for Quantum Communication},
  author = {Tomamichel, Marco and Wilde, Mark M. and Winter, Andreas},
  date = {2017-01},
  journaltitle = {IEEE Transactions on Information Theory},
  shortjournal = {IEEE Trans. Inform. Theory},
  volume = {63},
  pages = {715--727},
  doi = {10.1109/TIT.2016.2615847},
  url = {http://arxiv.org/abs/1406.2946},
  urldate = {2020-09-14},
  abstract = {We revisit a fundamental open problem in quantum information theory, namely whether it is possible to transmit quantum information at a rate exceeding the channel capacity if we allow for a non-vanishing probability of decoding error. Here we establish that the Rains information of any quantum channel is a strong converse rate for quantum communication: For any sequence of codes with rate exceeding the Rains information of the channel, we show that the fidelity vanishes exponentially fast as the number of channel uses increases. This remains true even if we consider codes that perform classical post-processing on the transmitted quantum data. As an application of this result, for generalized dephasing channels we show that the Rains information is also achievable, and thereby establish the strong converse property for quantum communication over such channels. Thus we conclusively settle the strong converse question for a class of quantum channels that have a non-trivial quantum capacity.},
  archivePrefix = {arXiv},
  eprint = {1406.2946},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Quantum Physics},
  number = {1}
}

@online{wang_revisiting_2020,
  title = {Revisiting the Equality Conditions of the Data Processing Inequality for the Sandwiched {{R}}\textbackslash 'enyi Divergence},
  author = {Wang, Jinzhao and Wilming, Henrik},
  date = {2020-10-01},
  url = {http://arxiv.org/abs/2009.14197},
  urldate = {2020-10-13},
  abstract = {We provide a transparent, simple and unified treatment of recent results on the equality conditions for the data processing inequality (DPI) of the sandwiched quantum R\textbackslash 'enyi divergence, including the statement that equality in the data processing implies recoverability via the Petz recovery map for the full range of \$\textbackslash alpha\$ recently proven by Jen\textbackslash v cov\textbackslash 'a. We also obtain a new set of equality conditions, generalizing a previous result by Leditzky et al.},
  archivePrefix = {arXiv},
  eprint = {2009.14197},
  eprinttype = {arxiv},
  keywords = {Mathematical Physics,Quantum Physics},
  primaryClass = {math-ph, physics:quant-ph},
  version = {2}
}

@article{wilde_amortized_2020,
  ids = {wilde_amortized_2020-1},
  title = {Amortized {{Channel Divergence}} for {{Asymptotic Quantum Channel Discrimination}}},
  author = {Wilde, Mark M. and Berta, Mario and Hirche, Christoph and Kaur, Eneet},
  date = {2020-08},
  journaltitle = {Letters in Mathematical Physics},
  shortjournal = {Lett Math Phys},
  volume = {110},
  pages = {2277--2336},
  doi = {10.1007/s11005-020-01297-7},
  url = {http://arxiv.org/abs/1808.01498},
  urldate = {2020-09-14},
  abstract = {It is well known that for the discrimination of classical and quantum channels in the finite, non-asymptotic regime, adaptive strategies can give an advantage over non-adaptive strategies. However, Hayashi [IEEE Trans. Inf. Theory 55(8), 3807 (2009)] showed that in the asymptotic regime, the exponential error rate for the discrimination of classical channels is not improved in the adaptive setting. We extend this result in several ways. First, we establish the strong Stein's lemma for classical-quantum channels by showing that asymptotically the exponential error rate for classical-quantum channel discrimination is not improved by adaptive strategies. Second, we recover many other classes of channels for which adaptive strategies do not lead to an asymptotic advantage. Third, we give various converse bounds on the power of adaptive protocols for general asymptotic quantum channel discrimination. Intriguingly, it remains open whether adaptive protocols can improve the exponential error rate for quantum channel discrimination in the asymmetric Stein setting. Our proofs are based on the concept of amortized distinguishability of quantum channels, which we analyse using data-processing inequalities.},
  archivePrefix = {arXiv},
  eprint = {1808.01498},
  eprinttype = {arxiv},
  keywords = {Computer Science - Information Theory,Quantum Physics},
  number = {8}
}

@book{wilde_quantum_2013,
  title = {Quantum {{Information Theory}}},
  author = {Wilde, Mark M.},
  date = {2013},
  publisher = {{Cambridge University Press}},
  location = {{Cambridge}},
  doi = {10.1017/CBO9781139525343},
  url = {https://www.cambridge.org/core/books/quantum-information-theory/9DC2CA59F45636D4F0F30D971B677623},
  urldate = {2020-09-14},
  abstract = {Finally, here is a modern, self-contained text on quantum information theory suitable for graduate-level courses. Developing the subject 'from the ground up' it covers classical results as well as major advances of the past decade. Beginning with an extensive overview of classical information theory suitable for the non-expert, the author then turns his attention to quantum mechanics for quantum information theory, and the important protocols of teleportation, super-dense coding and entanglement distribution. He develops all of the tools necessary for understanding important results in quantum information theory, including capacity theorems for classical, entanglement-assisted, private and quantum communication. The book also covers important recent developments such as superadditivity of private, coherent and Holevo information, and the superactivation of quantum capacity. This book will be warmly welcomed by the upcoming generation of quantum information theorists and the already established community of classical information theorists.}
