Improving continuous-variable entanglement distribution by separable states
Ladislav Mi\v{s}ta
Jr.
We investigate the physical mechanism behind the counterintuitive phenomenon, the distribution of continuous-variable entanglement between two distant modes by sending a third separable auxiliary mode between them. For this purpose, we propose a new more simple and more efficient protocol resulting in distributed entanglement with more than an order of the magnitude higher logarithmic negativity than in the previously proposed protocol. This new protocol shows that the distributed entanglement originates from the entanglement of one mode and the auxiliary mode used for distribution which is first destroyed by local correlated noises and restored subsequently by the interference of the auxiliary mode with the second distant separable correlated mode.
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\begin{document}
\title{Improving continuous-variable entanglement distribution by separable states}
\author{Ladislav Mi\v{s}ta, Jr.}
\affiliation{Department of Optics, Palack\' y University, 17.
listopadu 50,  772~07 Olomouc, Czech Republic}
\author{Natalia Korolkova}
\affiliation{School of Physics and Astronomy, University of St.
Andrews, North Haugh, St. Andrews, Fife, KY16 9SS, Scotland}

\date{\today}


\begin{abstract}
We investigate the physical mechanism behind the counterintuitive
phenomenon, the distribution of continuous-variable entanglement
between two distant modes by sending a third separable auxiliary
mode between them. For this purpose, we propose a new more simple
and more efficient protocol resulting in distributed entanglement
with more than an order of the magnitude higher logarithmic negativity than in the
previously proposed protocol. This new protocol shows that the
distributed entanglement originates from the entanglement of one
mode and the auxiliary mode used for distribution which is first
destroyed by local correlated noises and restored subsequently
by the interference of the auxiliary mode with the second distant
separable correlated mode.
\end{abstract}
\pacs{03.67.-a}


\maketitle
\section{Introduction}

Quantum entanglement is a quintessence of quantum mechanics that
nowadays finds applications in the field of quantum communication
and information processing. Entanglement assisted communication
requires the communicating parties, Alice and Bob, to establish an
entangled state between their systems $a$ and $b$. This can be
done {\it only} by using a global operation, e.g. sending a
quantum system $c$ between them, as entanglement cannot be created
by local operations and classical communication (LOCC)
\cite{Werner_89}. This, however, does not necessarily imply that
the system $c$ must be entangled with the other two systems as
everybody would intuitively expect. Namely, for the quantum mixed
states only classical correlations and quantum interference
suffice to create entanglement \cite{Cubitt_03}. Specifically, for
mixed states one can establish a distillable entanglement between
two qubits $a$ and $b$ by using a third qubit $c$ being always
separable from the two-qubit system $(ab)$ \cite{Cubitt_03}. This
phenomenon has a number of implications for quantum communication,
which still have to be fully explored. For instance, quantum
teleportation \cite{ Bennett_93} does not require transmission of
an entangled quantum system between the sender and the receiver.

Distribution of entanglement without sending entanglement was
first suggested for qubits \cite{Cubitt_03} and only recently, an
alternative protocol for entanglement distribution by separable
states was proposed for Gaussian states of infinitely-dimensional
quantum systems and quantum variables with continuous
spectra-continuous variables (CVs) \cite{Mista_08}.

In this paper we study how the CV protocol works.
To that aim we propose a new protocol distributing entanglement with
more than an order of the magnitude higher logarithmic negativity
than the previous one \cite{Mista_08}. The proposed
protocol is more simple and allows to fully explain the physical mechanism underlying
the phenomenon of CV entanglement distribution by separable states.

The paper is organized as follows. Section~\ref{sec_1} contains a
brief introduction into the formalism of Gaussian states. In
Sec.~\ref{sec_2} we remind the original scheme for distribution of
CV entanglement by separable Gaussian states. Section \ref{sec_3}
is dedicated to the proposal of a new improved scheme for
establishing entanglement using separable ancilla. In
Sec.~\ref{sec_4} we explain the physics behind the distribution of
CV entanglement by separable states. Section~\ref{sec_5} covers
conclusion and comments on the experimental feasibility of the
protocol.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Gaussian states}\label{sec_1}

We consider three distinguishable infinite-dimensional quantum
systems $A,B$ and $C$, referred to as ``mode $A$,'' ``mode $B$,''
and ``mode $C$.'' The modes are described by three pairs of
canonically conjugate quadrature operators $x_{j},p_{j}$,
$j=A,B,C$ satisfying the canonical commutation rules
$[x_{j},x_{k}]=[p_{j},p_{k}]=0$, $[x_{j},p_{k}]=i\delta_{jk}$.
Defining the column vector
$\xi=(x_{A},p_{A},x_{B},p_{B},x_{C},p_{C})^{T}$ the
commutation rules can be written in the compact form
$[\xi_{j},\xi_{k}]=-i\Omega_{jk}$, where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{Omega}
\Omega=J\oplus J\oplus J,\quad J=\left(\begin{array}{cc}
0 & -1 \\
1 & 0 \\
\end{array}\right)
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
is the standard three-mode symplectic matrix. All states of our
three-mode system can be represented in the real six-dimensional
phase space by the Wigner function \cite{Wigner_32} and Gaussian states are defined
as those states for which the Wigner function is Gaussian. Any
three-mode Gaussian state $\rho$ can be therefore fully
characterized by the real six-dimensional vector
$\langle\xi\rangle=\mbox{Tr}(\rho\xi)$ of phase-space displacements and by the
$6\times 6$ real symmetric covariance matrix (CM) $\gamma$ with
elements $\gamma_{jk}=\mbox{Tr}\left(\rho\{\xi_{j}-\langle\xi\rangle_{j}\openone,\xi_{k}-\langle\xi\rangle_{k}\openone\}\right)$,
$j,k=1,\ldots,6$, where $\{A,B\}\equiv AB+BA$. Due to the commutation rules the CM $\gamma$
satisfies the uncertainty relation \cite{Simon_87}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{uncertainty}
\gamma-i\Omega\geq0.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the following section we are interested in separability
properties of three-mode Gaussian states and therefore we need a
separability criterion for all possible $1\times 2$-mode
bipartitions, i.e. bipartitions  $A-(BC)$, $B-(AC)$ and $C-(AB)$. For
this purpose we can use the positive partial transpose (PPT)
criterion \cite{Peres_96,Horodecki_96} translated to Gaussian
states \cite{Duan_00,Simon_00,Werner_01,Giedke_01}. On the level
of CMs the partial transposition with respect to the mode $x$ transforms
the CM $\gamma$ to $\gamma^{(T_x)}=\Lambda_{x}\gamma\Lambda_{x}$
where $\Lambda_{x}$, $x=A,B,C$ are the diagonal matrices
$\Lambda_{A}=\sigma_{z}\oplus\openone\oplus\openone$,
$\Lambda_{B}=\openone\oplus\sigma_{z}\oplus\openone$,
$\Lambda_{C}=\openone\oplus\openone\oplus\sigma_{z}$, where
$\sigma_{z}$ is a Pauli diagonal matrix
$\sigma_{z}=\mbox{diag}(1,-1)$ and $\openone$ is a $2\times 2$
identity matrix. The PPT criterion then says that a three-mode
Gaussian state with CM $\gamma$ is separable with respect to
bipartition $x-(yz)$ (where $(x,y,z)$ is an even permutation of
$A,B,C$) if and only if the matrix $\gamma^{(T_x)}$ satisfies the
uncertainty relation (\ref{uncertainty}) \cite{Giedke_01}, i.e.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{criterion1}
\gamma^{(T_x)}-i\Omega\geq0.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The PPT criterion can be expressed in another equivalent form that
can be sometimes easier to use in practice. It relies on the fact
that for any matrix $\gamma^{(T_x)}$ of a partially transposed state
there exists a symplectic matrix $S$, i.e. a real $6\times 6$
matrix satisfying the condition $S\Omega S^{T}=\Omega$, such
that $S\gamma^{(T_x)}S^{T}=\mbox{diag}(s_1,s_1,s_2,s_2,s_3,s_3)$
\cite{Williamson_36}. The nonnegative quantities $s_i$, $i=1,2,3$
are the so called symplectic eigenvalues of $\gamma^{(T_x)}$ and
$\gamma$ is separable with respect to the splitting $x-(yz)$ if and only if
$s_{i}\geq1$ $\forall i$ \cite{Vidal_02}. The symplectic eigenvalues can be
computed via the eigenvalues of the matrix $\Omega\gamma^{(T_x)}$ which are
$\left\{\pm is_1,\pm is_{2},\pm is_{3}\right\}$. For
complex three-mode states the symplectic eigenvalues are involved and it
is easier to use the PPT criterion formulated in terms of
the symplectic invariants \cite{Serafini_06}. The matrix $\gamma^{(T_x)}$ possesses
three such invariants denoted $I_1,I_2$ and $I_3=\mbox{det}(\gamma^{(T_{x})})$ that
can be calculated easily as coefficients of the characteristic
polynomial of the matrix $\Omega\gamma^{(T_{x})}$, i.e.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{polynomial}
\mbox{det}(\Omega\gamma^{(T_{x})}-\mu\openone)=\mu^{6}+I_1\mu^{4}+I_2\mu^{2}+I_3.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The criterion then says that for CM $\gamma$ mode $x$ is
separable from modes $(yz)$ if and only if
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{Sigma}
\Sigma=\prod_{j=1}^{3}(s_{j}^2-1)=I_3-I_2+I_1-1\geq0
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
holds. Strictly speaking, the criterion is not sufficient for
separability of states having some symplectic eigenvalue equal to
unity \cite{Serafini_06}. However, the sharp inequality $\Sigma>0$
is already sufficient for separability.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The original  protocol}\label{sec_2}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
%\centerline{\psfig{width=6.5cm,angle=0,file=fig1-NK.eps}}
\centerline{\psfig{width=7.5cm,angle=0,file=fig1.eps}}
\caption{Two schemes of the protocol for distribution of CV
entanglement by separable Gaussian states. In the original scheme \cite{Mista_08}
(empty circle and ellipses) Alice's mode $A$ and Bob's distant mode $B$
are prepared in suitable rotated squeezed vacuum states while mode $C$ is
hold by Alice and it is in a vacuum state. In the improved scheme
of section \ref{sec_3} (hatched circle and ellipses) modes $A$ and $C$ hold by Alice are
in the momentum and position squeezed vacuum states, respectively, and Bob's mode $B$
is in a vacuum state. All the three modes are then displaced by displacements
$D_{A},D_{B}$ and $D_{C}$ distributed randomly with Gaussian distribution
with covariance matrix $Q$ ($\tilde{Q}$ for the improved protocol) after
which the modes are in a fully separable state (step 1). Mixing of modes $A$ and $C$ on a
balanced beam splitter $BS_{AC}$ entangles mode $A$ with the pair
of modes $(BC)$ while mode $B$ is separable from $(AC)$ and mode $C$
is separable from $(AB)$ (step 2). Mixing of modes $B$ and $C$ on a balanced
beam splitter $BS_{BC}$ finally entangles $A$ and $B$ wile $C$
still remains separable from $(AB)$ (step 3). See text for details.}
\label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The original protocol for distribution
of CV entanglement by separable Gaussian states \cite{Mista_08} is
depicted in Fig.~\ref{fig1} (see figure caption for details). For this protocol the contours of
the phase-space distributions of the input modes are shown as
empty circle and ellipses. The essential ingredient of the protocol is  a
fully separable three-mode Gaussian state $\rho_{ABC}$ that
transforms appropriately under beam splitter operations on
certain pairs of modes. More precisely, $\rho_{ABC}$ transforms under
$BS_{AC}$ operation on modes $A$
and $C$ onto a state separable with respect to two bipartitions
$B-(AC)$ and $C-(AB)$ (two-mode biseparable state \cite{Giedke_01}). This
state further transforms under $BS_{BC}$ operation on modes $B$ and
$C$ onto the state separable with respect to only one bipartition
$C-(AB)$ (one-mode biseparable state
\cite{Giedke_01}). The construction of such a state is therefore
the key nontrivial task that has to be solved in order to find the
protocol. The requirements of the protocol are counterintuitive:
by using LOCC and sending of a separable ancilla from Alice to Bob
to create entanglement between them. As no entanglement is shared
beforehand or sent by Alice to Bob, one can ask a natural
question where the entanglement in the protocol comes from.
In this original protocol,
there is no a straightforward simple explanation for the emergence of entanglement, in particular due to the complex structure of the quantum states involved.
For instance, the CMs occurring in the protocol possess correlations between position and
momentum quadratures and it is fairly possible that this kind of correlations is responsible for
the phenomenon. Similarly, it is not obvious which role is played by the squeezed state on
Bob's side and whether it has necessarily to be squeezed for the entanglement distribution to occur.

