---
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 code with a 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.

    Our result has several applications. Most importantly, 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. This for the first time conclusively settles the strong converse question for a class of quantum channels that have a non-trivial quantum capacity. Moreover, we show that the classical post-processing assisted quantum capacity of the quantum binary erasure channel satisfies the strong converse property.
author:
- 'Marco Tomamichel$^*$,'
- 'Mark M. Wilde$^\dagger$,'
- 'Andreas Winter$^\ddagger$ [^1] [^2][^3]'
bibliography:
- 'Ref.bib'
title: Strong Converse Rates for Quantum Communication
---

Introduction
============

The *quantum capacity* of a quantum channel ${\mathcal{N}}$, denoted $Q(\mathcal{N})$, is defined as the maximum rate (in qubits per channel use) at which it is possible to transmit quantum information over many memoryless uses of the channel, such that a receiver can recover the state with a fidelity that asymptotically converges to one as we increase the number of channel uses. (We will formally introduce capacity in Section \[sec:ent-gen-code\].) The question of determining the quantum capacity was set out by Shor in his seminal paper on quantum error correction [@S95]. Since then, a number of works established a multi-letter upper bound on the quantum channel capacity in terms of the coherent information [@SN96; @BKN98; @BNS98], and the coherent information lower bound on quantum capacity was demonstrated by a sequence of works [@L97; @capacity2002shor; @D05] which are often said to bear increasing standards of rigor.[^4] In more detail, the work in [@L97; @capacity2002shor; @D05] showed that the following *coherent information* of the channel$~\mathcal{N}$ is an *achievable rate for quantum communication*:$$I_{\rm c}\left(  \mathcal{N}\right)  :=  \max_{\phi_{RA}} I(R\rangle B)_{\rho},  \qquad \textrm{where} \qquad \rho_{RB} = \mathcal{N}_{A\rightarrow B}\left(  \phi_{RA}\right)
\label{eq:coh-info}$$ and the optimization is over all pure bipartite states $\phi_{RA}$. Here, the coherent information of a bipartite state $\rho_{RB}$ is defined as $I(R\rangle B)_{\rho}:= H(\rho_B)-H(\rho_{RB})$, with the von Neumann entropies $H(\rho) :=
-$Tr$\left\{  \rho \log\rho \right\}$.[^5] From the above result, we can also conclude that the rate $I_{\rm c}\left(  \mathcal{N}^{\otimes
l}\right)  /l$ is achievable for any positive integer $l$, simply by applying the formula in (\[eq:coh-info\]) to the superchannel $\mathcal{N}^{\otimes l}$ and normalizing. By a limiting argument, we find that the regularized coherent information $\lim_{l\rightarrow\infty}I_{c}\left(  \mathcal{N}^{\otimes l}\right)  /l$ is also achievable, and Refs. [@SN96; @BKN98; @BNS98] established that this regularized coherent information is also an upper bound on quantum capacity. This establishes that $$\begin{aligned}
  Q(\mathcal{N}) = \lim_{l\rightarrow\infty} \frac{I_{\rm c}(  \mathcal{N}^{\otimes l})}{l} . \label{eq:capaity-result}\end{aligned}$$

Clearly, the regularized coherent information is not a tractable characterization of quantum capacity. But the later work of Devetak and Shor proved that the single-letter coherent information formula in (\[eq:coh-info\]) is equal to the quantum capacity for the class of degradable quantum channels [@DS05]. Degradable channels are such that the receiver of the channel can simulate the channel to the environment by applying a degrading map to the channel output.

All of the above works established an understanding of quantum capacity in the following sense:

1.  (Achievability) If the rate of quantum communication is below the quantum capacity, then there exists a scheme for quantum communication such that the fidelity converges to one in the limit of many channel uses.

2.  (Weak Converse) If the rate of quantum communication is above the quantum capacity, then there cannot exist an asymptotically error-free quantum communication scheme.

It is crucial to note at this point that practical schemes for quantum communication are restricted to operate on finite block lengths, and it is thus in principle impossible to achieve exactly error-free communication (for most channels). Hence, the weak converse discussed above does not apply and we are left to wonder whether it is possible to transmit information at a larger rate than the regularized coherent information given in  if a non-vanishing error is permissible. Interestingly, it has been known since the early days of classical information theory that the capacity of a classical channel obeys the *strong converse property* [@Wolfowitz1964; @Arimoto73]: if the rate of communication exceeds the capacity, then the error probability necessarily converges to one in the limit of many channel uses. Furthermore, many works have now confirmed that the strong converse property holds for the classical capacity of several quantum channels [@ON1999; @W99; @KW09; @WWY13; @WW13] and also for the entanglement-assisted classical capacity of all quantum channels [@BDHSW12; @BCR09; @GW13].

Consequently, this work aims to sharpen the interpretation of quantum capacity in the same spirit.

Overview of Results and Outline
-------------------------------

In this paper, we define the *Rains information of a quantum channel* $\mathcal{N}_{A\rightarrow B}$ as follows:[^6]$$R\left(  \mathcal{N}\right)  := \max_{\rho_{RA}} \min_{\tau_{RB}\in{{\rm PPT}'}\left(
R:B\right)  }D\left(  {\mathcal{N}}(\rho_{RA}) \Vert\tau_{RB}\right)  , \label{eq:rains-bound}$$ where we maximize over input states $\rho_{RA}$ with $R$ a reference system, and minimize over a set of states (including all PPT states) first considered by Rains, namely the set $$\begin{aligned}
{{\rm PPT}'}\left(  R:B\right)   &  =\left\{  \tau_{RB}:\tau_{RB}\geq0\wedge\left\Vert T_{B}\left(  \tau_{RB}\right)  \right\Vert _{1}\leq1\right\} .$$ Here, $D(\rho\|\tau) := \operatorname{tr}\{ \rho (\log \rho - \log \tau) \}$ is the relative entropy and $T_{B}$ denotes the partial transpose. The quantity in (\[eq:rains-bound\]), often called the Rains bound, was first explored by Rains in the context of entanglement distillation [@R01] and later refined to the form above in [@AdMVW02].

Our main contribution is that the Rains information of a quantum channel is a *strong converse rate for quantum communication* (Theorem \[thm:strong-converse\]), even if we allow for classical pre- and post-processing. (The allowed codes are described in more detail in Section \[sec:ent-gen-code\].) That is, if the quantum communication rate of any protocol for a channel $\mathcal{N}_{A\rightarrow
B}$ exceeds its Rains information, then the fidelity of the scheme decays exponentially fast to zero as the number $n$ of channel uses increases. (We also mention that our main contribution here sets some previous claims from [@SS08; @SSW08] on a firm foundation. Namely, in Refs. [@SS08; @SSW08], the authors claim that the Rains bound for entanglement distillation leads to a weak converse upper bound on the quantum capacity of any channel. However, these papers do not appear to support this claim in any way. Our strong converse result in the present paper thus provides a proof of these claims.)

We establish this theorem by exploiting the generalized divergence framework of Sharma and Warsi [@SW12], which generalizes the related framework from the classical context [@PV10]. However, as discussed in Section \[sec:gen-div\], our main departing point from that work is to consider a different class of useless channels for quantum data transmission. That is, in [@SW12], the main idea was to exploit a generalized divergence to compare the output of the channel with an operator of the form $\pi_{R}\otimes\sigma_{B}$, where $\pi_R$ denotes the fully mixed state on the reference system $R$ and $\sigma_B$ is an arbitrary state on the output system. This state can be viewed as the output of a useless channel that replaces the input and reference system with the maximally mixed state and an uncorrelated state. The resulting information quantity is a generalization of the coherent information from (\[eq:coh-info\]), and one then invokes a data processing inequality to relate this quantity to communication rate and fidelity. Here, we instead compare the output of the channel with an operator in the set ${{\rm PPT}'}\left(  R:B\right)  $, which contains and is closely related to the positive partial transpose states, which in turn are well known to have no distillable entanglement (or equivalently, no quantum data transmission capabilities, so that they constitute another class of useless channels for quantum data transmission).

It turns out that a Rényi-like version of the Rains information of a quantum channel appears to be easier to manipulate, so that we can show that it obeys a weak subadditivity property (Theorem \[thm:weak-subadditivity\] in Section \[sec:weaksub\]). From there, some standard arguments from [@ON1999] conclude the proof of our main result (Theorem \[thm:strong-converse\] in Section \[sec:main\]).

The main application of this result is to establish the *strong converse property* for any generalized dephasing channel (Section \[sec:dephasing\]). The action of any channel in this class on an input state $\rho$ is as follows:$$\mathcal{N}\left(  \rho\right)  =\sum_{x,y=0}^{d-1}\left\langle
x\right\vert _{A}\rho\left\vert y\right\rangle _{A}\ \left\langle \psi
_{y}|\psi_{x}\right\rangle \ \left\vert x\right\rangle \left\langle
y\right\vert _{B},$$ where $\{\left\vert x\right\rangle _{A}\}$ and $\{\left\vert x\right\rangle
_{B}\}$ are orthonormal bases for the input and output systems, respectively, for some positive integer $d$, and $\{\left\vert \psi_{x}\right\rangle _{E}\}$ is a set of arbitrary pure quantum states. A particular example in this class is the qubit dephasing channel, whose action on a qubit density operator is$$\mathcal{N}\left(  \rho\right)  =\left(  1-p\right)  \rho+pZ\rho Z,$$ where the dephasing parameter $p\in\left[  0,1\right]  $ and $Z$ is the Pauli $\sigma_Z$ operator. We prove this result by showing that the Rains information of a generalized dephasing channel is equal to its coherent information (Proposition \[prop:I\_c=R-for-H\]).

Finally, in Section \[sec:erasure\], we establish the strong converse property for the classical post-processing assisted quantum capacity of the quantum erasure channel.

Related Work
------------

A few papers have made some partial progress on or addressed the strong converse for quantum capacity question [@BDHSW12; @BCR09; @BBCW13; @MW13; @SW12; @WW14], with none however establishing that the strong converse holds for any nontrivial class of channels. Refs. [@BDHSW12; @BCR09] prove a strong converse theorem for the entanglement-assisted quantum capacity of a channel, which in turn establishes this as a strong converse rate for the unassisted quantum capacity. Ref. [@BBCW13] proves that the entanglement cost of a quantum channel is a strong converse rate for the quantum capacity assisted by unlimited forward and backward classical communication, which then demonstrates that this quantity is a strong converse rate for unassisted quantum capacity.

Arguably the most important progress to date for establishing a strong converse for quantum capacity is from [@MW13]. These authors reduced the proof of the strong converse for the quantum capacity of degradable channels to that of establishing it for the simpler class of channels known as the symmetric channels (these are channels symmetric under the exchange of the receiver and the environment of the channel). Along the way, they also demonstrated that a pretty strong converse holds for degradable quantum channels, meaning that there is (at least) a jump in the quantum error from zero to $1/2$ as soon as the communication rate exceeds the quantum capacity.

Sharma and Warsi established the generalized divergence framework for understanding quantum communication and reduced the task of establishing the strong converse to a purely mathematical additivity question [@SW12], which hitherto has remained unsolved. The later work in [@WW14] demonstrated that randomly selected codes with a communication rate exceeding the quantum capacity of the quantum erasure channel lead to a fidelity that decreases exponentially fast as the number of channel uses increases. (We stress that a strong converse would imply this behavior for *all* codes whose rate exceeds capacity.)