}

@article{wilde_strong_2014,
  title = {Strong {{Converse}} for the {{Classical Capacity}} of {{Entanglement}}-{{Breaking}} and {{Hadamard Channels}} via a {{Sandwiched Rényi Relative Entropy}}},
  author = {Wilde, Mark M. and Winter, Andreas and Yang, Dong},
  date = {2014-10-01},
  journaltitle = {Communications in Mathematical Physics},
  shortjournal = {Commun. Math. Phys.},
  volume = {331},
  pages = {593--622},
  doi = {10.1007/s00220-014-2122-x},
  url = {https://doi.org/10.1007/s00220-014-2122-x},
  urldate = {2021-02-11},
  abstract = {A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a “sandwiched” Rényi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.},
  langid = {english},
  number = {2}
}



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\refsection{0}
  \datalist[entry]{none/global//global/global}
    \entry{fawzi_defining_2020}{online}{}
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        {{hash=55f394b61ad2322e8304d2c41d5e38aa}{%
           family={Fawzi},
           familyi={F\bibinitperiod},
           given={Omar},
           giveni={O\bibinitperiod}}}%
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      \field{abstract}{We introduce a new quantum R\textbackslash 'enyi divergence \$D\^\{\textbackslash\#\}\_\{\textbackslash alpha\}\$ for \$\textbackslash alpha \textbackslash in (1,\textbackslash infty)\$ defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum R\textbackslash 'enyi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched \$\textbackslash alpha\$-R\textbackslash 'enyi divergence between quantum channels for \$\textbackslash alpha {$>$} 1\$. Second it allows us to prove a chain rule property for the sandwiched \$\textbackslash alpha\$-R\textbackslash 'enyi divergence for \$\textbackslash alpha {$>$} 1\$ which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.}
      \field{day}{24}
      \field{eprintclass}{quant-ph}
      \field{eprinttype}{arxiv}
      \field{month}{7}
      \field{title}{Defining Quantum Divergences via Convex Optimization}
      \field{urlday}{19}
      \field{urlmonth}{11}
      \field{urlyear}{2020}
      \field{version}{1}
      \field{year}{2020}
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      \keyw{Mathematics - Optimization and Control,Quantum Physics}
    \endentry
    \entry{kubo_means_1979}{article}{}
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      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{journaltitle}{Mathematische Annalen}
      \field{langid}{und}
      \field{title}{Means of Positive Linear Operators.}
      \field{urlday}{3}
      \field{urlmonth}{12}
      \field{urlyear}{2020}
      \field{volume}{246}
      \field{year}{1979}
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    \entry{matsumoto_new_2013}{online}{}
      \name{author}{1}{}{%
        {{hash=aa79f7941de15bbd2db577b141141feb}{%
           family={Matsumoto},
           familyi={M\bibinitperiod},
           given={Keiji},
           giveni={K\bibinitperiod}}}%
      }
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      \field{abstract}{This paper proposes and studies new quantum version of \$f\$-divergences, a class of convex functionals of a pair of probability distributions including Kullback-Leibler divergence, Rnyi-type relative entropy and so on. There are several quantum versions so far, including the one by Petz. We introduce another quantum version (\$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$, below), defined as the solution to an optimization problem, or the minimum classical \$f\$- divergence necessary to generate a given pair of quantum states. It turns out to be the largest quantum \$f\$-divergence. The closed formula of \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$ is given either if \$f\$ is operator convex, or if one of the state is a pure state. Also, concise representation of \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$ as a pointwise supremum of linear functionals is given and used for the clarification of various properties of the quality. Using the closed formula of \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\$, we show: Suppose \$f\$ is operator convex. Then the\textbackslash{} maximum \$f\textbackslash,\$- divergence of the probability distributions of a measurement under the state \$ρ\$ and \$σ\$ is strictly less than \$\textbackslash mathrm\{D\}\_\{f\}\^\{\textbackslash max\}\textbackslash left( ρ\textbackslash Vertσ\textbackslash right) \$. This statement may seem intuitively trivial, but when \$f\$ is not operator convex, this is not always true. A counter example is \$f\textbackslash left( λ\textbackslash right) =\textbackslash left\textbackslash vert 1-λ\textbackslash right\textbackslash vert \$, which corresponds to total variation distance. We mostly work on finite dimensional Hilbert space, but some results are extended to infinite dimensional case.}
      \field{day}{19}
      \field{eprintclass}{quant-ph}
      \field{eprinttype}{arxiv}
      \field{langid}{english}
      \field{month}{11}
      \field{title}{A New Quantum Version of F-Divergence}
      \field{urlday}{24}