In the following section we show that neither of these properties accounts for the phenomenon.
We construct a more simple protocol where all the participating states have zero correlations
between position and momentum quadratures and where Bob's mode is initially in the vacuum state
(which is a completely classical state). Further, the new protocol exhibits a substantially better
performance than the original one as it establishes entanglement between modes $A$ and $B$ with
more than an order of the magnitude higher logarithmic negativity. Finally, due to its simplicity the protocol gives a clear physical
explanation of the counterintuitive phenomenon of entanglement distribution by separable states.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The improved protocol}\label{sec_3}

In contrast to the original protocol \cite{Mista_08} where the fully separable state of step 1 was constructed first,
we start with designing the state in step 2. That is, by local operations on a pair
of modes $(AC)$ and mode $B$ and classical communication we construct a three-mode mixed Gaussian state
in which mode $A$ is entangled with modes $(BC)$, mode $C$ is separable from modes $(AB)$ and mode $B$ is separable
from modes $(AC)$. Initially, we assume mode $B$ in the vacuum state and modes $A$ and $C$ in the two-mode squeezed
vacuum state. The corresponding CMs read as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gammaACgammaB}
\gamma_{0,AC}=\left(\begin{array}{cc}
\cosh(2t)\openone & \sinh(2t)\sigma_{z} \\
\sinh(2t)\sigma_{z} & \cosh(2t)\openone\\
\end{array}\right),\quad \gamma_{0,B}=\openone,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $t\geq0$ is the squeezing parameter. The matrix $\gamma_{0,AC}^{(T_{C})}$ has
 the lower symplectic eigenvalue equal to $\nu_{\rm in}=e^{-2t}<1$ for $t>0$ and thus modes
 $A$ and $C$ are entangled. Now we take the CM of the entire
 three-mode system,
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gammaABC}
\gamma_{ABC}=\left(\begin{array}{ccc}
\cosh(2t)\openone & 0 & \sinh(2t)\sigma_{z} \\
0 & \openone & 0 \\
\sinh(2t)\sigma_{z} & 0 & \cosh(2t)\openone\\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 and we add to it a nonnegative multiple of a suitable positive noise matrix
 $P$ such that the state described by CM
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \begin{equation}\label{tildegamma2a}
 \gamma_{2}=\gamma_{ABC}+xP, \qquad x\geq0,
 \end{equation}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 possesses entanglement between mode $A$ and modes $(BC)$ while it is separable
 with respect to the other two bipartitions. The matrix $P$ can be found using the method
 developed in \cite{Giedke_01} for construction of various three-mode entangled Gaussian states.
 It utilizes a two-mode version of the separability criterion represented by Eq.~(\ref{criterion1})
 according to which a two-mode CM $\gamma_{0,AC}$ is entangled if and only if the matrix
$\gamma_{0,AC}-iJ\oplus(-J)$ has a negative eigenvalue. By adding to the CM a sufficiently large
positive multiple of sum of the projectors onto the subspaces spanned by real and imaginary parts of the eigenvector
corresponding to the negative eigenvalue we can destroy the entanglement between modes $A$ and $C$. Extending
now the projectors suitably and adding a multiple of sum of the extensions to the CM $\gamma_{ABC}$ we can finally
construct a state with desired separability properties.
 The negative eigenvalue is easy to find and it reads as $\lambda=-(1-e^{-2t})<0$ for $t>0$ and the corresponding eigenvector is
 $p_{\lambda}=p_{1}+ip_{2}$, where $p_{1}=(0,-1,0,-1)^{T}$ and
 $p_{2}=(1,0,-1,0)^{T}$. The sought noise matrix then can be constructed from the following $6\times1$
 extensions of the vectors $p_{1}$ and $p_{2}$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{tildeq}
q_{1}&=&(0,-1,0,2,0,-1)^{T},\nonumber\\
q_{2}&=&(1,0,2,0,-1,0)^{T},
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
as $P=q_{1}q_{1}^{T}+q_{2}q_{2}^{T}$. The matrix
is by construction positive and the addition of its sufficiently large nonnegative multiple $xP$
to the CM (\ref{gammaABC}) smears the initial entanglement between modes $A$ and $C$
such that $A$ is entangled with $(BC)$ yet $C$ is separable from $(AB)$ and, furthermore, $B$ is separable from $(AC)$.
The separability of mode $B$ from the pair of modes $(AC)$ is obvious as the CM (\ref{tildegamma2a}) can be created
from the product state with CM $\gamma_{0,AC}\oplus\gamma_{0,B}$ by local random correlated displacements distributed
with Gaussian distribution described by the correlation matrix $xP$. The threshold value $x_{\rm sep}$
of $x$ such that for $x\geq x_{\rm sep}$ the state with CM (\ref{tildegamma2a}) is separable with respect
to $C-(AB)$ splitting can be found from the requirement of positivity of the partial transpose of the
state described by CM (\ref{tildegamma2a}) with respect to mode $C$ that is sufficient
for separability of the $1\times2$ system \cite{Werner_01}. The state has a positive partial transpose
if and only if the symplectic eigenvalues $\tau_{j}$, $j=1,2,3$,
of the matrix $\gamma_{2}^{(T_C)}$ satisfy $\tau_{j}\geq1$ for all $j$ \cite{Vidal_02}.
The lowest symplectic eigenvalue reads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{nu}
\tau_{3}=\frac{\sqrt{\left(1+6x+e^{-2t}\right)^{2}-32x^2}-\left(1+2x-e^{-2t}\right)}{2}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and the threshold value $x_{\rm sep}$ can be calculated from the condition $\tau_{3}=1$. After
some algebra one can show that the CM (\ref{tildegamma2a})
represents a state that is separable with respect to $C-(AB)$
splitting if $x\geq x_{\rm sep}$, where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{tildexsep}
x_{\rm sep}=\frac{e^{2t}-1}{2}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Entanglement of mode $A$ with a pair of modes $(BC)$ can be verified along the same lines. The lowest symplectic
eigenvalue of the matrix $\gamma_{2}^{(T_{A})}$ attains the form:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{omega}
\omega_{3}=\frac{1+6x+e^{-2t}-\sqrt{\left(1+2x-e^{-2t}\right)^{2}+32x^2}}{2}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and for $x>0$ and $t>0$ it satisfies the inequality $\omega_{3}<1$ which proves the entanglement between $A$ and $(BC)$.
Note that this entanglement is essential for the performance of the studied protocol as the second beam splitter $BS_{BC}$
alone cannot entangle mode $A$ with mode $B$ without $A$ being entangled with $(BC)$.