The methods given in the present paper easily lead to the Rains relative entropy of a quantum state being a strong converse rate for entanglement distillation. After the completion of the present paper, we learned that this result was already established by Hayashi in [@H06 Section 8.6].

Preliminaries {#sec:notation}
=============

States and Channels
-------------------

#### States {#states .unnumbered}

We denote different physical systems by capital letters (e.g. $A$, $B$) and we use these labels as subscripts to indicate with which physical system a mathematical object is associated. Let ${\mathcal{H}}_A$ denote the Hilbert space corresponding to the system $A$, where we restrict ourselves to finite-dimensional Hilbert spaces throughout this paper. We define $|A| = \dim \{ {\mathcal{H}}_A \}$. Moreover, let $\mathcal{B}(A)$ and $\mathcal{P}(A)$ denote the algebra of *bounded linear operators* acting on $\mathcal{H}_A$ and the subset of *positive semi-definite operators*, respectively. We also write $X_A \geq0$ if $X_A \in\mathcal{P}(A)$. An operator $\rho_A$ is in the set $\mathcal{S}( A)  $ of *quantum states* if $\rho_A \geq 0$ and $\operatorname{tr}\left\{
\rho_A \right\} =1$. We say that a quantum state $\rho_A$ is *pure* if $\text{rank}(\rho_A) = 1$. We denote the *identity operator* in ${\mathcal{B}}(A)$ by $I_A$ and the *fully mixed* state by $\pi_A = I_A / |A|$.

The fidelity between two density operators $\rho$ and $\sigma$ is defined as$$F\left(  \rho,\sigma\right)  :=\left\Vert \sqrt{\rho}\sqrt{\sigma
}\right\Vert _{1}^{2} = \operatorname{tr}\Big\{ \sqrt{\sqrt{\rho}\,\sigma\sqrt{\rho}\,} \Big\}^2.$$

A bipartite physical system $AB$ is described by the tensor-product Hilbert space ${\mathcal{H}}_{AB} = \mathcal{H}_{A}\otimes\mathcal{H}_{B}$ with the sets $\mathcal{B}(AB)$, ${\mathcal{P}}(AB)$, and ${\mathcal{S}}(AB)$ defined accordingly. Given a bipartite quantum state $\rho_{AB} \in{\mathcal{S}}(AB)$, we unambiguously write $\rho_{A}=\operatorname{tr}_{B}\left\{
\rho_{AB}\right\}$ for the reduced state on system $A$. A state $\rho_{AB}$ is called *maximally entangled* if $\rho_{AB}$ is pure and $\rho_A = \pi_A$ or $\rho_B = \pi_B$. A product state is a state that can be written in the form $\rho_{AB} = \rho_A \otimes \rho_B$.

A state $\rho_{AB}$ is called *separable* if it can be written in the form $$\begin{aligned}
  \rho_{AB} = \sum_{x \in {\mathcal{X}}} P(x) \rho_A^x \otimes \rho_B^x \,\end{aligned}$$ for a set ${\mathcal{X}}$, a probability mass function $P$ on ${\mathcal{X}}$ and $\rho_A^x \in {\mathcal{S}}(A)$, $\rho_B^x \in {\mathcal{S}}(B)$ for all $x \in {\mathcal{X}}$. We denote the set of all such states by ${{\rm SEP}}(A\!:\!B)$. Furthermore, we say that a bipartite state $\rho_{AB}$ is PPT if it has a positive partial transpose, namely if $T_{B}\left(  \rho_{AB}\right)  \geq0$, where $T_{B}$ indicates the partial transpose operation on system $B$. This is a necessary condition for a bipartite state to be separable [@P96].[^7] Let ${{\rm PPT}}(  A\!:\!B)  $ denote the set of all such states.[^8]

For our purposes it is useful to further enlarge ${{\rm PPT}}(A\!:\!B)$ to the set ${{\rm PPT}'}(  A:B)$, defined as $${{\rm PPT}'}\left(  A:B\right)  :=\left\{  \tau_{AB}:\tau_{AB}\geq
0\wedge\left\Vert T_{B}\left(  \tau_{AB}\right)  \right\Vert _{1}\leq1\right\}  .$$ The set ${{\rm PPT}'}(  A\!:\!B)$ includes all PPT states because $\left\Vert T_{B}\left(  \tau_{AB}\right)  \right\Vert _{1}=1$ if $\tau
_{AB}\in\ {{\rm PPT}}\left(  A\!:\!B\right)  $. Furthermore, all operators $\tau_{AB}\in{{\rm PPT}'}(  A\!:\!B)  $ are subnormalized, in the sense that Tr$\left\{  \tau_{AB}\right\}  \leq1$, because$$\operatorname{tr}\left\{  \tau_{AB}\right\}  =\operatorname{tr}\left\{  T_{B}\left(  \tau
_{AB}\right)  \right\}  \leq\left\Vert T_{B}\left(  \tau_{AB}\right)
\right\Vert _{1}\leq1.$$

#### Channels {#channels .unnumbered}

We denote linear maps from the set of bounded linear operators on one system to the bounded linear operators on another system by calligraphic letters. For example, ${\mathcal{N}}_{A\to B}$ denotes a map from ${\mathcal{B}}(A)$ to ${\mathcal{B}}(B)$, and we will drop the subscript $A \to B$ if it is clear from the context. Let id$_{A}$ denote the *identity map* acting on ${\mathcal{B}}(A)$. If a linear map is completely positive and trace-preserving (CPTP), we say that it is a *quantum channel*.

We say that a CPTP map ${\mathcal{N}}_{AB \to A'B'}$ consists of local operations (LO) from $A\!:\!B$ to $A'\!:\!B'$ if it has the form ${\mathcal{N}}_{AB \to A'B'} = \mathcal{L}_{A \to A'} \otimes \mathcal{M}_{B \to B'}$ for CPTP maps $\mathcal{L}_{A \to A'}$ and $\mathcal{M}_{B \to B'}$. Similarly, we say that ${\mathcal{N}}$ is LOCC if it consists of local operations and classical communication. Furthermore, LOCC maps are contained in the set of separability preserving maps from $A\!:\!B$ to $A'\!:\!B'$ that take ${{\rm SEP}}(A\!:\!B)$ to ${{\rm SEP}}(A'\!:\!B')$. Finally, $\mathcal{N}$ is a PPT preserving operation from $A\!:\!B$ to $A'\!:\!B'$ if the map $T_{B^{\prime}}\circ\mathcal{N}_{AB\rightarrow A^{\prime}B^{\prime}}\circ T_{B}$ is CPTP [@R01]. (So we can conclude that any LOCC map is a PPT-preserving map.)

Codes, Rates and Capacity {#sec:ent-gen-code}
-------------------------

#### Codes {#codes .unnumbered}

We consider a general class of classical pre- and post-processing (CPPP) assisted [entanglement generation (EG) codes]{}, for which the goal is for the sender (Alice) to use a quantum channel in order to share a state that is close to a maximally entangled state with the receiver (Bob).[^9] The [[CPPP ]{}]{}assisted codes allow classical pre- and post-processing before and after the quantum communication phase. In particular, the parties are allowed to prepare a separable resource state before the quantum communication commences. Note that any converse bounds for these classical communication assisted codes naturally also imply the same bounds for unassisted codes.

Formally, we define a *[[CPPP ]{}]{}assisted EG code* for a channel $\mathcal{N}_{A \to B}$ as a triple $$\begin{aligned}
{\mathcal{C}}= (M, {\mathcal{E}}_{\emptyset \to \tilde{A}A\tilde{B}}, {\mathcal{D}}_{\tilde{A}B\tilde{B} \to \hat{A}\hat{B}}) \,. \label{eq:code}\end{aligned}$$ Here, $\hat{A}$ and $\hat{B}$ are Hilbert spaces of dimension $M$, and $\tilde{A}$ and $\tilde{B}$ are auxiliary Hilbert spaces of arbitrary dimension. Moreover, ${\mathcal{E}}_{\emptyset \to \tilde{A}A\tilde{B}}$ is an LOCC quantum channel to $\tilde{A}A\!:\!\tilde{B}$, and ${\mathcal{D}}_{\tilde{A}B\tilde{B} \to \hat{A}\hat{B}}$ is an LOCC quantum channel from $\tilde{A}A\!:\!\tilde{B}$ to $\hat{A}\!:\!\hat{B}$. We write $|{\mathcal{C}}| = M$ for the *size* of the EG code. An *unassisted EG code* is defined in the same way, but ${\mathcal{E}}$ and ${\mathcal{D}}$ are restricted to be LO instead of LOCC.

The corresponding coding schemes begin with Alice and Bob preparing a bipartite state $\rho_{A\tilde{A}\tilde{B}} \in {{\rm SEP}}(A\tilde{A}\!:\!\tilde{B})$ using the quantum channel ${\mathcal{E}}$. Alice then sends the system $A$ through the channel $\mathcal{N}$, resulting in the state $$\rho_{\tilde{A}B\tilde{B}} = \mathcal{N}_{A\rightarrow B}\left(  \rho_{\tilde{A}A\tilde{B}}\right)  .
\label{eq:state-output-from-channel}$$ Finally, Alice and Bob perform a decoding $\mathcal{D}$, leading to the state $\omega_{\hat{A}\hat{B}} = \mathcal{D}_{\tilde{A}B\tilde{B}\rightarrow\hat{A}\hat{B}}(  \rho_{\tilde{A}B\tilde{B}}  )$.

The *fidelity* of the above code ${\mathcal{C}}$ on the channel ${\mathcal{N}}$ is given by$$F({\mathcal{C}}, {\mathcal{N}}) := \left\langle \Phi\right\vert _{\hat{A}\hat{B}}\, \omega_{\hat{A}\hat{B}}\, \left\vert
\Phi\right\rangle _{\hat{A}\hat{B}}, \label{eq:fidelity}$$ where $\left\vert \Phi\right\rangle _{\hat{A}\hat{B}}$ is a (fixed) maximally entangled state on $\hat{A}\hat{B}$.

#### Rates And Capacity {#rates-and-capacity .unnumbered}

The main focus of this paper is on EG codes for many parallel uses of a memoryless quantum channel. That is, we want to investigate product channels $\mathcal{N}^{\otimes n}$ for large $n$. Note that a [[CPPP ]{}]{}assisted EG code for ${\mathcal{N}}^{\otimes n}$ (as described above) allows for classical communication before and after the product channel is used, but does not allow for interactive schemes with classical communication between different channel uses.

We say that a rate $r$ is an *achievable rate* for ([[CPPP ]{}]{}assisted) quantum communication over the channel ${\mathcal{N}}$ if there exist a sequence of codes $\{{\mathcal{C}}_n \}_{n \in \mathbb{N}}$ where ${\mathcal{C}}_n$ is a ([[CPPP ]{}]{}assisted) EG code for ${\mathcal{N}}^{\otimes n}$, such that $$\begin{aligned}
  \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n| \geq r \qquad \textrm{and} \qquad\lim_{n\to\infty} F({\mathcal{C}}_n, {\mathcal{N}}^{\otimes n}) = 1 .\end{aligned}$$ The *quantum capacity* of ${\mathcal{N}}$, denoted $Q({\mathcal{N}})$ is the supremum of all achievable rates. Analogously, the *[[CPPP ]{}]{}assisted quantum capacity* of ${\mathcal{N}}$, denoted $Q_{{\rm pp}}({\mathcal{N}})$, is the supremum of all [[CPPP ]{}]{}assisted achievable rates.