      \field{urlmonth}{1}
      \field{urlyear}{2021}
      \field{year}{2013}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \verb{eprint}
      \verb 1311.4722
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      \verb{url}
      \verb https://arxiv.org/abs/1311.4722v4
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    \endentry
    \entry{hiai_different_2017}{article}{}
      \name{author}{2}{}{%
        {{hash=39238e527184d9060eb43bbeda4f173e}{%
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        {{hash=a4f461b8552b99e7cc906d07be32437d}{%
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           familyi={M\bibinitperiod},
           given={Milan},
           giveni={M\bibinitperiod}}}%
      }
      \strng{namehash}{67f18797c08b9e6a8a3ab43c60c221e7}
      \strng{fullhash}{67f18797c08b9e6a8a3ab43c60c221e7}
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      \field{abstract}{The concept of classical \$f\$-divergences gives a unified framework to construct and study measures of dissimilarity of probability distributions; special cases include the relative entropy and the R\textbackslash 'enyi divergences. Various quantum versions of this concept, and more narrowly, the concept of R\textbackslash 'enyi divergences, have been introduced in the literature with applications in quantum information theory; most notably Petz' quasi-entropies (standard \$f\$-divergences), Matsumoto's maximal \$f\$-divergences, measured \$f\$-divergences, and sandwiched and \$\textbackslash alpha\$-\$z\$-R\textbackslash 'enyi divergences. In this paper we give a systematic overview of the various concepts of quantum \$f\$-divergences with a main focus on their monotonicity under quantum operations, and the implications of the preservation of a quantum \$f\$-divergence by a quantum operation. In particular, we compare the standard and the maximal \$f\$-divergences regarding their ability to detect the reversibility of quantum operations. We also show that these two quantum \$f\$-divergences are strictly different for non-commuting operators unless \$f\$ is a polynomial, and obtain some analogous partial results for the relation between the measured and the standard \$f\$-divergences. We also study the monotonicity of the \$\textbackslash alpha\$-\$z\$-R\textbackslash 'enyi divergences under the special class of bistochastic maps that leave one of the arguments of the R\textbackslash 'enyi divergence invariant, and determine domains of the parameters \$\textbackslash alpha,z\$ where monotonicity holds, and where the preservation of the \$\textbackslash alpha\$-\$z\$-R\textbackslash 'enyi divergence implies the reversibility of the quantum operation.}
      \field{eprinttype}{arxiv}
      \field{journaltitle}{Reviews in Mathematical Physics}
      \field{month}{8}
      \field{number}{07}
      \field{shortjournal}{Rev. Math. Phys.}
      \field{title}{Different Quantum F-Divergences and the Reversibility of Quantum Operations}
      \field{urlday}{15}
      \field{urlmonth}{10}
      \field{urlyear}{2020}
      \field{volume}{29}
      \field{year}{2017}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{1750023}
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      \verb{url}
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      \keyw{Computer Science - Information Theory,Mathematical Physics,Quantum Physics}
    \endentry
    \entry{katariya_geometric_2020}{online}{}
      \name{author}{2}{}{%
        {{hash=fea0a58d7883750fb6453c00c6ce5037}{%
           family={Katariya},
           familyi={K\bibinitperiod},
           given={Vishal},
           giveni={V\bibinitperiod}}}%
        {{hash=b2c684c07442be97b24571e88e8147e3}{%
           family={Wilde},
           familyi={W\bibinitperiod},
           given={Mark\bibnamedelima M.},
           giveni={M\bibinitperiod\bibinitdelim M\bibinitperiod}}}%
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      \strng{bibnamehash}{73989245e1bbc1fb634ea794d470bbd4}
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      \field{abstract}{Quantum channel estimation and discrimination are fundamentally related information processing tasks of interest in quantum information science. In this paper, we analyze these tasks by employing the right logarithmic derivative Fisher information and the geometric R\textbackslash 'enyi relative entropy, respectively, and we also identify connections between these distinguishability measures. A key result of our paper is that a chain-rule property holds for the right logarithmic derivative Fisher information and the geometric R\textbackslash 'enyi relative entropy for the interval \$\textbackslash alpha\textbackslash in(0,1) \$ of the R\textbackslash 'enyi parameter \$\textbackslash alpha\$. In channel estimation, these results imply a condition for the unattainability of Heisenberg scaling, while in channel discrimination, they lead to improved bounds on error rates in the Chernoff and Hoeffding error exponent settings. More generally, we introduce the amortized quantum Fisher information as a conceptual framework for analyzing general sequential protocols that estimate a parameter encoded in a quantum channel, and we use this framework, beyond the aforementioned application, to show that Heisenberg scaling is not possible when a parameter is encoded in a classical-quantum channel. We then identify a number of other conceptual and technical connections between the tasks of estimation and discrimination and the distinguishability measures involved in analyzing each. As part of this work, we present a detailed overview of the geometric R\textbackslash 'enyi relative entropy of quantum states and channels, as well as its properties, which may be of independent interest.}