Having found the desired CM of step 2 we can return back to step 1 by implementing the inverse
of the balanced $BS_{AC}$ on modes $A$ and $C$. The beam splitter is described by the matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{UAC}
U_{AC}=\left(\begin{array}{ccc}
\frac{1}{\sqrt{2}}\openone & 0 & \frac{1}{\sqrt{2}}\openone \\
0 & \openone & 0\\
\frac{1}{\sqrt{2}}\openone& 0 & -\frac{1}{\sqrt{2}}\openone \\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and the inverse operation $U_{AC}^{-1}=U_{AC}^{T}=U_{AC}$ transforms
the CM (\ref{tildegamma2a}) as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{tildegamma1}
\gamma_{1}=U_{AC}^{T}\gamma_{2}U_{AC}=\gamma_{A}\oplus\gamma_{B}\oplus\gamma_{C}+\tilde{Q},
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the CM $\gamma_{B}=\gamma_{0,B}=\openone$ is the vacuum CM and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{product}
\gamma_{A,C}=\left(\begin{array}{cc}
e^{\pm2t} & 0 \\
0 & e^{\mp2t} \\
\end{array}\right)
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
are CMs of the momentum and position squeezed vacuum states, respectively. The structure of CM (\ref{tildegamma1})
reveals that the state of step 1 can be created from the product of two orthogonally squeezed states
and a vacuum state by local random displacements distributed according to the Gaussian distribution with
correlation matrix $\tilde{Q}=xU_{AC}^{T}PU_{AC}$ that has the following form:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{tildeQ}
\tilde{Q}=x\left(\begin{array}{ccc}
\openone-\sigma_{z} & \sqrt{2}(\sigma_{z}-\openone) & 0 \\
\sqrt{2}(\sigma_{z}-\openone) & 4\cdot\openone & \sqrt{2}(\sigma_{z}+\openone) \\
0 & \sqrt{2}(\sigma_{z}+\openone) & \sigma_{z}+\openone\\
\end{array}\right).
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This implies immediately that the state is fully separable as is the basic requirement of our protocol.

The improved protocol for
distribution of CV entanglement by separable
Gaussian states is depicted in Fig.~\ref{fig1}. The phase-space
contours of the input modes for this protocol are represented by
hatched circle and ellipses. As follows from the CM
$\gamma_{1}$ designed in Eqs. (\ref{tildegamma1}) and
(\ref{product}), the input modes $A$ and $C$ are both in a squeezed
state with reduced noise in two orthogonal quadratures
respectively and mode $B$ is in a vacuum state. In the step 1
Alice holding a pair of modes $(AC)$ and Bob holding mode $B$
prepare by LOCC a Gaussian state characterized by the CM
(\ref{tildegamma1}). In the next step 2, Alice superimposes mode
$A$ and $C$ on a balanced beam splitter $BS_{AC}$. Thus she
creates a state with CM (\ref{tildegamma2a}) that reads explicitly
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{tildegamma2b}
\gamma_{2}=\left(\begin{array}{ccc}
a\openone & 2x\sigma_{z} & b\sigma_{z} \\
2x\sigma_{z} & (1+4x)\openone & -2x\openone \\
b\sigma_{z} & -2x\openone & a\openone\\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $a=\cosh(2t)+x$ and $b=\sinh(2t)-x$. As was shown above, the state is separable with
respect to $B-(AC)$ splitting and for $x\geq x_{\rm sep}$, where $x_{\rm sep}$ is
given in Eq.~(\ref{tildexsep}), also with respect to $C-(AB)$ splitting.

The protocol is finalized by step 3 in which Bob mixes the mode $C$ received from Alice
with his mode $B$ on another balanced beam splitter $BS_{BC}$ described by the matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{UBC}
U_{BC}=\left(\begin{array}{ccc}
\openone & 0 & 0\\
0 & \frac{1}{\sqrt{2}}\openone &  \frac{1}{\sqrt{2}}\openone\\
0 & \frac{1}{\sqrt{2}}\openone&  -\frac{1}{\sqrt{2}}\openone\\
\end{array}\right).
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The beam splitter transforms
the CM (\ref{tildegamma2b}) to CM $\gamma_{3}=U_{BC}\gamma_{2}U_{BC}^{T}$ that reads explicitly as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{tildegamma3}
\gamma_{3}=\left(\begin{array}{ccc}
a\openone & \frac{2x+b}{\sqrt{2}}\sigma_{z} & \frac{2x-b}{\sqrt{2}}\sigma_{z} \\
\frac{2x+b}{\sqrt{2}}\sigma_{z} & \frac{1+a}{2}\openone & \frac{1+4x-a}{2}\openone \\
\frac{2x-b}{\sqrt{2}}\sigma_{z} & \frac{1+4x-a}{2}\openone & \frac{1+8x+a}{2}\openone\\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $a$ and $b$ are defined as in Eq.~(\ref{tildegamma2b}). The interference on the beam splitter $BS_{BC}$
finally entangles modes $A$ and $B$. This can be seen by computing the lower symplectic eigenvalue
$\nu$ of the matrix $\gamma_{3,AB}^{(T_{B})}$ corresponding to the reduced state of modes $A$ and $B$.
We express the CM $\gamma_{3,AB}$ in the block form
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{block}
\gamma_{3,AB}=\left(\begin{array}{cc}
A & C \\
C^{T} & B \\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $A,B$ and $C$ are $2\times2$ submatrices.  The eigenvalue then reads as \cite{Vidal_02}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{ABsymplectic}
\nu=\sqrt{\frac{\kappa-\sqrt{\kappa^2-4\mbox{det}(\gamma_{3,AB})}}{2}},
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{ABinvariants}
\mbox{det}\left(\gamma_{3,AB}\right)&=&\left[\frac{1+\cosh\left(2t\right)}{2}+\left(e^{-2t}+\frac{1}{2}\right)x\right]^{2},\nonumber\\
\kappa&=&\mbox{det}A+\mbox{det}B-2\mbox{det}C\nonumber\\
&=&a^2+\left(b+2x\right)^2+\frac{\left(a+1\right)^2}{4}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and it can be used to quantify the amount of distributed
entanglement by calculating the logarithmic negativity given by
the formula $E_{\mathcal{N}}=-\log_{2}\nu$ \cite{Vidal_02}. For
$t>0$ and $x=x_{\rm sep}=(e^{2t}-1)/2$ we get $\nu<1$ and
therefore modes $A$ and $B$ are entangled for an arbitrarily small
nonzero squeezing. Numerical analysis further reveals that the
symplectic eigenvalue decreases monotonously with increasing
squeezing and approaches the value of $\nu=1/3$ corresponding to
$E_{\mathcal{N}}\approx1.585$ ebits in the limit of infinite
squeezing  while all the separability properties of the states
involved remain preserved. To illustrate this, we have calculated
the output entanglement for some realistic values of squeezing
using Eqs.~(\ref{ABsymplectic}) and (\ref{ABinvariants}). Thus
$e^{2t}=2$ corresponding to 3dB of input squeezing results in the
symplectic eigenvalue of $\nu\approx0.6589$ which gives the
logarithmic negativity of $E_{\cal N}\approx0.6019$ ebits. For
$e^{2t}=10$ corresponding to 10dB squeezing that was recently
achieved experimentally \cite{Schnabel_08} we get
$\nu\approx0.3968$ and $E_{\cal N}\approx1.3334$ ebits.