On the other hand, $r$ is a *strong converse rate* for ([[CPPP ]{}]{}assisted) quantum communication if for every sequence of codes $\{ C_n \}_{n \in \mathbb{N}}$ as above, we have $$\begin{aligned}
  \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n| > r \qquad \implies \qquad\lim_{n\to\infty} F({\mathcal{C}}_n, {\mathcal{N}}^{\otimes n}) = 0 .\end{aligned}$$ The *strong converse quantum capacity*, denoted $Q^{\dagger}({\mathcal{N}})$, is the infimum of all strong converse rates. Analogously, the *[[CPPP ]{}]{}assisted strong converse quantum capacity* of ${\mathcal{N}}$, denoted $Q_{{\rm pp}}^{\dagger}({\mathcal{N}})$, is the infimum of all [[CPPP ]{}]{}assisted strong converse rates.

Clearly, the following inequalities hold by definition: $$\begin{aligned}
Q({\mathcal{N}}) \leq Q_{{\rm pp}}({\mathcal{N}}) \leq Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \qquad \textrm{and} \qquad Q({\mathcal{N}}) \leq  Q^{\dagger}({\mathcal{N}}) \leq Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \,.\end{aligned}$$ Finally, we say that a channel ${\mathcal{N}}$ satisfies the *strong converse property for quantum communication* if $Q({\mathcal{N}}) = Q^{\dagger}({\mathcal{N}})$. Similarly, we say that a channel ${\mathcal{N}}$ satisfies the *strong converse property for [[CPPP ]{}]{}assisted quantum communication* if $Q_{{\rm pp}}({\mathcal{N}}) = Q_{{\rm pp}}^{\dagger}({\mathcal{N}})$.

Generalized Divergence Framework {#sec:gen-div}
================================

A functional $\mathbf{D}:\mathcal{S}  \times \mathcal{P}
\rightarrow \mathbb{R}$ is a generalized divergence if it satisfies the monotonicity inequality$$\mathbf{D}\left(  \rho \Vert\sigma \right)  \geq\mathbf{D}\left(  \mathcal{N}(  \rho)  \Vert\mathcal{N}(  \sigma)  \right)  ,$$ where $\mathcal{N}$ is a CPTP map. It follows directly from monotonicity that any generalized divergence is invariant under unitaries, in the sense that $\mathbf{D}\left(  \rho\Vert\sigma\right)  =\mathbf{D}\mathcal{(}U\rho U^{\dag
}\Vert U\sigma U^{\dag})$, where $U$ is a unitary operator, and that it is invariant under tensoring with another quantum state $\tau$, namely $\mathbf{D}\left(  \rho\Vert\sigma\right)  =\mathbf{D}\mathcal{(}\rho \otimes\tau\Vert\sigma\otimes\tau)$.

Rains Relative Entropy and Rains Information
--------------------------------------------

We now define some information measures which play a central role in this paper. They are inspired by the Rains bound on distillable entanglement [@R01] and the subsequent reformulation of it in [@AdMVW02].

We define the *generalized Rains relative entropy* of a bipartite state $\rho_{AB}$ as follows:$$R_{\mathbf{D}}(A\!:\!B)_{\rho}  :=\min_{\tau_{AB}\in
{{\rm PPT}'}(  A:B)  }\mathbf{D}\left(  \rho_{AB}\Vert\tau
_{AB}\right)  ,$$ We sometimes abuse notation and write $R_{\mathbf{D}}(A\!:\!B)_{\rho} = R_{\mathbf{D}}(\rho_{AB})$ if the bipartition is obvious in context.

One property of $R_{\mathbf{D}}$, critical for our application here, is that it is monotone under PPT preserving operations, in the sense that $$R_{\mathbf{D}}(A\!:\!B)_{\rho}  \geq R_{\mathbf{D}}(A'\!:\!B')_{\tau} , \qquad \textrm{where} \qquad \tau_{A'B'} = \mathcal{P}_{AB\to A'B'}(  \rho_{AB})$$ and $\mathcal{P}_{AB\to A'B'}$ is any PPT-preserving operation from $A\!:\!B$ to $A'\!:\!B'$. This is because PPT-preserving operations do not take operators $\tau_{AB}$ in ${{\rm PPT}'}\left(  A\!:\!B\right)  $ outside of this set, which follows from $$\Vert T_{B'} (\mathcal{P}_{AB\to A'B'}(\tau_{AB})) \Vert_{1} = \Vert T_{B'}
(\mathcal{P}_{AB}(T_{B} (T_{B}(\tau_{AB})))) \Vert_{1} \leq\Vert T_{B}(\tau_{AB}) \Vert_{1} \leq1.$$ In the above, the first equality follows because the partial transpose $T_{B}$ is its own inverse, and the first inequality follows because the map $T_{B}
\circ\mathcal{P}_{AB} \circ T_{B}$ is CPTP (as it is PPT preserving) and the fact that the trace norm is monotone decreasing under CPTP maps. Since the set of PPT preserving operations includes LOCC operations, $R_{\mathbf{D}}\left(  A\!:\!B\right)_\rho  $ is also monotone under LOCC operations.

We also note that the Rains information is *subadditive*, namely for any two bipartite states $\rho_{AB}$ and $\sigma_{A'B'}$, we have $$\begin{aligned}
\label{lm:sub-add}
  \widetilde{R}_{\mathbf{D}}(AA'\!:\!BB')_{\rho \otimes \sigma} \leq
  \widetilde{R}_{\mathbf{D}}(A\!:\!B)_{\rho} + \widetilde{R}_{\mathbf{D}}(A'\!:\!B')_{\sigma} .\end{aligned}$$ This follows directly from the definition, since we can always restrict the minimization to product states.

Finally, we define the *generalized Rains information of a quantum channel* as $$R_{\mathbf{D}}\left(  \mathcal{N}\right)  := \max_{\rho_{RA}} R_{\mathbf{D}}(R\!:\!B)_{\omega}, \qquad \textrm{where} \qquad \omega_{RB} = \mathcal{N}_{A\rightarrow B}(  \rho_{RA}) \, .$$

Another critical property of the above quantities has to do with the set ${{\rm PPT}'}\left(  A:B\right)  $ of operators over which we are optimizing. That is, all operators in this set satisfy the property given in Lemma 2 of [@R99], which we recall now:

\[Lemma 2 of [@R99]\]\[lem:overlap\]Let $\tau_{AB}\in{{\rm PPT}'}(
A\!:\!B)$. Then the overlap of $\tau_{AB}$ with any maximally entangled state $\Phi_{AB}$ of Schmidt rank $M$ is at most $1/M$, i.e.$\operatorname{Tr}\left\{  \Phi_{AB}\tau_{AB}\right\}  \leq\frac{1}{M}.$

The same is true for $\sigma_{AB}\in\ \operatorname{PPT}(  A\!:\!B)  $ simply because $\operatorname{PPT}(  A\!:\!B)  \subseteq{{\rm PPT}'}(  A\!:\!B)  $.

Covariance of Quantum Channels
------------------------------

Covariant quantum channels have symmetries which allow us to simplify the set of states over which we need to optimize their generalized Rains information. Let $G$ be a finite group, and for every $g\in G$, let $g\rightarrow
U_{A}\left(  g\right)  $ and $g\rightarrow V_{B}\left(  g\right)  $ be unitary representations acting on the input and output spaces of the channel, respectively. Then a quantum channel $\mathcal{N}_{A\rightarrow B}$ is covariant with respect to these representations if the following relation holds for all input density operators $\rho_A \in {\mathcal{S}}(A)$ and group elements $g\in G$:$$\mathcal{N}_{A\rightarrow B}\left(  U_{A}\left(  g\right)  \rho_A U_{A}^{\dag
}\left(  g\right)  \right)  =V_{B}\left(  g\right)  \mathcal{N}_{A\rightarrow
B}\left(  \rho_A \right)  V_{B}^{\dag}\left(  g\right)  .$$ We then have the following proposition which allows us to restrict the form of the input states needed to optimize the generalized Rains information of a covariant channel:

\[prop:covariance\] Let $\mathcal{N}_{A\rightarrow B}$ be a covariant channel with group $G$ as above and let $\rho_{A} \in {\mathcal{S}}(A)$, $\phi^{\rho}_{RA}$ a purification of $\rho_A$, and $\rho_{RB} = \mathcal{N}_{A\rightarrow B}(\phi^\rho_{RA})$. Let $\bar{\rho}_{A}$ be the group average of $\rho_{A}$, i.e.,$$\bar{\rho}_{A} = \frac{1}{\left\vert G\right\vert }\sum_{g}U_{A}\left(  g\right)  \rho_A U_{A}^{\dag}\left(  g\right)  ,$$ and let $\phi^{\bar{\rho}}_{RA}$ be a purification of $\bar{\rho}_A$ and $\bar{\rho}_{RB} = \mathcal{N}_{A\rightarrow B}(\phi^{\bar{\rho}}_{RA})$. Then, $R_{\mathbf{D}}(R\!:B)_{\bar{\rho}} \geq R_{\mathbf{D}}(R\!:B)_{{\rho}}$