      \field{day}{22}
      \field{eprintclass}{math-ph, physics:quant-ph, stat}
      \field{eprinttype}{arxiv}
      \field{month}{4}
      \field{title}{Geometric Distinguishability Measures Limit Quantum Channel Estimation and Discrimination}
      \field{urlday}{15}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{year}{2020}
      \field{dateera}{ce}
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      \keyw{Computer Science - Information Theory,Mathematical Physics,Mathematics - Statistics Theory,Quantum Physics}
    \endentry
    \entry{petz_contraction_1998}{article}{}
      \name{author}{2}{}{%
        {{hash=f3207d1ab0a68b2245be8e7af1f4a561}{%
           family={Petz},
           familyi={P\bibinitperiod},
           given={Dénes},
           giveni={D\bibinitperiod}}}%
        {{hash=93acf59bab7fd7cc10f95e1c6ec630dc}{%
           family={Ruskai},
           familyi={R\bibinitperiod},
           given={Mary\bibnamedelima Beth},
           giveni={M\bibinitperiod\bibinitdelim B\bibinitperiod}}}%
      }
      \list{publisher}{1}{%
        {World Scientific Publishing Co.}%
      }
      \strng{namehash}{73281a1d9024fb80ba1121a297f6a53e}
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      \field{sortinithash}{5dd416adbafacc8226114bc0202d5fdd}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{abstract}{The contraction coefficient of stochastic mappings between full matrix algebras is introduced with respect to a generalized relative entropy labeled by an operator convex function g. It is proved that the coefficient is actually independent of g, in particular, it can be most conveniently computed by means of the square function.}
      \field{day}{1}
      \field{journaltitle}{Infinite Dimensional Analysis, Quantum Probability and Related Topics}
      \field{month}{1}
      \field{number}{01}
      \field{shortjournal}{Infin. Dimens. Anal. Quantum. Probab. Relat. Top.}
      \field{title}{Contraction of {{Generalized Relative Entropy Under Stochastic Mappings}} on {{Matrices}}}
      \field{urlday}{15}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{volume}{01}
      \field{year}{1998}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{83\bibrangedash 89}
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      \verb{url}
      \verb https://www.worldscientific.com/doi/abs/10.1142/S0219025798000077
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    \endentry
    \entry{fang_geometric_2019}{online}{}
      \name{author}{2}{}{%
        {{hash=f41256f59f78a2ec05bd0a13b0cdac9b}{%
           family={Fang},
           familyi={F\bibinitperiod},
           given={Kun},
           giveni={K\bibinitperiod}}}%
        {{hash=51aeaf4d123815795c788a59a36b494f}{%
           family={Fawzi},
           familyi={F\bibinitperiod},
           given={Hamza},
           giveni={H\bibinitperiod}}}%
      }
      \strng{namehash}{c725f8c60c62cb58c466e89389fa7e1c}
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      \strng{bibnamehash}{c725f8c60c62cb58c466e89389fa7e1c}
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      \field{abstract}{We present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties. For example we prove a chain rule inequality that immediately implies the "amortization collapse" for the geometric Rényi divergence, addressing an open question by Berta et al. [arXiv:1808.01498, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously best-known bounds based on the max-relative entropy while still keeping the new bounds single-letter efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.}
      \field{day}{12}
      \field{eprintclass}{math-ph, physics:quant-ph}
      \field{eprinttype}{arxiv}
      \field{month}{9}
      \field{title}{Geometric {{Rényi Divergence}} and Its {{Applications}} in {{Quantum Channel Capacities}}}
      \field{urlday}{14}
      \field{urlmonth}{9}
      \field{urlyear}{2020}
      \field{year}{2019}
      \field{dateera}{ce}
      \field{urldateera}{ce}
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      \keyw{Computer Science - Information Theory,Mathematical Physics,Quantum Physics}
    \endentry
    \entry{tomamichel_quantum_2016}{book}{}
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           family={Tomamichel},
           familyi={T\bibinitperiod},
           given={Marco},
           giveni={M\bibinitperiod}}}%
      }
      \list{location}{1}{%
        {Cham}%
      }
      \list{publisher}{1}{%
        {Springer International Publishing}%
      }
      \strng{namehash}{eee2caa0ee47fcf094b78e3bf7b78dd5}
      \strng{fullhash}{eee2caa0ee47fcf094b78e3bf7b78dd5}
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      \field{labeltitlesource}{title}
      \field{isbn}{978-3-319-21891-5}
      \field{langid}{english}
      \field{series}{{{SpringerBriefs}} in {{Mathematical Physics}}}
      \field{title}{Quantum {{Information Processing}} with {{Finite Resources}}}
      \field{urlday}{14}
      \field{urlmonth}{9}
      \field{urlyear}{2020}
      \field{volume}{5}
      \field{year}{2016}
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      \verb{url}
      \verb http://link.springer.com/10.1007/978-3-319-21891-5
      \endverb
    \endentry
    \entry{belavkin_c-algebraic_1982}{article}{}