It is interesting to compare the latter logarithmic negativity with
the maximum logarithmic negativity that can be obtained with
the original protocol \cite{Mista_08}. Numerical optimization of the original
protocol with respect to the noise parameter $x$ and the initial
squeezings (denoted by $d\pm r$ in \cite{Mista_08}) reveals that
the minimal lower symplectic eigenvalue of the partially
transposed final entangled state distributed between Alice and Bob
reads as $\nu\approx0.9299$ and the corresponding logarithmic
negativity is $E_{\cal N}\approx0.1048$ ebits.
Thus the logarithmic negativity of the entangled state created
between distant locations in the improved protocol is more than an order of
the magnitude higher than in the original protocol. Furthermore, also in this
improved protocol the mode $C$ remains separable from modes $(AB)$
after the step 3. This can be shown using the separability criterion based on symplectic
invariants \cite{Serafini_06}. For example, for $e^{2t}=2$ the criterion gives
$\Sigma=1>0$ which implies separability of mode $C$ from $(AB)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Explanation of the protocol}\label{sec_4}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
\centerline{\psfig{width=5.5cm,angle=0,file=fig2.eps}}
\caption{Scheme of the protocol for recovery of the entanglement between
modes $A$ and $C$. Mode $A$ in a suitable momentum-squeezed vacuum
state and mode $C$ in a position-squeezed vacuum state with the same level
of squeezing are displaced by displacements $D_{A}$ and $D_{C}$ distributed randomly
with Gaussian distribution characterized by the covariance matrix $\tilde{Q}$ (\ref{tildeQ}).
Next, the two modes are mixed on a balanced beam splitter $BS_{AC}$ that prepares
them in a two-mode separable state. Classical information traveling to Bob
is then used for displacement $D_{B}^{(G)}$ of mode $C$ with a suitable electronic gain $G$.
Provided that the gain is adjusted appropriately, the entanglement between modes $A$ and $C$ can be
perfectly recovered. See text for details.}
\label{fig3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The previous sections reveal that there indeed exist three-mode
mixed Gaussian fully separable states allowing to entangle two
modes of the state by mixing them stepwise with the third always
separable mode on two beam splitters. Therefore, once such a
tripartite state is established between distant Alice and Bob,
they can create, without need of any additional knowledge,
entanglement between them by sending a separable mode from one
site to another remote location. There is, however, another
interesting aspect in this protocol. When looking at the scheme in
Fig.~\ref{fig1}, one can imagine a different scenario when Bob has
some additional information about the preparation of the shared
state. Specifically, suppose that in each run of the protocol he
knows classical displacements $\bar{x}_{B},\bar{p}_{B}$ used to
displace his vacuum mode $B$. The natural question arises whether
Alice and Bob can then create a higher entanglement than when
using solely the shared fully separable state. In what follows we
give an affirmative answer to this question by showing that they
can in fact restore the whole amount of entanglement contained in
the two-mode squeezed vacuum state with CM $\gamma_{0,AC}$
defined in Eq.~(\ref{gammaACgammaB}). Moreover, here we provide an
explanation where the entanglement comes from in the improved protocol
depicted in Fig.~\ref{fig1}.

Let us consider the scheme in Fig.~\ref{fig3}. There, the
situation on Alice's side remains the same as in the protocol in
Fig.~\ref{fig1}. Bob, however, does not use an auxiliary vacuum
mode $B$ appropriately displaced with $D_{\rm B}$ to achieve the
desired final CM. Instead, he directly displaces mode $C$
received from Alice with a suitable electronic gain $G$. The CM of
the resulting state is most easily calculated in Heisenberg
picture. Let us denote by $x_{A}^{(0)}$, $p_{A}^{(0)}$,
$x_{C}^{(0)}$ and $p_{C}^{(0)}$ position and momentum quadratures
of initially vacuum modes $A$ and $C$, respectively. Arranging the
quadratures of the squeezed vacuum modes $A$ and $C$ into the
column vectors
$\xi_{A}=(x_{A},p_{A})^{T}=(e^{t}x_{A}^{(0)},e^{-t}p_{A}^{(0)})^{T}$
and
$\xi_{C}=(x_{C},p_{C})^{T}=(e^{-t}x_{C}^{(0)},e^{t}p_{C}^{(0)})^{T}$,
the quadratures of modes $A$ and $C$ after the displacements
$D_{A}$ and $D_{C}$ read $\xi_{A}+\bar{\xi}_{A}$ and
$\xi_{C}+\bar{\xi}_{C}$, where the vectors
$\bar{\xi}_{A}=(\bar{x}_{A},\bar{p}_{A})^{T}$ and
$\bar{\xi}_{C}=(\bar{x}_{C},\bar{p}_{C})^{T}$ define classical
displacements. Next, the two modes are mixed on the balanced beam
splitter $BS_{AC}$. Further, mode $C$ undergoes the displacement
$D_{B}^{(G)}$ defined by the displacement vector
$\bar{\xi}_{B}=(\bar{x}_{B},\bar{p}_{B})^{T}$ of mode $B$ and
$2\times2$ gain matrix $G$. This yields finally the transformed
quadratures in the following form:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{transform}
\xi_{A}''&=&\frac{\xi_{A}+\bar{\xi}_{A}+\xi_{C}+\bar{\xi}_{C}}{\sqrt{2}},\nonumber\\
\xi_{C}''&=&\xi_{C}'+G\bar{\xi}_{B}=\frac{\xi_{A}+\bar{\xi}_{A}-\xi_{C}-\bar{\xi}_{C}}{\sqrt{2}}+G\bar{\xi}_{B}.\nonumber\\
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here $\xi_{C}'$ is the vector of quadratures of mode $C$ behind
the beam splitter $BS_{AC}$. To determine the final state of a
two-mode subsystem $(AC)$, one has first to calculate the
correlations
$A_{ij}=\langle\{\xi_{A,i}'',\xi_{A,j}''\}\rangle$,
$C_{ij}=\langle\{\xi_{C,i}'',\xi_{C,j}''\}\rangle$ for
modes $A$ and $C$, respectively, and the intermodal correlations
$\Gamma_{ij}=\langle\{\xi_{A,i}'',\xi_{C,j}''\}\rangle$,
$i,j=1,2$, where the angle brackets denote averaging of the quantum quadrature
operators over the quantum state with CM
$\gamma_{A}\oplus\openone_{B}\oplus\gamma_{C}$,
where $\gamma_{A,C}$ are defined in Eq.~(\ref{product}) and the
classical displacements are averaged over the Gaussian distribution with covariance (\ref{tildeQ}).
Using Eqs.~(\ref{transform}), CMs (\ref{product}) and
the correlation matrix (\ref{tildeQ}) and taking into account the
fact that the vectors of quantum quadratures $\xi_{A},\xi_{C}$ are
uncorrelated with the vectors of classical displacements
$\bar{\xi}_{A},\bar{\xi}_{B}$ and $\bar{\xi}_{C}$, we arrive at
the CM of the state of modes $A$ and $C$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gamma2ACa}
\gamma_{2,AC}^{(G)}=\left(\begin{array}{cc}
A & \Gamma \\
\Gamma^{T} & C \\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{tildeACGamma}
A&=&a\openone,\quad\Gamma=\sigma_{z}(b\openone+2xG^{T}),\nonumber\\
C&=&a\openone+4xGG^{T}-2x(G+G^{T}),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the parameters $a$ and $b$ are defined below Eq.(\ref{tildegamma2b}).
The structure of the CM indicates the optimal choice of the gain matrix $G$. We see that by choosing
$G=G_{\rm opt}=\openone$ the noise in the diagonal block $C$ cancels and the CM transforms to
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gamma2ACb}
\gamma_{2,AC}^{(G_{\rm opt})}=\left(\begin{array}{cc}
a\openone & (b+2x)\sigma_{z} \\
(b+2x)\sigma_{z} & a\openone\\
\end{array}\right).
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To characterize the separability properties of CM
$\gamma_{2,AC}^{(G_{\rm opt})}$, it remains to calculate
the lower symplectic eigenvalue $\nu_{AC}$ for the matrix
$\left(\gamma_{2,AC}^{(G_{\rm opt})}\right)^{(T_{C})}$.
Using Eq.~(\ref{ABsymplectic}) one finds that it attains the
simple form $\nu_{AC}=a-b-2x=e^{-2t}$. Hence we
immediately see that the eigenvalue coincides with the lower
symplectic eigenvalue $\nu_{\rm in}=e^{-2t}$ of the
partially transposed pure two-mode squeezed vacuum state with CM
(\ref{gammaACgammaB}) that can be prepared by Alice from
her pure squeezed states. Since any Gaussian operation on mode $C$
cannot increase the entanglement of the state with CM (\ref{gamma2ACb})
\cite{Eisert_02} we see that Bob's displacement on mode $C$ with
the gain matrix $G_{\rm opt}$ is indeed an optimal Gaussian operation.
It is worth mentioning that while the entanglement of the two-mode squeezed
vacuum state figuring in the protocol can be restored perfectly,
the purity of the state cannot because
$\mbox{det}(\gamma_{2,AC}^{(G_{\rm
opt})})=\left(1+2xe^{-2t}\right)^{2}$.