Given the purification $\phi^\rho_{RA}$, consider the following state$$\left\vert \psi\right\rangle _{PRA}:=\sum_{g}\frac{1}{\sqrt{\left\vert
G\right\vert }}\left\vert g\right\rangle _{P}\left[  I_{R}\otimes U_{A}\left(
g\right)  \right]  \left\vert \phi^{\rho}\right\rangle _{RA}\text{.}$$ Observe that $\left\vert \psi\right\rangle _{PRA}$ is a purification of $\overline{\rho}_{A}$ with purifying systems $P$ and $R$. Let $\tau_{PRB}$ be an arbitrary operator in ${{\rm PPT}'}\left(  PR:B\right)  $. Then the following chain of inequalities holds:$$\begin{aligned}
&  \mathbf{D}\left(  \mathcal{N}_{A\rightarrow B}(\psi_{PRA})\middle\Vert\tau
_{PRB}\right) \nonumber\\
& \qquad \geq\mathbf{D}\left(  \sum_{g}\frac{1}{\left\vert G\right\vert }\left\vert
g\right\rangle \left\langle g\right\vert _{P}\otimes\mathcal{N}_{A\rightarrow
B}(U_{A}\left(  g\right)  \phi_{RA}^{\rho}U_{A}^{\dag}\left(  g\right)
)\middle\Vert\sum_{g}p\left(  g\right)  \left\vert g\right\rangle \left\langle
g\right\vert _{P}\otimes\tau_{RB}^{g}\right) \\
& \qquad =\mathbf{D}\left(  \sum_{g}\frac{1}{\left\vert G\right\vert }\left\vert
g\right\rangle \left\langle g\right\vert _{P}\otimes V_{B}\left(  g\right)
\mathcal{N}_{A\rightarrow B}(\phi_{RA}^{\rho})V_{B}^{\dag}\left(  g\right)
\middle\Vert\sum_{g}p\left(  g\right)  \left\vert g\right\rangle \left\langle
g\right\vert _{P}\otimes\tau_{RB}^{g}\right) \label{eq:classical-on-P}\\
& \qquad =\mathbf{D}\left(  \sum_{g}\frac{1}{\left\vert G\right\vert }\left\vert
g\right\rangle \left\langle g\right\vert _{P}\otimes\mathcal{N}_{A\rightarrow
B}(\phi_{RA}^{\rho})\middle\Vert\sum_{g}p\left(  g\right)  \left\vert g\right\rangle
\left\langle g\right\vert _{P}\otimes V_{B}^{\dag}\left(  g\right)  \tau
_{RB}^{g}V_{B}\left(  g\right)  \right) \\
& \qquad \geq\mathbf{D}\left(  \mathcal{N}_{A\rightarrow B}(\phi_{RA}^{\rho})\middle\Vert\sum_{g}p\left(  g\right)  V_{B}^{\dag}\left(  g\right)  \tau_{RB}^{g}V_{B}\left(  g\right)  \right) \\
& \qquad \geq\min_{\tau_{RB}\in{{\rm PPT}'}\left(  R:B\right)  }\mathbf{D}\left(
\mathcal{N}_{A\rightarrow B}(\phi_{RA}^{\rho})\middle\Vert\tau_{RB}\right)   =R_{\mathbf{D}}(R\!:B)_{{\rho}} \,.\end{aligned}$$ The first inequality follows from monotonicity of the generalized divergence $\mathbf{D}$ under a dephasing of the $P$ register (where the dephasing operation is given by $\sum_{g}\left\vert g\right\rangle \left\langle
g\right\vert \cdot\left\vert g\right\rangle \left\langle g\right\vert $). The first equality follows from the assumption of channel covariance. The second equality follows from invariance of the generalized divergence under unitaries, with the unitary chosen to be$$\sum_{g}\left\vert g\right\rangle \left\langle g\right\vert _{P}\otimes
V_{B}^{\dag}\left(  g\right)  .$$ Furthermore, this unitary does not take the state out of the class ${{\rm PPT}'}$, i.e. $$\sum_{g}p\left(  g\right)  \left\vert g\right\rangle \left\langle g\right\vert
_{P}\otimes V_{B}^{\dag}\left(  g\right)  \tau_{RB}^{g}V_{B}\left(  g\right)
\in{{\rm PPT}'}\left(  PR\!:\!B\right)  .$$ This is because, in this case, one could also implement this operation as a classically controlled LOCC operation, i.e., a von Neumann measurement $\left\{  \left\vert g\right\rangle \left\langle g\right\vert \right\}  $ of the register $P$ followed by a rotation $V_{B}^{\dag}\left(  g\right)  $ of the $B$ register. One can do so here because both arguments to $\mathbf{D}$ in (\[eq:classical-on-P\]) are classical on $P$. The second inequality follows because the generalized divergence $\mathbf{D}$ is monotone under the discarding of the register $P$. The final inequality results from taking a minimization, and the final equality is by definition. Since $\tau_{PRB}$ is chosen to be an arbitrary operator in ${{\rm PPT}'}(  PR\!:\!B)  $, it follows that$$\min_{\tau_{PRB} \in {{\rm PPT}'}(PR:B)} \mathbf{D}\left(  \mathcal{N}_{A\rightarrow B}(\psi_{PRA}) \middle\| \tau_{PRB} \right) \geq R_{\mathbf{D}}(R\!:\!B)_{{\rho}}$$ The conclusion then follows because all purifications are related by a unitary on the purifying system and the quantity $R_{\mathbf{D}}$ is invariant under unitaries on the purifying system.

Specializing to Rényi Divergence
--------------------------------

At this point we specialize our discussion to a particular type of Rényi divergence. For $\rho\in\mathcal{S}$, $\sigma\in\mathcal{P}$ and $\alpha \in (0,1) \cup (1,\infty)$, we define the *sandwiched Rényi relative entropy* of order $\alpha$ as [@MDSFT13; @WWY13]$$\widetilde{D}_{\alpha}\left(  \rho\Vert\sigma\right)  := \left\{
\begin{array}
[c]{cc}\frac{1}{\alpha
-1}\log  \operatorname{tr}\left\{  \left(  \sigma^{\left(  1-\alpha\right)  /2\alpha}\rho
\sigma^{\left(  1-\alpha\right)  /2\alpha}\right)  ^{\alpha}\right\}  &
\text{if supp}\left(  \rho\right)  \subseteq\text{supp}\left(  \sigma\right)
\text{ or } \alpha\in(0,1)\\
\infty & \text{else}\end{array}
\right.  .$$ The sandwiched Rényi relative entropy is defined for $\alpha\in\{0,1,\infty\}$ by taking the respective limit. In particular, $\lim_{\alpha \to 1} \widetilde{D}_{\alpha}(\rho\|\sigma) = D(\rho\|\sigma)$. In the following, we restrict our attention to the regime for which $\alpha \geq 1$. The sandwiched Rényi relative entropy is monotone under CPTP maps $\mathcal{N}$ for such values of $\alpha$ [@FL13; @B13monotone; @MO13] and thus constitutes a generalized divergence as discussed above. Moreover, the Rényi divergence is *quasi-convex* in the first argument, i.e. if $\rho$ decomposes as $\rho = \lambda \rho^1 + (1-\lambda) \rho^2$ for $\lambda \in [0,1]$ and $\rho^1, \rho^2 \in {\mathcal{S}}$, then $\max_{i \in \{0,1\}} \widetilde{D}_{\alpha}(\rho^i\|\sigma) \geq \widetilde{D}_{\alpha}(\rho\|\sigma)$.

We denote the corresponding Rains relative entropy by $\widetilde{R}_{\alpha}(A\!:\!B)_{\rho}$ (or $\widetilde{R}_{\alpha}(\rho_{AB})$ if the bipartition is obvious). Consequently, the *Rényi Rains information of a quantum channel* ${\mathcal{N}}_{A\to B}$ is defined as $$\widetilde{R}_{\alpha}(  \mathcal{N})  :=\max_{\rho_{RA}}\min_{\tau_{RB}\in{{\rm PPT}'}(  R:B)  }\widetilde{D}_{\alpha
}\left(  \mathcal{N}_{A\rightarrow B}(  \rho_{RA})  \middle\Vert\tau
_{RB}\right)  .$$ It suffices to perform the maximization in $\widetilde{R}_{\alpha}\left(  \mathcal{N}\right)$ over pure bipartite states $\rho_{RA}$, due to the quasi-convexity of $\widetilde{D}_\alpha$ whenever $\alpha > 1$. As a result, it suffices for the dimension of the reference system $R$ to be no larger than the dimension of the channel input $A$, due to the well known Schmidt decomposition.

The Rényi Rains information converges to $R(
\mathcal{N})  $ in the limit as $\alpha$ approaches one from above. This is shown in the following Lemma, whose proof is provided in Appendix \[sec:ogawa-nagaoka\].

\[lm:continuity\] For any quantum channel $\mathcal{N}$ and $\alpha > \beta > 1$, we have $$\widetilde{R}_{\alpha}(  \mathcal{N})
 \geq \widetilde{R}_{\beta}(  \mathcal{N}) \geq R(  \mathcal{N}) \qquad \textrm{and} \qquad
\lim_{\alpha\rightarrow1^{+}}\widetilde{R}_{\alpha}(  \mathcal{N})
=R\left(  \mathcal{N}\right)  .$$

Relating Fidelity of an Entanglement Generation Code to the Rényi Rains Information of a Channel
------------------------------------------------------------------------------------------------

The power of the generalized divergence framework is that it allows us to relate rate and fidelity to an information quantity. The usual approach is to compare the states resulting from any code to a set of states resulting from a useless channel. For the transmission of classical information, the only set of useless channels are those which trace out the input to the channel and replace it with an arbitrary density operator, effectively cutting the communication line. However, for the transmission of quantum information, there are more interesting classes of useless channels [@SS12]. For example, it is well known that a PPT entanglement binding channel has zero quantum capacity [@HHH00]. More generally, the bound in Lemma \[lem:overlap\] establishes that if both the input to the channel and the reference system are replaced with an operator $\tau_{RB}\in{{\rm PPT}'}(R\!:\!B)$, then the fidelity with a maximally entangled state can never be larger than $1/M$. Since for a memoryless channel we are taking $M=2^{nQ}$, this overlap will be exponentially small with the number of channel uses, so that channels that replace with $\tau_{RB}$ cannot send any quantum information reliably.

The following proposition gives a one-shot bound on the fidelity of any CPPP assisted EG code:

\[lem:one-shot\] Let ${\mathcal{N}}$ be a quantum channel. Any [[CPPP ]{}]{}assisted EG code ${\mathcal{C}}$ on ${\mathcal{N}}$ obeys the following bound. For all $\alpha>1$,$$F({\mathcal{C}}, {\mathcal{N}}) \leq2^{-\left(  \frac{\alpha-1}{\alpha}\right)  \left(  \log |{\mathcal{C}}| -  \widetilde
{R}_{\alpha}\left(  \mathcal{N} \right)  \right)  } \, .
\label{eq:fidelity-bound-rains}$$

Let ${\mathcal{C}}= (M, {\mathcal{E}}_{\emptyset \to \tilde{A}A\tilde{B}}, {\mathcal{D}}_{\tilde{A}B\tilde{B} \to \hat{A}\hat{B}})$ as in  and recall the state $\rho_{\tilde{A}B\tilde{B}}$ in  prior to decoding and the state $\omega_{\hat{A}\hat{B}} = {\mathcal{D}}_{\tilde{A}B\tilde{B}\to\hat{A}\hat{B}}(\rho_{\tilde{A}B\tilde{B}})$. The binary test channel $\mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}$ outputs a flag indicating whether a state is maximally entangled or not:$$\mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}\left(  \cdot\right)  :=\operatorname{tr}\left\{  \Phi_{\hat{A}\hat{B}}\left(  \cdot\right)  \right\}  \left\vert
1\right\rangle \left\langle 1\right\vert +\operatorname{tr}\left\{  \left(
I_{\hat{A}\hat{B}}-\Phi_{\hat{A}\hat{B}}\right)  \left(  \cdot\right)  \right\}
\left\vert 0\right\rangle \left\langle 0\right\vert .
\label{eq:entanglement-test}$$