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           given={V.\bibnamedelimi P.},
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        {{hash=f9bbb57216f97780ce2de5f9bf967aaa}{%
           family={Staszewski},
           familyi={S\bibinitperiod},
           given={P.},
           giveni={P\bibinitperiod}}}%
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      \strng{namehash}{6fb47032b3ba16bcec9c3d813b1a6438}
      \strng{fullhash}{6fb47032b3ba16bcec9c3d813b1a6438}
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      \field{journaltitle}{Annales de l'I.H.P. Physique théorique}
      \field{langid}{french}
      \field{number}{1}
      \field{title}{C\textsuperscript{*}-algebraic generalization of relative entropy and entropy}
      \field{urlday}{24}
      \field{urlmonth}{1}
      \field{urlyear}{2021}
      \field{volume}{37}
      \field{year}{1982}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{51\bibrangedash 58}
      \range{pages}{8}
      \verb{urlraw}
      \verb http://www.numdam.org/item/AIHPA_1982__37_1_51_0/
      \endverb
      \verb{url}
      \verb http://www.numdam.org/item/AIHPA_1982__37_1_51_0/
      \endverb
    \endentry
    \entry{demyanov_theory_1971}{article}{}
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        {{hash=5a37764119c8b7e32f06e81c13c3c212}{%
           family={Dem'yanov},
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           given={V.\bibnamedelimi F.},
           giveni={V\bibinitperiod\bibinitdelim F\bibinitperiod}}}%
        {{hash=bc4d1a748f13a831b50f6456f7a76863}{%
           family={Malozemov},
           familyi={M\bibinitperiod},
           given={V.\bibnamedelimi N.},
           giveni={V\bibinitperiod\bibinitdelim N\bibinitperiod}}}%
      }
      \list{publisher}{1}{%
        {IOP Publishing}%
      }
      \strng{namehash}{4e60ea1e21774b0b6c6bcb06d43d7f89}
      \strng{fullhash}{4e60ea1e21774b0b6c6bcb06d43d7f89}
      \strng{bibnamehash}{4e60ea1e21774b0b6c6bcb06d43d7f89}
      \strng{authorbibnamehash}{4e60ea1e21774b0b6c6bcb06d43d7f89}
      \strng{authornamehash}{4e60ea1e21774b0b6c6bcb06d43d7f89}
      \strng{authorfullhash}{4e60ea1e21774b0b6c6bcb06d43d7f89}
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      \field{sortinithash}{50c6687d7fc80f50136d75228e3c59ba}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{day}{30}
      \field{journaltitle}{Russian Mathematical Surveys}
      \field{langid}{english}
      \field{month}{6}
      \field{number}{3}
      \field{shortjournal}{Russ. Math. Surv.}
      \field{title}{On the {{Theory}} of {{Non}}-{{Linear Minimax Problems}}}
      \field{urlday}{23}
      \field{urlmonth}{1}
      \field{urlyear}{2021}
      \field{volume}{26}
      \field{year}{1971}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{57}
      \range{pages}{1}
      \verb{doi}
      \verb 10.1070/RM1971v026n03ABEH003834
      \endverb
      \verb{urlraw}
      \verb https://iopscience.iop.org/article/10.1070/RM1971v026n03ABEH003834/meta
      \endverb
      \verb{url}
      \verb https://iopscience.iop.org/article/10.1070/RM1971v026n03ABEH003834/meta
      \endverb
    \endentry
    \entry{horn_topics_1991}{book}{}
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        {{hash=472bb8160d835c69b57bcf72a2e976f2}{%
           family={Horn},
           familyi={H\bibinitperiod},
           given={Roger\bibnamedelima A.},
           giveni={R\bibinitperiod\bibinitdelim A\bibinitperiod}}}%
        {{hash=dac60894a8d7e4e7a13519d50f5fbf9a}{%
           family={Johnson},
           familyi={J\bibinitperiod},
           given={Charles\bibnamedelima R.},
           giveni={C\bibinitperiod\bibinitdelim R\bibinitperiod}}}%
      }
      \list{location}{1}{%
        {Cambridge}%
      }
      \list{publisher}{1}{%
        {Cambridge University Press}%
      }
      \strng{namehash}{eecf59942273a63031b3fcbcfd5fa22f}
      \strng{fullhash}{eecf59942273a63031b3fcbcfd5fa22f}
      \strng{bibnamehash}{eecf59942273a63031b3fcbcfd5fa22f}
      \strng{authorbibnamehash}{eecf59942273a63031b3fcbcfd5fa22f}
      \strng{authornamehash}{eecf59942273a63031b3fcbcfd5fa22f}
      \strng{authorfullhash}{eecf59942273a63031b3fcbcfd5fa22f}
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      \field{sortinithash}{50c6687d7fc80f50136d75228e3c59ba}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{abstract}{Building on the foundations of its predecessor volume, Matrix Analysis, this book treats in detail several topics in matrix theory not included in the previous volume, but with important applications and of special mathematical interest. As with the previous volume, the authors assume a background knowledge of elementary linear algebra and rudimentary analytical concepts. Many examples and exercises of varying difficulty are included.}
      \field{isbn}{978-0-521-46713-1}
      \field{title}{Topics in {{Matrix Analysis}}}
      \field{urlday}{10}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{year}{1991}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \verb{doi}
      \verb 10.1017/CBO9780511840371
      \endverb
      \verb{urlraw}
      \verb https://www.cambridge.org/core/books/topics-in-matrix-analysis/B988495A235F1C3406EA484A2C477B03
      \endverb
      \verb{url}
      \verb https://www.cambridge.org/core/books/topics-in-matrix-analysis/B988495A235F1C3406EA484A2C477B03
      \endverb
    \endentry
    \entry{khatri_principles_2020}{online}{}
      \name{author}{2}{}{%