Previous analysis reveals the origin of the entanglement in the
improved protocol in Fig.~\ref{fig1}. The displacements $D_{A}$ and
$D_{C}$ before the beam splitter $BS_{AC}$ can be transformed to
another displacements $D_{A}'$ and $D_{C}'$ behind the beam
splitter. Consequently, the state of modes $A$ and $C$ behind the
beam splitter can be viewed as a pure two-mode squeezed vacuum
state with CM (\ref{gammaACgammaB}). The entanglement and purity
of this state is then subsequently destroyed by local correlated
displacements $D_{A}'$ and $D_{C}'$. The classical information on
displacements $\bar{x}_{B}$ and $\bar{p}_{B}$ traveling to Bob,
that as we have seen allows him to perfectly restore the
entanglement between modes $A$ and $C$ if he uses the protocol of
Fig.~\ref{fig3}, is instead used to displace his vacuum mode $B$.
Mixing of the displaced vacuum mode $B$ with mode $C$ on the
second balanced beam splitter $BS_{BC}$ leads finally to the
following vector of Bob's quadratures:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{transformb}
\xi_{B,{\rm out}}=\frac{\xi_{C}'+\bar{\xi}_{B}+\xi_{B}^{(0)}}{\sqrt{2}},
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\xi_{B}^{(0)}=(x_{B}^{(0)},p_{B}^{(0)})^{T}$ is the vector of Bob's vacuum quadratures and $\xi_{C}'$ denotes
the vector of quadratures of mode $C$ behind the beam splitter $BS_{AC}$. Compare the last equation with
the equation for the vector of optimal quadratures $\xi_{C}''=\xi_{C}'+\bar{\xi}_{B}$
obtained by setting $G=G_{\rm opt}=\openone$ in equation for $\xi_{C}''$ in Eqs.~(\ref{transform}).
We find out that the vector of output Bob's quadratures
(\ref{transformb}) is obtained from the vector of optimal quadratures by mixing it with the vector of vacuum
quadratures of Bob's mode on a balanced beam splitter described by Eq.~(\ref{UBC}).
Thus, mixing of mode $C$ with mode $B$ on a beam splitter $BS_{BC}$ in
Fig.~\ref{fig1} implements effectively optimal displacement of mode $C$ restoring perfectly entanglement between modes $A$ and $C$ followed
by mixing of the mode with the vacuum mode on a balanced beam splitter which reduces the amount of the entanglement. This finally
explains the origin of the distributed entanglement in the improved protocol as well as the fact that it is weaker in comparison
with the maximum entanglement that can be restored by Bob using the scheme in Fig.~\ref{fig3} with optimal gain matrix $G_{\rm opt}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusion}\label{sec_5}
In conclusion, we performed a detailed investigation of the origin of entanglement in
the protocol for entanglement distribution by separable Gaussian states \cite{Mista_08}.
For this purpose we proposed a more simple and efficient protocol that enables to distribute
entanglement with more than an order of the magnitude higher logarithmic negativity than the older protocol
\cite{Mista_08}. Based on the new protocol we then found that the distributed entanglement
originates from the entanglement of sender's mode with the auxiliary mode that is used for distribution.
The entanglement is first destroyed by local correlated displacements that make the
auxiliary mode separable from sender's mode. The separable mode is then sent to the
receiver who partially restores the entanglement by mixing it with his suitably classically
correlated mode.

We emphasize that the proposed protocol for establishing entanglement between two remote
parties using separable ancilla mode consists of experimentally feasible Gaussian
states and operations involving single-mode squeezed states, correlated displacements and
beam splitters, dispensing with the technically challenging CNOT gates of the qubit case.
The input squeezed states with CMs (\ref{product}) need not to be minimum uncertainty states
and the protocol works well also with states with a large noise excess in the antisqueezed
quadratures, e.g., for $e^{2t}=2$ and $200$ vacuum units of noise excess in each quadrature the
logarithmic negativity of the distributed entanglement is $E_{\cal N}\approx0.5851$ ebits compared to
$E_{\cal N}\approx0.6019$ ebits for the minimum uncertainty states.

The verification of the effect, however, can become an issue, as
the possibility to demonstrate experimentally the proposed
protocol crucially relies on our ability to check separability of
mode $C$ from the pair of modes $(AB)$ in step 2. In theory we
assumed Gaussianity of states involved in the protocol. This
enabled us to prove the separability using relatively simple
sufficient conditions for separability of Gaussian states
\cite{Vidal_02,Serafini_06}. In an experiment, however, in order
to check whether the state prepared by us is Gaussian we would
have to perform complete tomography of the state on
infinite-dimensional state space which amounts to infinite number
of measurements. In this sense the Gaussian states are idealized
limit states and for the experimental demonstration of the
protocol we therefore need a sufficient condition for separability
of a generic generally non-Gaussian state or a reliable
indication, that the Gaussian character of the states is preserved
throughout the protocol. Nevertheless, a less ambitious goal can
be achieved by measuring only the three-mode CM of the state in
step 2. By showing that the measured CM satisfies some Gaussian
sufficient condition for separability of mode $C$ from modes
$(AB)$ we know that the Gaussian entanglement of modes $A$ and $B$
was distributed by sending an ancilla that was separable in
Gaussian sense.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\acknowledgments

L. M. would like to thank Radim Filip and Jarom\'{\i}r Fiur\'a\v{s}ek for fruitful discussions.
The research has been supported by the research projects ``Measurement and Information in Optics,'' (MSM 6198959213),
Center of Modern Optics (LC06007) of the Czech Ministry of Education and from GACR (Grant No. 202/08/0224).
We also acknowledge the financial support of the Future and Emerging Technologies (FET) programme
within the Seventh Framework Programme for Research of the European Commission, under the FET-Open grant agreement
COMPAS, number 212008. N. K. is grateful to the Alexander von Humboldt Foundation for the financial support.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%CUTTED PARTS OF THE FIRST SUBMISSION%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The positivity of partial transpose is not sufficient for full
separability of three-mode Gaussian states as there can exist the
so called PPT entangled states \cite{Horodecki_97,Werner_01}, i.e. states satisfying the
inequality (\ref{criterion1}) for all $j=A,B,C$ that however cannot
be written as a convex mixture of tripartite product states.
The three-mode Gaussian PPT entangled states can be
distinguished from fully separable states using criterion derived in
\cite{Giedke_01}. The criterion is based on the expression of the
CM $\gamma$ in the block form
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{block}
\gamma=\left(\begin{array}{cc}
A & C \\
C^{T} & B \\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $A$ is a $2\times2$ submatrix and $B$ is a $4\times4$
submatrix. It is formulated in terms of two $2\times2$ matrices
$N$ and $\tilde{N}$ defined as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{NtildeN}
N&=&A-C\frac{1}{B-i\Omega_2}C^{T},\\
\tilde{N}&=&A-C\frac{1}{B-i\tilde{\Omega}_2}C^{T},
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\Omega_{2}=J\oplus J$, $\tilde{\Omega}_2=J\oplus(-J)$ and
the matrix inversions are understood in the sense of inversion on
the range. According to that criterion \cite{Giedke_01} a
PPT three-mode Gaussian state with CM $\gamma$ is fully separable
if and only if there exists a point $\zeta=(\zeta_1,\zeta_2)^{T}\in\mathbb{R}^{2}$
fulfilling the following inequalities:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{subequations}\label{inequalities}
\begin{eqnarray}
\mbox{min}\{\mbox{Tr}N,\mbox{Tr}\tilde{N}\}&\geq&2\sqrt{1+\zeta_{1}^2+\zeta_2^{2}},\label{C}\\
\mbox{det}N+1+L^{T}\zeta&\geq&\mbox{Tr}N\sqrt{1+\zeta_{1}^2+\zeta_2^{2}},\label{calE}\\
\mbox{det}\tilde{N}+1+{\tilde
L}^{T}\zeta&\geq&\mbox{Tr}\tilde{N}\sqrt{1+\zeta_{1}^2+\zeta_2^{2}},\label{tildecalE}
\end{eqnarray}
\end{subequations}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $L=(N_{11}-N_{22},2\mbox{Re}(N_{12}))^{T}$,
$\tilde{L}=(\tilde{N}_{11}-\tilde{N}_{22},2\mbox{Re}(\tilde{N}_{12}))^{T}$.