Furthermore, consider an arbitrary state $\tau_{\tilde{A}B\tilde{B}} \in {{\rm PPT}'}(\tilde{A}\!:\!B\tilde{B})$ and observe that $\mathcal{D}_{\tilde{A}B\tilde{B}\to\hat{A}\hat{B}}(\tau_{\tilde{A}B\tilde{B}})
\in {{\rm PPT}'}(\hat{A}\!:\!\hat{B})$ because the decoding operator is restricted to be LOCC. For ease of presentation, we set $$p = \operatorname{tr}\left\{  \Phi_{\hat{A}\hat{B}}\, \mathcal{D}_{\tilde{A}B\tilde{B}\rightarrow\hat{A}\hat{B}} (  \tau_{\tilde{A}B\tilde{B}})  \right\}  \qquad \textrm{and} \qquad F = F({\mathcal{C}},{\mathcal{N}}) = \operatorname{tr}\left\{ \Phi_{\hat{A}\hat{B}}\, \omega_{\hat{A}\hat{B}} \right\} . \label{eq:overlap-decoded-PPT}$$ Applying monotonicity of the divergence under the decoding map ${\mathcal{D}}$ and the test ${\mathcal{B}}$, we find $$\begin{aligned}
\widetilde{D}_{\alpha}\left(  \rho_{\tilde{A}B\tilde{B}}\middle\Vert\tau_{\tilde{A}B\tilde{B}}\right)   &
\geq\widetilde{D}_{\alpha}\left(  \mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}\left(
\omega_{\hat{A}\hat{B}}\right)  \middle\Vert\mathcal{B}_{\hat{A}\hat{B}\rightarrow Z}\left(
\mathcal{D}_{\tilde{A}B\tilde{B}\rightarrow\hat{A}\hat{B}}\left(  \tau_{\tilde{A}B\tilde{B}}\right)  \right)  \right)
\label{eq:renyi-develop-1}\\
&  =\frac{1}{\alpha-1}\log\left[  F^{\alpha}p^{1-\alpha}+\left(  1-F\right)
^{\alpha}\left[  \operatorname{tr}\left\{  \tau_{\tilde{A}B\tilde{B}} \right\}  -p\right]  ^{1-\alpha
}\right] \\
&  \geq\frac{1}{\alpha-1}\log\left[  F^{\alpha}p^{1-\alpha}\right]   \\
& \geq\frac{1}{\alpha-1}\log\left[  F^{\alpha}\left(  1/M\right)  ^{1-\alpha
}\right]  \\
& =\frac{\alpha}{\alpha-1}\log F+\log M \label{eq:renyi-develop-last} .$$ The second inequality follows by discarding the second term $\left(
1-F\right)  ^{\alpha}\left[  \operatorname{tr}\left\{  \tau_{RB}\right\}  -p\right]
^{1-\alpha}$ (recall that we are considering $\alpha>1$). The third inequality follows from (\[eq:overlap-decoded-PPT\]) and Lemma \[lem:overlap\].

Now, recall that the state $\rho_{\tilde{A}A\tilde{B}}$ is in ${{\rm SEP}}(\tilde{A}A\!:\!\tilde{B})$ and can thus be decomposed into products of pure states. Using the quasi-convexity of $\widetilde{D}_{\alpha}$ in the first argument, we find that there exist pure states $\sigma_{\tilde{A}A}$ and $\sigma_{\tilde{B}}$ such that for every $\tau_{\tilde{A}B} \in {{\rm PPT}'}(\tilde{A}:B)$, we have $$\begin{aligned}
  \widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \sigma_{\tilde{A}A} \right)  \middle\| \tau_{\tilde{A}B}\right) 
  &\geq
   \widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \sigma_{\tilde{A}A} \right) \otimes \sigma_{\tilde{B}} \middle\| \tau_{\tilde{A}B} \otimes \sigma_{\tilde{B}} \right) \\
  &\geq  \widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \rho_{\tilde{A}A\tilde{B}} \right) \middle\| \tau_{\tilde{A}B} \otimes \sigma_{\tilde{B}} \right)\\
  & \geq \frac{\alpha}{\alpha-1}\log
F+\log M,\end{aligned}$$ where in the last line we apply the development in - given that $\tau_{\tilde{A}B} \otimes \sigma_{\tilde{B}} \in
{{\rm PPT}'}(  \tilde{A}\!:\!B\tilde{B}) $. Since the above bound holds for all $\tau_{\tilde{A}B} \in {{\rm PPT}'}(\tilde{A}:B)$, we can conclude that $$\min_{\tau_{\tilde{A}B} \in {{\rm PPT}'}(\tilde{A}:B) }\widetilde{D}_{\alpha}\left( {\mathcal{N}}_{A\to B} \left( \sigma_{\tilde{A}A} \right)  \middle\| \tau_{\tilde{A}B}\right) 
\geq \frac{\alpha}{\alpha-1}\log
F+\log M.$$ We can finally remove the dependence on any particular code by optimizing over all inputs to the channel. Identifying $\tilde{A}$ with $R$ to simplify notation, we find $$\widetilde{R}_{\alpha}\left(  \mathcal{N}\right)  =\max_{\rho_{RA}}\min
_{\tau_{RB}\in{{\rm PPT}'}\left(  R:B\right)  }\widetilde{D}_{\alpha}\left(
\mathcal{N}_{A\rightarrow B}\left(  \rho_{RA}\right)  \middle\Vert\tau_{RB}\right)
\geq\frac{\alpha}{\alpha-1}\log F+\log M.$$ This bound is then equivalent to (\[eq:fidelity-bound-rains\]).

Weak Subadditivity of the $\alpha$-Rains Information for Memoryless Channels {#sec:weaksub}
============================================================================

In this section, we prove an important theorem, which is critical for concluding that the Rains information of a channel is a strong converse rate for quantum communication. Before we commence, we need the following technical property.

\[lem:mono-inc\] Let $\alpha > 1$, $\rho, \rho' 
\in \mathcal{S}$ and $\sigma \in \mathcal{P}$. If $\rho \leq \gamma \rho'$ for some $\gamma \geq 1$, then $$\widetilde{D}_{\alpha} \left(  \rho\Vert\sigma\right)  \leq \frac{\alpha}{\alpha-1} \log \gamma + \widetilde{D}_{\alpha}\left(  \rho'\Vert\sigma\right) \,.$$

From the assumption that $\rho\leq \gamma \rho^{\prime}$, we get $\sigma^{\left(
1-\alpha\right)  /2\alpha}\rho\sigma^{\left(  1-\alpha\right)  /2\alpha}\leq \gamma \sigma^{\left(  1-\alpha\right)  /2\alpha}\rho^{\prime}\sigma^{\left(
1-\alpha\right)  /2\alpha}$. Then we have that $$\operatorname{tr}\{ (\sigma^{\left(
1-\alpha\right)  /2\alpha}\rho\sigma^{\left(  1-\alpha\right)  /2\alpha})^\alpha\}
\leq
\gamma^\alpha \operatorname{tr}\{ (\sigma^{\left(  1-\alpha\right)  /2\alpha}\rho^{\prime}\sigma^{\left(
1-\alpha\right)  /2\alpha})^\alpha \}$$ because Tr$\left\{  f\left(  P\right)  \right\}  \leq\ $Tr$\left\{  f\left(  Q\right)
\right\}  $ for $P\leq Q$ and $f$ a monotone increasing function (see, e.g., [@MO13 Lemma III.6]). Taking logarithms and dividing by $\alpha -1$ gives the statement of the lemma.

Now we are ready to prove that the Rains information of the channel obeys a weak subadditivity property.

\[thm:weak-subadditivity\] Let ${\mathcal{N}}_{A\to B}$ be a quantum channel. For all $\alpha>1$ and $n \in \mathbb{N}$, we have $$\widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\right)  \leq
n\widetilde{R}_{\alpha}\left(  \mathcal{N}\right)  +\frac{\alpha\left\vert
A\right\vert ^{2}}{\alpha-1}\log n \label{eq:weak-subadd} .$$

To begin with, we observe that a tensor-power channel is covariant with respect to permutations of the input and output systems, in the sense that$$\forall\pi\in S_{n}:W_{B^{n}}^{\pi}\mathcal{N}^{\otimes n}\left(  \rho_{A^{n}}\right)  \left(  W_{B^{n}}^{\pi}\right)  ^{\dag}=\mathcal{N}^{\otimes
n}\left(  W_{A^{n}}^{\pi}\rho_{A^{n}}\left(  W_{A^{n}}^{\pi}\right)  ^{\dag
}\right)  ,$$ where $W_{A^{n}}^{\pi}$ and $W_{B^{n}}^{\pi}$ are unitary representations of the permutation $\pi$, acting on the input space$~A^{n}$ and the output space $B^{n}$, respectively. So, letting $\left\vert \phi^{\rho}\right\rangle
_{RA^{n}}$ denote a purification of $\rho_{A^{n}}$, we can apply Proposition \[prop:covariance\] to find that $$\widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\left(
\phi_{RA^{n}}^{\rho}\right)  \right)  \leq\widetilde{R}_{\alpha}\left(
\mathcal{N}^{\otimes n}\left(  \phi_{RA^{n}}^{\overline{\rho}}\right)  \right)  , \label{eq:1st-proof-line}$$ where $\phi_{RA^{n}}^{\overline{\rho}}$ is a purification of the permutation invariant state $\overline{\rho}_{A^{n}}$. Now, this purification $\phi_{RA^{n}}^{\overline{\rho}}$ is related by a unitary on the reference system $R$ to a state $\left\vert \psi\right\rangle _{\hat{A}^{n}A^{n}}\in\text{Sym}((\hat{A}\otimes
A)^{\otimes n})$, where $\hat{A}\simeq A$ [@RennerThesis Lemma 4.3.1]. So it follows that$$\widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\left(
\phi_{RA^{n}}^{\rho}\right)  \right)  \leq\widetilde{R}_{\alpha}\left(
\mathcal{N}^{\otimes n}\left(  \phi_{RA^{n}}^{\overline{\rho}}\right)  \right)  =\widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\left(  \psi_{\hat{A}^{n}A^{n}}\right)  \right)  .
\label{eq:2nd-proof-line}$$ For such a state in Sym$((\hat{A}\otimes A)^{\otimes n})$, we observe (see, e.g., [@CKR09]) that $$\psi_{\hat{A}^{n}A^{n}}\leq n^{\left\vert A\right\vert ^{2}}\omega_{\hat{A}^{n}A^{n}}^{(n)  }, 
\quad \textrm{where} \quad
\omega_{\hat{A}^{n}A^{n}}^{\left(  n\right)  } := \int d\mu\left(  \varphi\right)
\, \varphi_{\hat{A}A}^{\otimes n}, \label{eq:de-finetti-state} \,.$$ with $\mu\left(  \varphi\right)  $ denoting the uniform probability measure on the unit sphere consisting of pure bipartite states $\varphi
_{\hat{A}A}$.

Employing Lemma \[lem:mono-inc\], we find that$$\begin{aligned}
  \widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\left(
\psi_{\hat{A}^{n}A^{n}}\right)  \right) \leq \frac{\alpha\left\vert A\right\vert ^{2}}{\alpha-1}\log n + \widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\left(
\omega_{\hat{A}^{n}A^{n} }\right)  \right) .\end{aligned}$$ and since the right hand side does not depend on the state $\rho_{A^n}$ anymore, this yields directly $$\begin{aligned}
  \widetilde{R}_{\alpha}({\mathcal{N}}^{\otimes n}) \leq \frac{\alpha\left\vert A\right\vert ^{2}}{\alpha-1}\log n + \widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\left(
\omega_{\hat{A}^{n}A^{n} }\right)  \right) 
\leq \frac{\alpha\left\vert A\right\vert ^{2}}{\alpha-1}\log n + \max_{\phi_{\hat{A}A}} \widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\left(
\phi_{\hat{A}A}^{\otimes n} \right)  \right) \label{eq:use-this}\end{aligned}$$ where we used the quasi-convexity of $\widetilde{D}_{\alpha}$ and the definition of $\omega_{\hat{A}^{n}A^{n}}^{(n)}$ in  to establish the second inequality. Now simply note that $\widetilde{R}_{\alpha}$ is subadditive for product states, i.e.$\widetilde{R}_{\alpha}\big(  \mathcal{N}^{\otimes n}(
\phi_{\hat{A}A}^{\otimes n} )  \big) \leq n \widetilde{R}_{\alpha}(  \mathcal{N}(\phi_{\hat{A}A})  )$. Combining this with  concludes the proof.