        {{hash=f6ffff405a57ccb12a243e1dcdf7ce53}{%
           family={Khatri},
           familyi={K\bibinitperiod},
           given={Sumeet},
           giveni={S\bibinitperiod}}}%
        {{hash=b2c684c07442be97b24571e88e8147e3}{%
           family={Wilde},
           familyi={W\bibinitperiod},
           given={Mark\bibnamedelima M.},
           giveni={M\bibinitperiod\bibinitdelim M\bibinitperiod}}}%
      }
      \strng{namehash}{f9e238081eb016b52e488a457e07e08a}
      \strng{fullhash}{f9e238081eb016b52e488a457e07e08a}
      \strng{bibnamehash}{f9e238081eb016b52e488a457e07e08a}
      \strng{authorbibnamehash}{f9e238081eb016b52e488a457e07e08a}
      \strng{authornamehash}{f9e238081eb016b52e488a457e07e08a}
      \strng{authorfullhash}{f9e238081eb016b52e488a457e07e08a}
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      \field{sortinithash}{50c6687d7fc80f50136d75228e3c59ba}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{shorttitle}
      \field{abstract}{This is a preliminary version of a book in progress on the theory of quantum communication. We adopt an information-theoretic perspective throughout and give a comprehensive account of fundamental results in quantum communication theory from the past decade (and earlier), with an emphasis on the modern one-shot-to-asymptotic approach that underlies much of today's state-of-the-art research in this field. In Part I, we cover mathematical preliminaries and provide a detailed study of quantum mechanics from an information-theoretic perspective. We also provide an extensive and thorough review of the quantum entropy zoo, and we devote an entire chapter to the study of entanglement measures. Equipped with these essential tools, in Part II we study classical communication (with and without entanglement assistance), entanglement distillation, quantum communication, secret key distillation, and private communication. In Part III, we cover the latest developments in feedback-assisted communication tasks, such as quantum and classical feedback-assisted communication, LOCC-assisted quantum communication, and secret key agreement.}
      \field{day}{9}
      \field{eprintclass}{cond-mat, physics:hep-th, physics:math-ph, physics:quant-ph}
      \field{eprinttype}{arxiv}
      \field{month}{11}
      \field{shorttitle}{Principles of {{Quantum Communication Theory}}}
      \field{title}{Principles of {{Quantum Communication Theory}}: {{A Modern Approach}}}
      \field{urlday}{11}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{year}{2020}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \verb{eprint}
      \verb 2011.04672
      \endverb
      \verb{urlraw}
      \verb http://arxiv.org/abs/2011.04672
      \endverb
      \verb{url}
      \verb http://arxiv.org/abs/2011.04672
      \endverb
      \keyw{Computer Science - Information Theory,Condensed Matter - Statistical Mechanics,High Energy Physics - Theory,Mathematical Physics,Quantum Physics}
    \endentry
    \entry{petz_quasi-entropies_1986}{article}{}
      \name{author}{1}{}{%
        {{hash=f3207d1ab0a68b2245be8e7af1f4a561}{%
           family={Petz},
           familyi={P\bibinitperiod},
           given={Dénes},
           giveni={D\bibinitperiod}}}%
      }
      \strng{namehash}{f3207d1ab0a68b2245be8e7af1f4a561}
      \strng{fullhash}{f3207d1ab0a68b2245be8e7af1f4a561}
      \strng{bibnamehash}{f3207d1ab0a68b2245be8e7af1f4a561}
      \strng{authorbibnamehash}{f3207d1ab0a68b2245be8e7af1f4a561}
      \strng{authornamehash}{f3207d1ab0a68b2245be8e7af1f4a561}
      \strng{authorfullhash}{f3207d1ab0a68b2245be8e7af1f4a561}
      \field{sortinit}{1}
      \field{sortinithash}{50c6687d7fc80f50136d75228e3c59ba}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{abstract}{Convexity properties of entropy-like functionals on states of a finite dimensional algebra are discussed. The treatment covers both the quantum mechanical and the classical cases. The purpose is to generalize Lieb's convexity theorem and the monotonicity of the relative entropy using the Jensen inequality of operator convex functions. From the quasi-entropies defined here the quantum version of Rényi's α- entropies can be deduced.}
      \field{day}{1}
      \field{journaltitle}{Reports on Mathematical Physics}
      \field{langid}{english}
      \field{month}{2}
      \field{number}{1}
      \field{shortjournal}{Reports on Mathematical Physics}
      \field{title}{Quasi-Entropies for Finite Quantum Systems}
      \field{urlday}{11}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{volume}{23}
      \field{year}{1986}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{57\bibrangedash 65}
      \range{pages}{9}
      \verb{doi}
      \verb 10.1016/0034-4877(86)90067-4
      \endverb
      \verb{urlraw}
      \verb https://www.sciencedirect.com/science/article/pii/0034487786900674
      \endverb
      \verb{url}
      \verb https://www.sciencedirect.com/science/article/pii/0034487786900674
      \endverb
    \endentry
    \entry{muller-lennert_quantum_2013}{article}{}
      \name{author}{5}{}{%
        {{hash=01d5886fbf8186a990680d9d51257fdb}{%
           family={Müller-Lennert},
           familyi={M\bibinithyphendelim L\bibinitperiod},
           given={Martin},
           giveni={M\bibinitperiod}}}%
        {{hash=1f44b53a0edef91ac301834ba7d25e8e}{%
           family={Dupuis},
           familyi={D\bibinitperiod},
           given={Frédéric},
           giveni={F\bibinitperiod}}}%
        {{hash=556c613cc63e7ce63ef019e6575969a5}{%
           family={Szehr},