The separability criterion can be interpreted geometrically \cite{Giedke_01}. The
inequality (\ref{C}) determines a circular disk ${\cal C}$ of
radius
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{radius}
r_{\cal C}=\mbox{min}\left\{\sqrt{\frac{(\mbox{Tr}N)^2}{4}-1},
\sqrt{\frac{(\mbox{Tr}\tilde{N})^2}{4}-1}\right\}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and center $m_{\cal C}=(0,0)^{T}$, whereas inequality (\ref{calE}) determines the
ellipse ${\cal E}$ described by the inequality
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{ellipsisE}
(\zeta-m_{\cal E})^{T}K(\zeta-m_{\cal E})\leq m,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{K}
K=\left(\begin{array}{cc}
(\mbox{Tr}N)^{2}-L_{1}^{2} & -L_1L_2 \\
-L_1L_2 &  (\mbox{Tr}N)^{2}-L_{2}^{2} \\
\end{array}\right)
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{m}
m=\frac{k_2}{k_1}\left[(\mbox{det}N+1)^{2}-k_1\right],\quad
m_{\cal E}=\frac{\mbox{det}N+1}{k_1}L.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Here $k_1=4[\mbox{det}N+(\mbox{Im}N_{12})^{2}]$ and
$k_2=(\mbox{Tr}N)^{2}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Similarly, the inequality (\ref{tildecalE}) determines the ellipse
$\tilde{\cal E}$ that is described by the same inequality as
(\ref{ellipsisE}) for tilted quantities. From the geometrical
considerations we see that a PPT state with CM $\gamma$ is
separable if and only if the sets ${\cal C}$, ${\cal E}$ and
$\tilde{\cal E}$ have a common nonempty intersection $I$. This
geometrical picture allows to further reformulate the
separability criterion in a more practical form utilizing only the
quantities computable directly from the CM $\gamma$ \cite{Giedke_01}.
It relies on the observation that if the intersection $I$ is
nonempty then it necessarily contains some of the nine points
$\{m_{\cal C},m_{\cal E},m_{\tilde{\cal E}},i_{{\cal C}{\cal
E}}^{\pm}, i_{{\cal C}{\tilde{\cal E}}}^{\pm},i_{{\cal
E}{\tilde{\cal E}}}^{\pm}\}$, where $m_{\cal C}$, $m_{\cal E}$ and
$m_{\tilde{\cal E}}$ denote centers of the sets ${\cal C}$, ${\cal
E}$ and ${\tilde{\cal E}}$ and by $i_{{\cal C}{\cal E}}^{\pm}$, $,
i_{{\cal C}{\tilde{\cal E}}}^{\pm}$ and $i_{{\cal E}{\tilde{\cal
E}}}^{\pm}$ we denoted the intersections of the borders of sets
${\cal C}$ and ${\cal E}$, ${\cal C}$ and ${\tilde{\cal E}}$, and
${\cal E}$ and ${\tilde{\cal E}}$, respectively. The final version
of the criterion that is used in this paper reads as follows
\cite{Giedke_01}:

\noindent
{\it Separability criterion.} A three-mode PPT Gaussian state described by the CM $\gamma$ is
fully separable if and only if the inequalities
$\mbox{det}N>0$ and $\mbox{det}\tilde{N}>0$
hold and there exists a point $\zeta_{\rm sol}\in\{m_{\cal
C},m_{\cal E},m_{\tilde{\cal E}},i_{{\cal C}{\cal E}}^{\pm},
i_{{\cal C}{\tilde{\cal E}}}^{\pm},i_{{\cal E}{\tilde{\cal
E}}}^{\pm}\}$ fulfilling the inequalities (\ref{inequalities}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}
%\centerline{\psfig{width=6.5cm,angle=0,file=fig2.eps}}
\centerline{\psfig{width=7.5cm,angle=-90,file=fig2.ps}}
\caption{Substituting scheme of the protocol for distribution of CV
entanglement by separable Gaussian states. Modes $A$ and $B$ in
suitable different momentum-squeezed states are superimposed on a
balanced beam splitter $BS_{AB}$. The two modes and the vacuum
mode $C$ are then displaced randomly with Gaussian distribution
with covariance matrix $xP$ by displacements $D_{A},D_{B}$ and
$D_{C}$ after which the modes $A,B$ and $C$ are
in a fully separable state. Mixing of modes $A$ and $C$ on a
balanced beam splitter $BS_{AC}$ entangles mode $A$ with the pair
of modes $(BC)$ while mode $B$ is separable from $(AC)$ and mode $C$
is separable from $(AB)$. Mixing of modes $B$ and $C$ on a balanced
beam splitter $BS_{BC}$ finally entangles $A$ and $B$ while $C$
still remains separable from $(AB)$. See text for details.}
\label{fig2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This
question can be answered using an alternative substituting scheme
depicted in Fig.~\ref{fig2}. The scheme not only sheds light on
the physical mechanism underlying the phenomenon under study but
it in fact displays the way in which the sought fully separable
state with complex separability properties was originally
engineered.

\noindent
{\it The substituting nonlocal scheme.}
The substituting scheme starts with three pure Gaussian states (Fig.~\ref{fig2}).
The modes $A$ and $B$ are both in a momentum squeezed vacuum state with squeezing
parameters $r_{A}$ and $r_{B}$, respectively, and mode $C$ is in a vacuum state.
The entire state is therefore described by the diagonal CM
$\gamma_{\rm in}=\mbox{diag}(e^{2r_A},e^{-2r_A},e^{2r_B},e^{-2r_B},1,1)$ and we assume $r_B> r_A\geq0$.
Next, the modes $A$ and $B$ are superimposed on a balanced beam splitter $BS_{AB}$
described by the matrix $U_{AB}$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{UAB}
U_{AB}=\left(\begin{array}{ccc}
\frac{1}{\sqrt{2}}\openone &  \frac{1}{\sqrt{2}}\openone & 0 \\
\frac{1}{\sqrt{2}}\openone&  -\frac{1}{\sqrt{2}}\openone & 0 \\
0 & 0 & \openone\\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\openone$ and $0$ are the $2\times2$ identity and zero matrix, respectively.
The resultant CM is $\gamma_{0}=U_{AB}\gamma_{\rm in}U_{AB}^{T}=\gamma_{AB}\oplus\openone_{C}$, where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gammaAB}
\gamma_{AB}=\left(\begin{array}{cccc}
e^{2d}a & 0 & -e^{2d}c & 0 \\
0 & e^{-2d}a & 0 & e^{-2d}c \\
-e^{2d}c & 0 & e^{2d}a & 0 \\
0 & e^{-2d}c & 0 & e^{-2d}a \\
\end{array}\right)
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
with $a=\cosh(2r)$, $c=\sinh(2r)$, $r=(r_B-r_A)/2$ and
$d=(r_B+r_A)/2$. For $r>0$ the CM $\gamma_{AB}$ describes an
entangled state. With respect to what follows it is useful to
verify the entanglement by using the separability criterion
\cite{Simon_00} according to which the state is entangled if and
only if the matrix $\gamma_{AB}-i\Omega_{2}^{(T_A)}$ has a
negative eigenvalue, where
$\Omega_{2}^{(T_A)}=(\sigma_3\oplus\openone)\Omega_{2}(\sigma_{3}\oplus\openone)=(-J)\oplus
J$. The matrix has four eigenvalues one of which can be negative.
Provided that $r>0$, the negative eigenvalue reads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{lambda}
\lambda=e^{-2r}\cosh(2d)-\sqrt{1-e^{-4r}+e^{-4r}\cosh^{2}(2d)},
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and thus CM $\gamma_{AB}$ indeed represents an entangled state.