A consequence of Theorem \[thm:weak-subadditivity\] is that the Rains information of a channel is weakly subadditive. This corollary is required in order to set some of the claims in [@SS08; @SSW08] on a firm foundation. We provide its proof in Appendix \[sec:appendix\].

\[cor:von-Neumann-subadd\] The Rains information of a quantum channel is weakly subadditive, in the sense that$$\limsup_{n \to\infty} \frac{1}{n}R\left(  \mathcal{N}^{\otimes n}\right)  \leq
R\left(  \mathcal{N}\right)  .$$

The Rains Information is a Strong Converse Rate for Quantum Communication {#sec:main}
=========================================================================

We are ready to state our main result, which states that the Rains information of the channel is an upper bound on the [[CPPP ]{}]{}assisted quantum capacity.

\[thm:strong-converse\] For any quantum channel ${\mathcal{N}}$, we have $Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \leq \inf_{l \in \mathbb{N}} \frac{1}{l} R({\mathcal{N}}^{\otimes l})$.

Before we prove this result, we first state a technical proposition which implies that the fidelity decreases exponentially fast as the number of channel uses increases, with the exponent bounded in  below. (However, we we do not know if the exponent in  is optimal.)

\[pr:strong-converse\] Let ${\mathcal{N}}$ be a quantum channel. Consider any sequence of codes $\{ {\mathcal{C}}_n \}_{n \in \mathbb{N}}$, where ${\mathcal{C}}_n$ is a [[CPPP ]{}]{}assisted EG code for ${\mathcal{N}}^{\otimes n}$. Then the rate of this sequence, $r = \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n|$, satisfies $$\begin{aligned}
   \liminf_{n\to\infty} \left\{ - \frac{1}{n} \log F({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}) \right\} \geq \sup_{\alpha > 1} \left\{ \frac{\alpha-1}{\alpha} \big( r - \widetilde{R}_{\alpha}({\mathcal{N}}) \big) \right\}. \label{eq:exponent}\end{aligned}$$

The first statement is shown as follows. First, by definition of $r$, for any $\delta > 0$, there exists an $N_0 \in \mathbb{N}$ such that $\frac{1}{n} \log |{\mathcal{C}}_n| \geq r - \delta$ for all $n \geq N_0$. For such $n$ and any $\alpha > 1$, we employ Proposition \[lem:one-shot\] to find the following bound: $$\begin{aligned}
F\left({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}\right) &\leq2^{-\left(  \frac{\alpha-1}{\alpha}\right)  \left(  \log
|{\mathcal{C}}_n|-\widetilde{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\right)  \right)  }\\
&\leq 2^{-\left(  \frac{\alpha-1}{\alpha}\right)  \left(  n(r-\delta) -\widetilde
{R}_{\alpha}\left(  \mathcal{N}^{\otimes n}\right)  \right)  }\\
&  \leq2^{-\big(  \frac{\alpha-1}{\alpha}\big)  \big(  n(r -\delta) - n 
\widetilde{R}_{\alpha}(\mathcal{N})  -\frac{\alpha\left\vert
A\right\vert^{2}}{\alpha-1}\log n \big)  } \label{eq:use-weak}   
= n^{\left\vert A\right\vert^{2}}2^{-n\left(  \frac{\alpha-1}{\alpha
}\right)  \left(  r-\delta-\widetilde{R}_{\alpha}(  \mathcal{N})  \right)  } .$$ Here, we used the weak subadditivity result from Theorem \[thm:weak-subadditivity\] to establish . Hence, we find $$\begin{aligned}
   \liminf_{n\to\infty} \left\{ - \frac{1}{n} \log F({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}) \right\} \geq \frac{\alpha-1}{\alpha} \big( r - \delta - \widetilde{R}_{\alpha}({\mathcal{N}}) \big) \end{aligned}$$ and the first statement of the theorem follows since we can choose $\delta$ and $\alpha$ arbitrarily.

We briefly sketch a proof of the main theorem here for the case in which we want to show that $Q_{{\rm pp}}({\mathcal{N}}) \leq R({\mathcal{N}})$ and provide the full proof below. Consider any code with $r$ defined as in Proposition \[pr:strong-converse\]. If $r > R({\mathcal{N}})$, then by continuity of $\widetilde{R}_{\alpha}({\mathcal{N}})$ as $\alpha \to 1^+$, there always exists an $\alpha > 1$ such that $r > \widetilde{R}_{\alpha}({\mathcal{N}})$ as well. Thus, the right hand side of  is strictly positive and the fidelity thus vanishes.

Consider any sequence of codes $\{ {\mathcal{C}}_n \}_{n \in \mathbb{N}}$, where ${\mathcal{C}}_n$ is a [[CPPP ]{}]{}assisted EG code for ${\mathcal{N}}^{\otimes n}$ such that $r = \liminf_{n\to\infty} \frac{1}{n} \log |{\mathcal{C}}_n| > \inf_{l \in \mathbb{N}} \frac{1}{l} R({\mathcal{N}}^{\otimes l})$. Then, by definition of the infimum there exists a value $l \in \mathbb{N}$ such that $l r > R({\mathcal{N}}^{\otimes l})$. Furthermore, by continuity of $\widetilde{R}_{\alpha}(\mathcal{N}^{\otimes l})$ as $\alpha \to 1^+$ (cf. Lemma \[lm:continuity\]), there exists an $\alpha > 1$ such that $l r > \widetilde{R}_{\alpha}({\mathcal{N}}^{\otimes l})$. We now consider the subsequence of codes $\{ {\mathcal{C}}_{kl} \}_{k \in \mathbb{N}}$ for the channels ${\mathcal{N}}^{\otimes kl} =\big({\mathcal{N}}^{\otimes l}\big)^{\otimes k}$. Then, Proposition \[pr:strong-converse\] applied to the channel ${\mathcal{N}}^{\otimes l}$ yields $$\begin{aligned}
   \liminf_{k\to\infty} \left\{ - \frac{1}{k} \log F({\mathcal{C}}_{kl},{\mathcal{N}}^{\otimes kl}) \right\} \geq \frac{\alpha-1}{\alpha } \big( lr - \widetilde{R}_{\alpha}({\mathcal{N}}^{\otimes l}) \big) > 0 .  \label{eq:nonzero-lower}
 \end{aligned}$$ Here we used that $\liminf_{k\to\infty} \frac{1}{k} \log |{\mathcal{C}}_{kl}| \geq lr$ since the $\liminf$ of the subsequence $\{ \frac{1}{kl} \log |{\mathcal{C}}_{kl}| \}_{k}$ is lower bounded by the $\liminf$ of the sequence $\{ \frac{1}{n} \log |{\mathcal{C}}_n| \}_{n}$. On the other hand, we have $$\begin{aligned}
   \liminf_{n\to\infty} \left\{ - \frac{1}{n} \log F({\mathcal{C}}_n,{\mathcal{N}}^{\otimes n}) \right\} \geq \min_{j \in \{0,1, \dots l-1\}} \liminf_{k\to\infty} \left\{ - \frac{1}{k l+j} \log F({\mathcal{C}}_{kl+j},{\mathcal{N}}^{\otimes kl}) \right\} \,.
 \end{aligned}$$ Now note that  implies that the $\liminf$ for the subsequence with $j=0$ is strictly positive. In fact, an adaption of the argument leading to  reveals that the same is true for all $j \in \{0, 1, \ldots, l-1\}$. Hence, we conclude that the fidelity vanishes and that $\inf_{l \in \mathbb{N}} \frac{1}{l} R({\mathcal{N}}^{\otimes l})$ is a strong converse rate.

Theorem \[thm:strong-converse\] establishes the Rains information of a quantum channel as an upper bound on the strong converse capacity $Q_{{\rm pp}}^{\dagger}({\mathcal{N}})$ for [[CPPP ]{}]{}assisted quantum communication over a channel ${\mathcal{N}}$. Thus, in summary, the following inequalities hold for all quantum channels: $$\begin{aligned}
I_{\rm c}({\mathcal{N}}) \leq \lim_{l \to \infty} \frac{1}{l} I_{\rm c}({\mathcal{N}}^{\otimes l})  = Q({\mathcal{N}}) \leq Q^{\dagger}({\mathcal{N}}) \leq Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \leq 
\inf_{l\geq 1} \frac{1}{l} R({\mathcal{N}}^{\otimes l})
\leq R({\mathcal{N}}) . \label{eq:hier}\end{aligned}$$

Strong Converse Property for Quantum Communication over Dephasing Channels {#sec:dephasing}
==========================================================================

In this section, we show that the Rains information of a generalized dephasing channel [@DS05; @YHD05MQAC; @itit2008hsieh; @BHTW10] (also known as Hadamard diagonal channels [@KMNR07] and Schur multiplier channels [@MW13]) is equal to the coherent information of this channel. Consequently, the hierarchy in  collapses for generalized dephasing channels. In particular, we establish that the quantum capacity of this class of channels obeys the strong converse property and also that classical pre- and post-processing does not increase the capacity for these channels.

A generalized dephasing channel is any channel with an isometric extension of the form $$U_{A\rightarrow BE}^{\mathcal{N}}:=\sum_{x=0}^{d-1}\left\vert
x\right\rangle _{B}\left\langle x\right\vert _{A}\otimes\left\vert \psi
_{x}\right\rangle _{E}, \label{eq:deph}$$ where the states $\left\vert \psi_{x}\right\rangle $ are arbitrary (not necessarily orthonormal).

\[prop:I\_c=R-for-H\]Let $\mathcal{N}$ be a generalized dephasing channel of the form . Then $I_{c}\left(  \mathcal{N}\right)  =R\left(  \mathcal{N}\right)$.