           familyi={S\bibinitperiod},
           given={Oleg},
           giveni={O\bibinitperiod}}}%
        {{hash=a5cf36db5a8bff12419022f6f31208ed}{%
           family={Fehr},
           familyi={F\bibinitperiod},
           given={Serge},
           giveni={S\bibinitperiod}}}%
        {{hash=eee2caa0ee47fcf094b78e3bf7b78dd5}{%
           family={Tomamichel},
           familyi={T\bibinitperiod},
           given={Marco},
           giveni={M\bibinitperiod}}}%
      }
      \list{publisher}{1}{%
        {American Institute of Physics}%
      }
      \strng{namehash}{6f2459188a929bfe757e428103a3b989}
      \strng{fullhash}{b340df391c6cedaae8e3f28a36ea9267}
      \strng{bibnamehash}{6f2459188a929bfe757e428103a3b989}
      \strng{authorbibnamehash}{6f2459188a929bfe757e428103a3b989}
      \strng{authornamehash}{6f2459188a929bfe757e428103a3b989}
      \strng{authorfullhash}{b340df391c6cedaae8e3f28a36ea9267}
      \field{sortinit}{1}
      \field{sortinithash}{50c6687d7fc80f50136d75228e3c59ba}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{shorttitle}
      \field{abstract}{The Rényi entropies constitute a family of information measures that generalizes the well-known Shannon entropy, inheriting many of its properties. They appear in the form of unconditional and conditional entropies, relative entropies, or mutual information, and have found many applications in information theory and beyond. Various generalizations of Rényi entropies to the quantum setting have been proposed, most prominently Petz's quasi-entropies and Renner's conditional min-, max-, and collision entropy. However, these quantum extensions are incompatible and thus unsatisfactory. We propose a new quantum generalization of the family of Rényi entropies that contains the von Neumann entropy, min-entropy, collision entropy, and the max-entropy as special cases, thus encompassing most quantum entropies in use today. We show several natural properties for this definition, including data-processing inequalities, a duality relation, and an entropic uncertainty relation.}
      \field{day}{1}
      \field{journaltitle}{Journal of Mathematical Physics}
      \field{month}{12}
      \field{number}{12}
      \field{shortjournal}{Journal of Mathematical Physics}
      \field{shorttitle}{On Quantum {{Rényi}} Entropies}
      \field{title}{On Quantum {{Rényi}} Entropies: {{A}} New Generalization and Some Properties}
      \field{urlday}{11}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{volume}{54}
      \field{year}{2013}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{122203}
      \range{pages}{1}
      \verb{doi}
      \verb 10.1063/1.4838856
      \endverb
      \verb{urlraw}
      \verb https://aip.scitation.org/doi/full/10.1063/1.4838856
      \endverb
      \verb{url}
      \verb https://aip.scitation.org/doi/full/10.1063/1.4838856
      \endverb
    \endentry
    \entry{wilde_strong_2014}{article}{}
      \name{author}{3}{}{%
        {{hash=b2c684c07442be97b24571e88e8147e3}{%
           family={Wilde},
           familyi={W\bibinitperiod},
           given={Mark\bibnamedelima M.},
           giveni={M\bibinitperiod\bibinitdelim M\bibinitperiod}}}%
        {{hash=5c89b6062e88d9d71cbcf213a288a5b4}{%
           family={Winter},
           familyi={W\bibinitperiod},
           given={Andreas},
           giveni={A\bibinitperiod}}}%
        {{hash=43bfecf0db835905ffe98dadaa0b057f}{%
           family={Yang},
           familyi={Y\bibinitperiod},
           given={Dong},
           giveni={D\bibinitperiod}}}%
      }
      \strng{namehash}{06a0a732ff2c19280991790072ecd4d6}
      \strng{fullhash}{06a0a732ff2c19280991790072ecd4d6}
      \strng{bibnamehash}{06a0a732ff2c19280991790072ecd4d6}
      \strng{authorbibnamehash}{06a0a732ff2c19280991790072ecd4d6}
      \strng{authornamehash}{06a0a732ff2c19280991790072ecd4d6}
      \strng{authorfullhash}{06a0a732ff2c19280991790072ecd4d6}
      \field{sortinit}{1}
      \field{sortinithash}{50c6687d7fc80f50136d75228e3c59ba}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{abstract}{A strong converse theorem for the classical capacity of a quantum channel states that the probability of correctly decoding a classical message converges exponentially fast to zero in the limit of many channel uses if the rate of communication exceeds the classical capacity of the channel. Along with a corresponding achievability statement for rates below the capacity, such a strong converse theorem enhances our understanding of the capacity as a very sharp dividing line between achievable and unachievable rates of communication. Here, we show that such a strong converse theorem holds for the classical capacity of all entanglement-breaking channels and all Hadamard channels (the complementary channels of the former). These results follow by bounding the success probability in terms of a “sandwiched” Rényi relative entropy, by showing that this quantity is subadditive for all entanglement-breaking and Hadamard channels, and by relating this quantity to the Holevo capacity. Prior results regarding strong converse theorems for particular covariant channels emerge as a special case of our results.}
      \field{day}{1}
      \field{journaltitle}{Communications in Mathematical Physics}
      \field{langid}{english}
      \field{month}{10}
      \field{number}{2}
      \field{shortjournal}{Commun. Math. Phys.}
      \field{title}{Strong {{Converse}} for the {{Classical Capacity}} of {{Entanglement}}-{{Breaking}} and {{Hadamard Channels}} via a {{Sandwiched Rényi Relative Entropy}}}