At the end of step 1, the initial entanglement of modes $A$ and
$B$ has to be destroyed, and the specific mixed fully  separable
Gaussian state of modes $A, B, C$ is designed by using correlated
displacements $D_{A, B, C}$. In step 2, a separable mode $B$ is
sent to Bob, and Alice holding the mode $A$ superimposes it with
the ancilla $C$ on a beam splitter. In step 3, mode $C$ interacts
with $B$ on another beam splitter and it results in entanglement
of the output modes $A$ and $B$. The mode $C$ remains separable
throughout all the steps of the protocol (see also the caption of
Fig.~\ref{fig2} )



\noindent {\it Construction of a mixed tri-partite Gaussian state
with desired separability properties.} Step 2 requires a Gaussian
state that is separable with respect to two bipartitions which is
impossible with pure Gaussian states \cite{Adesso_06}.
Consequently, in step 1 we have to construct a suitable mixed
fully separable Gaussian state. Following the qubit protocol
\cite{Cubitt_03} this can be done by adding a properly designed
noise into the pure state CM $\gamma_{0}$. It is natural to seek
the noise in the form of a multiple of a positive matrix $P\geq0$
such that the resulting CM $\gamma_1=\gamma_{0}+xP$ with $x\geq0$
describes a fully separable Gaussian state. For this purpose we
will follow the qubit protocol \cite{Cubitt_03} and require the
reduced state of the subsystem $(AB)$ to be separable lying on the
boundary of separable and entangled states. This requirement is
only necessary for full separability of the CM $\gamma_{1}$ and
determines the structure of the $4\times4$ submatrix of the matrix
$P$ corresponding to the subsystem $(AB)$ and denoted as $P_{AB}$.
We find matrices $P_{AB}$ and $P$ using the method developed in
\cite{Giedke_01} for construction of various three-mode entangled
Gaussian states. The method is based on the utilization of the
eigenvector of the matrix $\gamma_{AB}-i\Omega_{2}^{(T_A)}$
corresponding to its negative eigenvalue (\ref{lambda}) that reads
explicitly as
$p_{\lambda}=(i\cos\phi,\sin\phi,i\cos\phi,-\sin\phi)^{T}$, where
$\sin\phi,\cos\phi>0$ and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{tanphi}
\tan\phi=e^{-2r}\sinh(2d)+\sqrt{1+e^{-4r}\sinh^{2}(2d)}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Expressing now the vector $p_{\lambda}$ as $p_{\lambda}=p_1+ip_2$,
where $p_1=(0,\sin\phi,0,-\sin\phi)^{T}$ and
$p_2=(\cos\phi,0\cos\phi,0)^{T}$, we can define a positive matrix
$P_{AB}=p_1p_1^{T}+p_2p_2^{T}$. By adding a multiple of the matrix
$P_{AB}$ to the CM $\gamma_{AB}$ we can create a new CM
$\sigma_{AB}=\gamma_{AB}+xP_{AB}$ where $x\geq0$ that can describe
a separable state if $x$ is sufficiently large. The least value of
$x$, denoted by $x_{0}$, for which the state becomes separable can
be calculated from the symplectic eigenvalues of the matrix
$\sigma_{AB}^{(T_A)}=(\sigma_3\oplus\openone)\sigma_{AB}(\sigma_{3}\oplus\openone)$
that can be found by diagonalizing the matrix
$\Omega_{2}\sigma_{AB}^{(T_A)}$. Hence we find the symplectic
eigenvalues to be $s_1=e^{2r}$ and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{s2}
s_2=\sqrt{[e^{2(d-r)}+2x\cos^{2}\phi][e^{-2(d+r)}+2x\sin^{2}\phi]}.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As $s_1>1$ one can determine $x_{0}$ from the separability
condition $s_2=1$ that gives
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{xsep}
x_{0}=\frac{\sqrt{e^{-4r}\delta^{2}+(1-e^{-4r})\sin^{2}(2\phi)}-e^{-2r}\delta}{\sin^{2}(2\phi)},
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\delta=e^{2d}\sin^{2}\phi+e^{-2d}\cos^{2}\phi$. By setting
$x=x_{0}$ we destroy the entanglement between modes $A$ and $B$.
However, further analysis reveals that this is not sufficient to
destroy the entanglement in the three-mode state with CM
$\gamma_{1}$ entirely and one has to take sufficiently large
$x>x_{0}$ to guarantee full separability of the desired three-mode
state.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Next we have to complete the $4\times4$ matrix $P_{AB}$ to the
$6\times6$ positive matrix $P$ and add its multiple to the CM
$\gamma_0$. Similar to \cite{Giedke_01}, we extend the $4\times1$
vectors $p_1$ and $p_2$ to $6\times1$ vectors
$q_1=(p_1^{T},y^{T})^{T}$ and
$q_2=(p_2^{T},y^{T})^{T}$, where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{y}
y=\sqrt{2}\left(1,1\right)^{T}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and we define the matrix $P$ as $P=q_1q_1^{T}+q_2q_2^{T}$.
The three-mode CM we are interested in then reads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gamma1}
\gamma_{1}=\gamma_{0}+xP.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%proves to be
This CM has to satisfy the criterion of full separability
\cite{Giedke_01}. This involves, first, that it corresponds to a
PPT state, i.e. $\gamma_{1}$ satisfies the inequality
(\ref{criterion1}) for all $j=A,B,C$. For $j=C$ the inequality
(\ref{criterion1}) is satisfied by construction as the addition of
the noise $xP$ is implemented by local displacements of modes $A$,
$B$ and $C$ \cite{Werner_01} which cannot entangle mode $C$ with
$(AB)$. For $j=A,B$ Eq.~(\ref{criterion1}) can be verified by
calculating the symplectic eigenvalues of $\gamma_{1}^{(T_j)}$,
$j=A,B$. They can be computed easily from the matrix
$\chi_{j}=U_{AB}\gamma_{1}^{(T_j)}U_{AB}^{T}$ having the same
symplectic eigenvalues that reads explicitly
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\chi_{j}=\left(\begin{array}{ccc}
\alpha & 0 & \varepsilon_{j}\\
0 & \beta & 0 \\
\varepsilon_{j}^{T} & 0 & \theta \\
\end{array}\right),\quad j=A,B,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\alpha=\mbox{diag}(e^{2(d-r)}+2x\cos^{2}\phi,
e^{-2(d+r)}+2x\sin^{2}\phi)$,
$\beta=\mbox{diag}(e^{2(d+r)},e^{-2(d-r)})$, and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}
\theta&=&\left(\begin{array}{cc}
1+4x & 4x \\
4x & 1+4x \\
\end{array}\right),\nonumber\\
\varepsilon_{j}&=&2x\left(\begin{array}{cc}
\cos\phi_{j} & \cos\phi_{j} \\
-\sin\phi_{j} & -\sin\phi_{j} \\
\end{array}\right),\quad j=A,B,
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\phi_{A,B}=\pm\phi$. By calculating the eigenvalues of the
matrix $\Omega\chi_{j}$ one finds the matrices
$\gamma_{1}^{(T_A)}$ and $\gamma_{1}^{(T_B)}$ to have the same
symplectic eigenvalues of the form:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{s1s2s3}
\tilde s_1=e^{2r},\quad \tilde
s_{2,3}=\sqrt{\frac{X\pm\sqrt{X^2-4Y}}{2}}
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
with $X=1+8x+s_2^{2}$ and $Y=8xe^{-2r}(e^{-2r}+\delta x)+s_2^{2}$,
where $s_2$ is defined in Eq.~(\ref{s2}). Apparently, since
$\tilde s_1>1$ and $\tilde s_2\geq \tilde s_3$ the separability of CM
$\gamma_1$ can be examined based on the value of  $\tilde s_3$. If
$\tilde s_3<1$, mode $A$ is entangled with $(BC)$ and $B$ is
entangled with  $(AC)$. For $\tilde s_3\geq1$ the CM
(\ref{gamma1}) is PPT. The least value of $x$ for which this
happens, $x_{\rm PPT}$, can be determined from the equation $\tilde s_3=1$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{xPPT}
x_{\rm PPT}=\frac{2\sinh(2r)}{\delta}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
It is of interest to look at the relation between the entanglement
of the two-mode CM $\sigma_{AB}$ corresponding to the reduced state of modes $A$ and $B$ and the
entanglement of mode $A$ with the subsystem $(BC)$ and mode $B$ with the subsystem $(AC)$ in
the CM $\gamma_{1}$. Comparison of the threshold values $x_{0}$ and
$x_{\rm PPT}$ given in Eqs.~(\ref{xsep}) and (\ref{xPPT}) reveals
that $x_{\rm PPT}>x_{0}$ for all $r$. Hence
in order the CM $\gamma_1$ to correspond to the PPT state
one has to set at least $x=x_{\rm PPT}$ as although the choice
$x=x_{0}$ destroys the entanglement between modes $A$ and $B$
it is not sufficient to destroy the entanglement
between $A$ and $(BC)$ and $B$ and $(AC)$. In what follows
we further show that the choice $x=x_{\rm PPT}$ also suffices to
destroy the entanglement completely so that the CM (\ref{gamma1})
is fully separable. Nevertheless, we have to take $x>x_{\rm PPT}$
in order the ancillary mode $C$ to remain separable from the modes $(AB)$
in the step 2 of the protocol.