We have already seen in that $I_{c}\left(  \mathcal{N}\right)  \leq R\left(  \mathcal{N}\right)$ holds for all channels. We now establish that the opposite inequality holds for a generalized dephasing channel $\mathcal{N}$. Consider that any generalized dephasing channel $\mathcal{N}$ obeys the following covariance property:[^10] $$\mathcal{N}\left(  Z_{A}\left(  z\right)  \rho Z_{A}^{\dag}\left(
z\right)  \right)  =Z_{B}\left(  z\right)  \mathcal{N}\left(  \rho\right)
Z_{B}^{\dag}\left(  z\right)  , \label{eq:covariance-dephasing}$$ for $z\in\left\{  0,\cdots,d-1\right\}  $, where$$\begin{aligned}
Z_{A}\left(  z\right)  \left\vert x\right\rangle _{A}    =\exp\left\{  2\pi
ixz/d\right\}  \left\vert x\right\rangle _{A},\qquad
Z_{B}\left(  z\right)  \left\vert x\right\rangle _{B}    =\exp\left\{  2\pi
ixz/d\right\}  \left\vert x\right\rangle _{B}. \label{eq:phase-op-B}$$ Furthermore, a uniform mixing of these operators is equivalent to a completely dephasing channel:$$\frac{1}{d}\sum_{i=0}^{d-1}Z_{A}\left(  z\right)  \left(  \cdot\right)
Z_{A}^{\dag}\left(  z\right)  =\sum_{i=0}^{d-1}\left\vert x\right\rangle
\left\langle x\right\vert _{A}\left(  \cdot\right)  \left\vert x\right\rangle
\left\langle x\right\vert _{A},$$ with the same true for the operators $\left\{  Z_{B}\left(
z\right)  \right\}  _{z\in\left\{  0,\ldots,d-1\right\}  }$. Then we can apply Proposition \[prop:covariance\] to conclude that the Rains information of a generalized dephasing channel is maximized by a state with a Schmidt decomposition of the following form:$$\left\vert \varphi^{p}\right\rangle _{RA}:=\sum_{x}\sqrt{p_{X}\left(
x\right)  }\left\vert x\right\rangle _{R}\left\vert x\right\rangle _{A},
\label{eq:aligned-schmidt}$$ for some probability distribution $p_{X}\left(  x\right)  $ and some orthonormal basis $\left\{  \left\vert x\right\rangle _{R}\right\}  $ for the reference system $R$ (with the key result being that the basis $\left\{
\left\vert x\right\rangle _{A}\right\}  $ is aligned with the basis of the channel). That is,$$R\left(  \mathcal{N}\right)  =\max_{p_X}\min_{\tau_{RB}\in{{\rm PPT}'}\left(  R:B\right)  }D\left(  \mathcal{N}  \left(  \varphi_{RA}^{p}\right)  \middle\Vert\tau
_{RB}\right)  . \label{eq:rains-hadamard}$$ Let $\Delta_{P}$ be a CPTP map constructed as follows:$$\begin{aligned}
\Delta_{P}\left(  \cdot\right)   & = P\left(  \cdot\right)  P+\left(
I-P\right)  \left(  \cdot\right)  \left(  I-P\right)  , \qquad \textrm{with} \qquad P  =\sum_{x}\left\vert x\right\rangle \left\langle x\right\vert
_{R}\otimes\left\vert x\right\rangle \left\langle x\right\vert _{B} \,.\end{aligned}$$ Then the following chain of inequalities holds:$$\begin{aligned}
I_{c}\left(  \mathcal{N}\right)   &=\max_{\varphi_{RA}}\min_{\sigma_{B}}D\left(  \mathcal{N}  \left(
\varphi_{RA}\right)  \middle\Vert I_{R}\otimes\sigma_{B}\right) \\
&  \geq\max_{p_X}\min_{\sigma_{B}}D\left( \mathcal{N} \left(  \varphi_{RA}^{p}\right)  \middle\Vert
I_{R}\otimes\sigma_{B}\right) \\
&  \geq\max_{p_X}\min_{\sigma_{B}}D\left(  \Delta_{P}\left(
\mathcal{N}  \left(  \varphi_{RA}^{p}\right)  \right)  \middle\Vert\Delta_{P}\left(  I_{R}\otimes\sigma_{B}\right)
\right) \\
&  =\max_{p_X}\min_{\sigma_{B}}
 D\left(  P\left(  \mathcal{N}  \left(  \varphi_{RA}^{p}\right)
\right)  P\middle\Vert P\left(  I_{R}\otimes\sigma_{B}\right)  P\right) \\
&  =\max_{p_X}\min_{q}D\left(  \mathcal{N}  \left(  \varphi_{RA}^{p}\right)  \middle\Vert\sum
_{x}q\left(  x\right)  \left\vert x\right\rangle \left\langle x\right\vert
_{R}\otimes\left\vert x\right\rangle \left\langle x\right\vert _{B}\right) \\
&  \geq\max_{p_X}\min_{\tau_{RB}\in{{\rm PPT}'}\left(  R:B\right)
}D\left(  \mathcal{N} \left(
\varphi_{RA}^{p}\right)  \middle\Vert\tau_{RB}\right)  =R\left(  \mathcal{N}\right)  .\end{aligned}$$ The first inequality follows by restricting the maximization to be over pure bipartite vectors of the form in (\[eq:aligned-schmidt\]). The second inequality follows from monotonicity of the relative entropy under the CPTP map $\Delta_{P}$. The second equality follows because$$\begin{gathered}
D\left(  \Delta_{P}\left[  \left(  \text{id}_{R}\otimes\mathcal{N}\right)
\left(  \varphi_{RA}^{p}\right)  \right]  \middle\Vert\Delta_{P}\left(  I_{R}\otimes\sigma_{B}\right)  \right)  =D\left(  P\left[  \left(  \text{id}_{R}\otimes\mathcal{N}\right)  \left(  \varphi_{RA}^{p}\right)  \right]
P\middle\Vert P\left(  I_{R}\otimes\sigma_{B}\right)  P\right) \\
+D\left(  \left(  I-P\right)  \left[  \left(  \text{id}_{R}\otimes
\mathcal{N}\right)  \left(  \varphi_{RA}^{p}\right)  \right]  \left(
I-P\right)  \middle\Vert\left(  I-P\right)  \left(  I_{R}\otimes\sigma_{B}\right)
\left(  I-P\right)  \right)\end{gathered}$$ since $P\perp I-P$, and because the state $\left(  \text{id}_{R}\otimes\mathcal{N}\right)  \left(  \varphi_{RA}^{p}\right)  $ has no support in the subspace onto which $I-P$ projects, so that$$D\left(  \left(  I-P\right)  \left[  \left(  \text{id}_{R}\otimes
\mathcal{N}\right)  \left(  \varphi_{RA}^{p}\right)  \right]  \left(
I-P\right)  \middle\Vert\left(  I-P\right)  \left(  I_{R}\otimes\sigma_{B}\right)
\left(  I-P\right)  \right)  =0.$$ The third equality follows because $$\begin{aligned}
P\left[  \left(  \text{id}_{R}\otimes\mathcal{N}\right)  \left(
\varphi_{RA}^{p}\right)  \right]  P & =\left(  \text{id}_{R}\otimes
\mathcal{N}\right)  \left(  \varphi_{RA}^{p}\right), \\
P\left(  I_{R}\otimes\sigma_{B}\right)  P & =\sum_{x}q\left(  x\right)
\left\vert x\right\rangle \left\langle x\right\vert _{R}\otimes\left\vert
x\right\rangle \left\langle x\right\vert _{B},\end{aligned}$$ for some distribution $q\left(  x\right)  =\left\langle x\right\vert
_{B}\sigma_{B}\left\vert x\right\rangle _{B}$. The last inequality follows because the state $\sum_{x}q\left(  x\right)  \left\vert x\right\rangle \left\langle x\right\vert
_{R}\otimes\left\vert x\right\rangle \left\langle x\right\vert _{B}\in{{\rm PPT}'}\left(  R\!:\!B\right)$, and the final equality follows from (\[eq:rains-hadamard\]).

Strong Converse for Classical Communication\
Assisted Capacity of Erasure Channels {#sec:erasure}
============================================

In this section, we consider a class of erasure channels of the form $$\begin{aligned}
  {\mathcal{N}}_{A \to B} : \rho_A \mapsto (1-p) \rho_{B} + p | e \rangle\!\langle e|_{B}, \label{eq:erasure}\end{aligned}$$ where $p \in [0,1]$ is the erasure probability, $\rho_B$ is an isometric embedding of $\rho_A$ into $B$, and $\vert e\rangle$ is a quantum state orthogonal to $\rho_{B}$.

Let ${\mathcal{N}}$ be an erasure channel of the form . Then, $Q_{{\rm pp}}({\mathcal{N}}) = Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) = (1-p) \log |A|$.

First note that erasure channels are covariant under the full unitary group acting on $A$, and thus Proposition \[prop:covariance\] implies that $$\begin{aligned}
  R({\mathcal{N}}) = R({\mathcal{N}}(\psi_{RA})) \leq D \big( {\mathcal{N}}(\psi_{RA}) \big\| \tau_{RB} \big)
  = H(\tau_{RB}) - H(\rho_{RB})\end{aligned}$$ where $\psi_{RA}$ is a maximally entangled state, $\rho_{RB} = {\mathcal{N}}(\psi_{RA})$, and $\tau_{RB}$ is chosen as follows: $$\begin{aligned}
  |\psi\rangle_{RA} = \sum_x \frac{1}{\sqrt{|A|}} |x\rangle_R \otimes |x\rangle_A , \qquad \textrm{and} \qquad \tau_{RB} = \sum_x \frac{1}{|A|} |x\rangle\!\langle x|_R \otimes {\mathcal{N}}\big(|x\rangle\!\langle x|_A\big)  \,.\end{aligned}$$ Evaluating this, we find $$\begin{aligned}
H(RB)_\tau & = H(R)_\tau + H(B|R)_\tau \\
 & = \log |A| + \sum_x \frac{1}{|A|} H(B)_{{\mathcal{N}}(|x\rangle\!\langle x|_A)} \\
 & = \log |A| + \sum_x \frac{1}{|A|} [H((1-p)|x\rangle\!\langle x|_A + p |e\rangle\langle e|)] \\
 & = \log |A| + h_2(p) ,\\
H(RB)_\rho & = H( (1-p) \psi_{RA} + p \pi_R \otimes |e\rangle \langle e |) \\
& = h_2(p) + (1-p) H( \psi_{RA}) + p H(\pi_R \otimes |e\rangle \langle e |) \\
& = h_2(p) + p \log|A| .\end{aligned}$$ In the above, $H(B|R) := H(BR) - H(R)$ is the conditional entropy. Therefore, we conclude that $Q_{{\rm pp}}^{\dagger}({\mathcal{N}}) \leq R({\mathcal{N}}) \leq (1-p) \log |A|.$

Finally, it is well known that any rate $r < (1-p) \log |A|$ can be achieved with [[CPPP ]{}]{}assistance [@PhysRevLett.78.3217]. We review this argument briefly. Let Alice prepare and send $n$ maximally entangled states through ${\mathcal{N}}^{\otimes n}$. Bob then records which instances of the channel led to an erasure, and communicates this to Alice. They will then use the correctly transmitted states to distill a maximally entangled state of dimension $r n$. This will succeed whenever $r n / \log |A|$ is smaller than the number of correctly transmitted states, which will happen with probability 1 as $n \to \infty$ as long as $r < (1-p) \log |A|$. This establishes that $Q_{{\rm pp}}({\mathcal{N}}) \geq (1-p) \log |A|$ and concludes the proof.

Finally, recall that the erasure channel is degradable for $p\leq 1/2$ [@DS05] and we can thus calculate its unassisted quantum capacity to be $Q({\mathcal{N}}) = \max\{(1-2p) \log |A|, 0\}$. Thus, we have an example of a channel where $Q({\mathcal{N}}) < R({\mathcal{N}})$, which means that our upper bound in terms of the Rains information of the channel is not sufficient to establish the strong converse property for general degradable channels.

Conclusion {#sec:conclusion}
==========

This paper has established that the Rains information of a quantum channel is a strong converse rate for quantum communication. The main application of the first result is to establish the strong converse property for the quantum capacity of all generalized dephasing channels. Going forward from here, there are several questions to consider. First, are there any other channels besides the generalized dephasing ones for which the Rains information is equal to the coherent information? If true, the theorems established here would establish the strong converse property for these channels. For example, can we prove a strong converse theorem for the quantum capacity of general Hadamard channels?