      \field{urlday}{11}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{volume}{331}
      \field{year}{2014}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{593\bibrangedash 622}
      \range{pages}{30}
      \verb{doi}
      \verb 10.1007/s00220-014-2122-x
      \endverb
      \verb{urlraw}
      \verb https://doi.org/10.1007/s00220-014-2122-x
      \endverb
      \verb{url}
      \verb https://doi.org/10.1007/s00220-014-2122-x
      \endverb
    \endentry
    \entry{tomamichel_fully_2009}{article}{}
      \name{author}{3}{}{%
        {{hash=eee2caa0ee47fcf094b78e3bf7b78dd5}{%
           family={Tomamichel},
           familyi={T\bibinitperiod},
           given={Marco},
           giveni={M\bibinitperiod}}}%
        {{hash=6108c483c140c0ecd5ed21ecccf894f7}{%
           family={Colbeck},
           familyi={C\bibinitperiod},
           given={Roger},
           giveni={R\bibinitperiod}}}%
        {{hash=31ab9a997aaced32f65c72d1be34ac55}{%
           family={Renner},
           familyi={R\bibinitperiod},
           given={Renato},
           giveni={R\bibinitperiod}}}%
      }
      \strng{namehash}{0a8d2ef5455950854ea23ed90bfd8a51}
      \strng{fullhash}{0a8d2ef5455950854ea23ed90bfd8a51}
      \strng{bibnamehash}{0a8d2ef5455950854ea23ed90bfd8a51}
      \strng{authorbibnamehash}{0a8d2ef5455950854ea23ed90bfd8a51}
      \strng{authornamehash}{0a8d2ef5455950854ea23ed90bfd8a51}
      \strng{authorfullhash}{0a8d2ef5455950854ea23ed90bfd8a51}
      \field{sortinit}{2}
      \field{sortinithash}{ed39bb39cf854d5250e95b1c1f94f4ed}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{abstract}{The classical asymptotic equipartition property is the statement that, in the limit of a large number of identical repetitions of a random experiment, the output sequence is virtually certain to come from the typical set, each member of which is almost equally likely. In this paper, we prove a fully quantum generalization of this property, where both the output of the experiment and side information are quantum. We give an explicit bound on the convergence, which is independent of the dimensionality of the side information. This naturally leads to a family of Renyi-like quantum conditional entropies, for which the von Neumann entropy emerges as a special case.}
      \field{eprinttype}{arxiv}
      \field{journaltitle}{IEEE Transactions on Information Theory}
      \field{month}{12}
      \field{number}{12}
      \field{shortjournal}{IEEE Trans. Inform. Theory}
      \field{title}{A {{Fully Quantum Asymptotic Equipartition Property}}}
      \field{urlday}{15}
      \field{urlmonth}{2}
      \field{urlyear}{2021}
      \field{volume}{55}
      \field{year}{2009}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{5840\bibrangedash 5847}
      \range{pages}{8}
      \verb{doi}
      \verb 10.1109/TIT.2009.2032797
      \endverb
      \verb{eprint}
      \verb 0811.1221
      \endverb
      \verb{urlraw}
      \verb http://arxiv.org/abs/0811.1221
      \endverb
      \verb{url}
      \verb http://arxiv.org/abs/0811.1221
      \endverb
      \keyw{Quantum Physics}
    \endentry
    \entry{frank_monotonicity_2013}{article}{}
      \name{author}{2}{}{%
        {{hash=41797099fdaa2d7e4b6fd967dd4fa2af}{%
           family={Frank},
           familyi={F\bibinitperiod},
           given={Rupert\bibnamedelima L.},
           giveni={R\bibinitperiod\bibinitdelim L\bibinitperiod}}}%
        {{hash=298133a6b79dcce8ef5f6cb15b60f119}{%
           family={Lieb},
           familyi={L\bibinitperiod},
           given={Elliott\bibnamedelima H.},
           giveni={E\bibinitperiod\bibinitdelim H\bibinitperiod}}}%
      }
      \strng{namehash}{a7d48216ac30678c0e30b9014bdeedbf}
      \strng{fullhash}{a7d48216ac30678c0e30b9014bdeedbf}
      \strng{bibnamehash}{a7d48216ac30678c0e30b9014bdeedbf}
      \strng{authorbibnamehash}{a7d48216ac30678c0e30b9014bdeedbf}
      \strng{authornamehash}{a7d48216ac30678c0e30b9014bdeedbf}
      \strng{authorfullhash}{a7d48216ac30678c0e30b9014bdeedbf}
      \field{sortinit}{2}
      \field{sortinithash}{ed39bb39cf854d5250e95b1c1f94f4ed}
      \field{labelnamesource}{author}
      \field{labeltitlesource}{title}
      \field{abstract}{We show that a recent definition of relative R\textbackslash 'enyi entropy is monotone under completely positive, trace preserving maps. This proves a recent conjecture of M\textbackslash "uller-Lennert et al.}
      \field{eprinttype}{arxiv}
      \field{journaltitle}{Journal of Mathematical Physics}
      \field{month}{12}
      \field{number}{12}
      \field{shortjournal}{Journal of Mathematical Physics}
      \field{title}{Monotonicity of a Relative {{Renyi}} Entropy}
      \field{urlday}{7}
      \field{urlmonth}{12}
      \field{urlyear}{2020}
      \field{volume}{54}
      \field{year}{2013}
      \field{dateera}{ce}
      \field{urldateera}{ce}
      \field{pages}{122201}
      \range{pages}{1}
      \verb{doi}
      \verb 10.1063/1.4838835
      \endverb
      \verb{eprint}
      \verb 1306.5358
      \endverb
      \verb{urlraw}
      \verb http://arxiv.org/abs/1306.5358
      \endverb
      \verb{url}
      \verb http://arxiv.org/abs/1306.5358
      \endverb
      \keyw{Computer Science - Information Theory,Mathematical Physics,Mathematics - Functional Analysis,Quantum Physics}
    \endentry
  \enddatalist
  \keyalias{fawzi_defining_2020-1}{fawzi_defining_2020}
  \keyalias{tomamichel_fullyquantum_2009}{tomamichel_fully_2009}
\endrefsection
\endinput