Is it possible to show weak subadditivity of a Rényi coherent information quantity for some class of channels, addressing the original question posed in [@SW12]? To this end, the developments in [@TBH13] might be helpful. Next, is it possible to show that the Rains information of a quantum channel represents a strong converse rate for quantum communication assisted by interactive forward and backward classical communication? Similarly, can one show that the squashed entanglement of a channel [@TGW14] is a strong converse rate for this task? Recent work has proved that the squashed entanglement of a quantum channel is an upper bound on the quantum capacity with interactive forward and backward classical communication, and it could be that the quantities defined in [@BSW14] would be helpful for settling this question.

Finally, now that the strong converse holds for the classical capacity, the quantum capacity, and the entanglement-assisted capacity of all generalized dephasing channels, can we establish that the strong converse holds for trade-off capacities of these channels, in the sense of [@BHTW10; @WH10b]? Can we establish second-order characterizations for this class of channels, in the sense of [@TH12; @TT13; @DL14; @DTW14]?

**Acknowledgements.** We acknowledge many discussions with our colleagues regarding the strong converse for quantum capacity. This includes Mario Berta, Frederic Dupuis, Will Matthews, Ciara Morgan, Naresh Sharma, Stephanie Wehner, and Dong Yang. MT is funded by the Ministry of Education (MOE) and National Research Foundation Singapore, as well as MOE Tier 3 Grant Random numbers from quantum processes (MOE2012-T3-1-009). MMW acknowledges startup funds from the Department of Physics and Astronomy at LSU, support from the NSF through Award No. CCF-1350397, and support from the DARPA Quiness Program through US Army Research Office award W31P4Q-12-1-0019. MMW is also grateful to the quantum information theory group at the Universitat Autònoma de Barcelona and to Stephanie Wehner’s quantum information group at the Centre for Quantum Technologies, National University of Singapore, for hosting him for research visits during May-June 2014. AW acknowledges financial support by the Spanish MINECO, project FIS2008-01236 with the support of FEDER funds, the EC STREP RAQUEL, the ERC Advanced Grant IRQUAT, and the Philip Leverhulme Trust.

Proof of Lemma \[lm:continuity\] {#sec:ogawa-nagaoka}
================================

The first statement follows because the underlying sandwiched relative entropy, $\widetilde{D}_{\alpha}(\rho\|\sigma)$, is monotonically increasing in $\alpha$ [@MDSFT13 Theorem 7] for all $\rho \in \mathcal{S}$ and $\sigma \in \mathcal{P}$, i.e.$$\widetilde{D}_{\alpha}\left(  \rho\Vert\sigma\right)  \geq\widetilde{D}_{\beta}\left(  \rho\Vert\sigma\right)$$ for all $\alpha\geq\beta\geq 0$. This already establishes that the limit exists and satisfies $$\lim_{\alpha \to 1^+} \widetilde{R}_{\alpha}(\mathcal{N}) \geq R(\mathcal{N}) .$$

We would like to show the opposite inequality. Consider the following bound (from [@TCR09 Lemma 8] and [@T12 Lemma 6.3], see also [@WWY13 Eq. (19)]) $$\widetilde{D}_{1+\delta}\left(  \rho\Vert\sigma\right)  \leq D\left(  \rho
\Vert\sigma\right)  +4 \delta \left[  \log\nu\left(
\rho,\sigma\right)  \right]  ^{2},$$ which holds for $\delta\in\left(  0,\log3/\left(  4\log\nu\left(
\rho,\sigma\right)  \right)  \right)  $ where$$\nu\left(  \rho,\sigma\right)  := \operatorname{tr}(\rho^{3/2} \sigma^{- 1/2})
+ \operatorname{tr}(\rho^{1/2} \sigma^{1/2}) +1.$$ Let us for the moment assume that $\sigma > 0$ with $\operatorname{tr}(\sigma) \leq 1$, and its smallest eigenvalue is denoted as $\lambda$. In this case, we can bound $ v(\rho,\sigma) \leq 2 + 1/{\sqrt{\lambda}}$.

For any state $\rho_{RB}$ and $\delta > 0$ sufficiently small, we can apply the bound above to arrive at$$\begin{aligned}
&\min_{\sigma_{RB}\in{{\rm PPT}'}\left(  R:B\right)  }\widetilde{D}_{1+\delta}
\left( \rho_{RB} \Vert\sigma_{RB}\right)  \\
&\qquad \leq \min_{\tau_{RB}\in{{\rm PPT}'}\left(  R:B\right)  }\widetilde{D}_{1+\delta}
\left(  \rho_{RB}  \middle\Vert  (1- \delta) \tau
_{RB} + \delta  \pi_{RB} \right)  \\
&\qquad \leq \min_{\tau_{RB}\in{{\rm PPT}'}\left(  R:B\right)  } D\left(  \rho_{RB}  \middle\Vert  (1-\delta) \tau
_{RB} + \delta \pi_{RB} \right) + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2 \\
& \qquad \leq\min_{\tau_{RB}\in{{\rm PPT}'}\left(  R:B\right)  } D\left(  \rho_{RB}  \middle\Vert  \tau
_{RB} \right) + \log \frac1{1-\delta} + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2 \label{eq:uniform-bound}\end{aligned}$$ The first inequality follows by picking $\sigma_{RB}$ to be of the form $\sigma_{RB} = \left(  1- \delta
\right)  \tau_{RB} + \delta \pi_{RB}$ where $\pi_{RB}$ is the fully mixed state on $RB$. Also, note that $\left(  1-\delta\right)  \tau_{RB} + \delta \pi_{RB}\in T\left(  R:B\right)$. To verify the second inequality, note that the minimum eigenvalue of $\left(  1-\delta\right)  \tau_{RB} + \delta \pi_{RB}$ is always larger than $\delta/(|A| |B|)$. (The system $R$ can be chosen to be of size $|A|$ without loss of generality, where $|A|$ is the dimension of the channel input.) Finally, recall that $D\left(  \rho\Vert\sigma\right)  \leq D\left(  \rho
\Vert\sigma^{\prime}\right)$ whenever $\sigma^{\prime}\leq\sigma$ to verify the last inequality.

Since, crucially, the upper bound in  is uniform in $\rho_{RB}$, we can immediately conclude that $$\widetilde{R}_{1+\delta}(\mathcal{N}) \leq R(\mathcal{N}) + \log \frac1{1-\delta} + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2 \label{eq:nice-bound}$$ by maximizing over channel output states as in the definition of $\widetilde{R}_{1+\delta}$ and $R$. Thus, $\lim_{\alpha \to 1^+} \widetilde{R}_{\alpha}(\mathcal{N}) \leq R(\mathcal{N})$, concluding the proof.

Proof of Corollary \[cor:von-Neumann-subadd\] {#sec:appendix}
=============================================

We can focus only on operators $\tau_{RB}$ such that supp$\left(  \rho
_{RB}\right)  \subseteq\ $supp$\left(  \tau_{RB}\right)  $, where $\rho
_{RB}=\mathcal{N}_{A\rightarrow B}\left(  \phi_{RA}\right)  $. This is because the quantity of interest contains a minimization over all $\tau_{RB}$. For any sufficiently small $\delta>0$$$\begin{aligned}
R\left(  \mathcal{N}^{\otimes n}\right)  \leq\widetilde{R}_{1+\delta
}\left(  \mathcal{N}^{\otimes n}\right)  \leq\frac{1+\delta}{\delta}\left\vert A\right\vert ^{2}\log n+n\widetilde
{R}_{1+\delta}\left(  \mathcal{N}\right).\end{aligned}$$ The first inequality is from the monotonicity of the sandwiched Rényi relative entropy in the Rényi parameter [@MDSFT13 Theorem 7]. The second inequality follows from Theorem \[thm:weak-subadditivity\].

This invites an application of , which gives $$\frac{1}{n} R\left(  \mathcal{N}^{\otimes n}\right) \leq R(\mathcal{N}) + \frac{1+\delta}{\delta}\left\vert A\right\vert ^{2} \frac{\log n}{n} + \log \frac1{1-\delta} + 4 \delta \left( \log \left( 2 + \frac{\sqrt{|A| |B|}}{\sqrt{\delta}} \right) \right)^2$$ Choosing $\delta = 1/\sqrt{n}$ and taking the limit $n \to \infty$ then immediately yields the desired result.

[Marco Tomamichel]{} (M’13) received the M.Sc. in Electrical Engineering and Information Technology degree from ETH Zurich (Switzerland) in 2007. He then graduated with a Ph.D. in Physics at the Institute of Theoretical Physics at ETH Zurich in 2012. From 2012 to 2014 he was a Research Fellow and then Senior Research Fellow at the Centre for Quantum Technologies at the National University of Singapore. Currently he is a Lecturer in the School of Physics at the University of Sydney. His research interests include classical and quantum information theory, specifically finite resource analysis, as well as quantum cryptography.

[Mark M. Wilde]{} (M’99-SM’13) was born in Metairie, Louisiana, USA. He received the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, California, in 2008. He is an Assistant Professor in the Department of Physics and Astronomy and the Center for Computation and Technology at Louisiana State University. His current research interests are in quantum Shannon theory, quantum optical communication, quantum computational complexity theory, and quantum error correction.

[Andreas Winter]{} received a Diploma degree in Mathematics from the Freie Universität Berlin, Berlin, Germany, in 1997, and a Ph.D. degree from the Fakultät für Mathematik, Universität Bielefeld, Bielefeld, Germany, in 1999. He was Research Associate at the University of Bielefeld until 2001, and then with the Department of Computer Science at the University of Bristol, Bristol, UK. In 2003, still with the University of Bristol, he was appointed Lecturer in Mathematics, and in 2006 Professor of Physics of Information. Since 2012 he has been ICREA Research Professor with the Universitat Autònoma de Barcelona, Barcelona, Spain.

[^1]: $^*$ School of Physics, The University of Sydney, Sydney, NSW 2006, Australia. (Email: [marco.tomamichel@sydney.edu.sg]{}).

[^2]: $^\dagger$ Hearne Institute for Theoretical Physics, Department of Physics and Astronomy, Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA.

[^3]: $^\ddagger$ ICREA & Física Teórica, Informació i Fenomens Quántics, Universitat Autònoma de Barcelona, ES-08193 Bellaterra (Barcelona), Spain.

[^4]: However, see the later works [@qcap2008second] and [@qcap2008fourth], which set [@L97] and [@capacity2002shor], respectively, on a firm foundation.

[^5]: All logarithms in this paper are taken base two.

[^6]: It should be clear from the context whether $R$ refers to “Rains” or to a reference system.

[^7]: It is also sufficient for $2\times2$ and $2\times3$ systems, but otherwise only necessary [@Horodecki19961].

[^8]: Note that there is no need to have an asymmetric notation here because a state is PPT with respect to transpose on system $A$ if and only if it is PPT with respect to a partial transpose on system $B$.

[^9]: This generalizes the standard classical scenario in which we are interested in transmitting a uniformly distributed message.

[^10]: The covariance in (\[eq:covariance-dephasing\]) in fact holds for any operators of the form $\sum_{x=0}^{d-1}\exp\left\{  i\varphi_{x}\right\}
    \left\vert x\right\rangle \left\langle x\right\vert _{A}$ and $\sum
    _{x=0}^{d-1}\exp\left\{  i\varphi_{x}\right\}  \left\vert x\right\rangle
    \left\langle x\right\vert _{B}$ with $\varphi_{x}\in\mathbb{R}$, but it suffices to consider only the operators in (\[eq:phase-op-B\]) for our proof here.