To prove the full separability of the CM $\gamma_1$ it remains to show
that the matrices (\ref{NtildeN}) satisfy
$\mbox{det}N,\mbox{det}\tilde{N}>0$ and that among the nine points
$\{m_{\cal C},m_{\cal E},m_{\tilde{\cal E}},i_{{\cal C}{\cal
E}}^{\pm}, i_{{\cal C}{\tilde{\cal E}}}^{\pm},i_{{\cal
E}{\tilde{\cal E}}}^{\pm}\}$ there is a solution
of inequalities (\ref{inequalities}). For the CM $\gamma_{1}$ one gets
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{Nmatrix}
N=\tilde{N}=\frac{1}{\Delta}\left(\begin{array}{cc}
n_{11} & n_{12} \\
n_{12}^{\ast} & n_{22} \\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\Delta=2\sinh^{2}(2r)+\cosh(2r)\delta x$ and
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{Nelements}
n_{12}\!&=&\!(1-e^{4r})\sin(2\phi)\frac{x}{2}+i\sinh(2r)\left[\delta
x-2\sinh(2r)\right],\nonumber\\
n_{11}\!&=&\!\left(e^{4r}\cos^{2}\phi+e^{4d}\sin^{2}\phi\right)x,\nonumber\\
n_{22}\!&=&\!\left(e^{4r}\sin^{2}\phi+e^{-4d}\cos^{2}\phi\right)x.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As $N=\tilde{N}$, the quantities derived from $\tilde{N}$ are the
same as those derived from $N$ and hence it is sufficient to work
with the latter ones.  Making use of Eq.~(\ref{Nmatrix}) one finds
the determinant of the matrix $N$ to be
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{detN}
\mbox{det}N=\frac{\left[\cosh(2r)+2\sinh(2r)\right]\delta
x-2\sinh^{2}(2r)}{\Delta}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Substituting now $x_{\rm PPT}$ from Eq.~(\ref{xPPT}) for $x$ in
the nominator on the right hand side of Eq.~(\ref{detN}) one
obtains a strictly positive expression for any $r>0$ and therefore
$\mbox{det}N>0$ for $x\geq x_{\rm PPT}$ and $r>0$ as is required by
the criterion of full separability.

In order to complete the proof of full separability of the CM
$\gamma_1$ we have to show that the corresponding circle ${\cal
C}$ has a nonempty intersection with the ellipsis ${\cal E}$ (that
coincides with the ellipsis $\tilde{\cal E}$ due to the equation
$N=\tilde{N}$). The circle ${\cal C}$ has a center $m_{\cal
C}=(0,0)^{T}$ and radius
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{rC}
r_{\cal C}=\sqrt{\frac{(\mbox{Tr}N)^2}{4}-1},
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{TrN}
\mbox{Tr}N=\frac{\left[e^{4r}+\cosh(4d)-\sinh(4d)\cos(2\phi)\right]x}{\Delta},
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
as can be straightforwardly calculated from Eq.~(\ref{Nelements}).
For the ellipsis ${\cal E}$ one obtains $m=0$ using
Eqs.~(\ref{m}), (\ref{Nelements}) and (\ref{detN}) and hence the
ellipsis is just the following point:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{mcalE}
m_{\cal
E}&=&\frac{e^{-2r}}{2\delta}\left\{\left[e^{4r}-\cosh(4d)\right]\cos(2\phi)+\sinh(4d),\right.\nonumber\\
&&(\left.1-e^{4r})\sin(2\phi)\right\}^{T}.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Calculating now the radius (\ref{rC}) for $x=x_{\rm PPT}$ one
finds that $r_{\cal C}=\|m_{\cal E}\|$, where the symbol $\|.\|$
stands for the norm, and therefore the point $m_{\cal E}$ lies
exactly on the boundary of the circle ${\cal C}$. Moreover, the
position of the point $m_{\cal C}$ is independent of $x$ and the
radius $r_{\cal C}$ is an increasing function of $x$ because
$dr_{\cal C}/dx>0$ as can be easily verified using
Eq.~(\ref{TrN}). This immediately implies that for any $x\geq
x_{\rm PPT}$ the point $m_{\cal E}$ lies on the boundary or inside
the circle ${\cal C}$ which in other words means that for all
$x\geq x_{\rm PPT}$ the CM $\gamma_{1}$ represents a fully separable state
as we wanted to prove.

\noindent
{\it Entanglement distribution in the substituting scheme.}
Having a fully separable Gaussian state in hands we can proceed with
the step 2. In this step the modes $A$ and $C$ are superimposed on
a balanced beam splitter $BS_{AC}$ described by the matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{UAC}
U_{AC}=\left(\begin{array}{ccc}
\frac{1}{\sqrt{2}}\openone & 0 & \frac{1}{\sqrt{2}}\openone \\
0 & \openone & 0\\
\frac{1}{\sqrt{2}}\openone& 0 & -\frac{1}{\sqrt{2}}\openone \\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
that transforms the CM (\ref{gamma1}) as
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{gamma2}
\gamma_{2}=U_{AC}\gamma_{1}U_{AC}^{T}.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Obviously, the CM is separable with respect to partition $B-(AC)$
as the $BS$ acts locally on subsystem $(AC)$ and cannot therefore
entangle it with mode $B$. More interestingly, also mode $C$ can
remain separable from the subsystem $(AB)$ if we choose the
squeezings $r_{A},r_{B}$ and the parameter $x$ properly. This can
be proved by applying on the matrix $\gamma_{2}^{(T_{C})}$ the
separability criterion based on the symplectic invariants
\cite{Serafini_06}. The matrix possesses three such invariants
denoted $I_1,I_2$ and $I_3=\mbox{det}(\gamma_{2}^{(T_{C})})$ that
can be calculated easily as coefficients of the characteristic
polynomial of the matrix $\Omega\gamma_{2}^{(T_{C})}$, i.e.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{polynomial}
\mbox{det}(\Omega\gamma_{2}^{(T_{C})}-\mu\openone)=\mu^{6}+I_1\mu^{4}+I_2\mu^{2}+I_3.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The criterion then says that for CM $\gamma_{2}$ mode $C$ is
separable from modes $(AB)$ if and only if
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{Sigma}
\Sigma_{2}\equiv I_3-I_2+I_1-1\geq0
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
holds. Strictly speaking, the criterion is not sufficient for
separability of states having some symplectic eigenvalue equal to
unity \cite{Serafini_06}. Namely, such states, $\gamma_{2}$ with
$x=0$ being an example, can be entangled but can give $\Sigma_{2}=0$ .
However, the sharp inequality $\Sigma_{2}>0$ is already sufficient for
separability. Calculating the invariants using Eqs.~(\ref{gamma2})
and (\ref{polynomial}) we arrive at the following simple
expression:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{Sigma1}
\Sigma_{2}=x(ux+v),
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the explicit form of the parameters $u$ and $v$ can be found in the Appendix.
In the plane $(x,\Sigma_{2})$ Eq.~(\ref{Sigma1}) determines a parabola
that intersects the $x$-axis in the origin. Owing to the complex
dependence of parameters $u$ and $v$ on squeezings $r_A$ and $r_B$
we however cannot decide for general $r_{A}$ and $r_{B}$ whether
there exist $x\geq x_{\rm PPT}$ such that $\Sigma_{2}>0$.
Nevertheless, we can find particular values of squeezings for
which $\Sigma_{2}>0$ and therefore CM $\gamma_{2}$ is separable for
$x$ larger than a certain threshold value $x_{\rm th}$. For
instance, by taking $e^{2r_A}=3/2$ and $e^{2r_{B}}=2$ we get $u>0$
and $v<0$ whence the parabola is oriented upwards and the
threshold value is $x_{\rm th}=-v/u\approx1.04>x_{\rm
PPT}\approx0.2043$ as follows by direct calculation from
Eqs.~(\ref{xPPT}) and (\ref{Sigma1}). As a consequence, for
$x>x_{\rm th}$ we have $\Sigma_{2}>0$ and the CM (\ref{gamma2}) is
separable with respect to the partition $C-(AB)$.

The protocol is finalized by the step 3 in which the modes $B$ and
$C$ are superimposed on a balanced beam splitter $BS_{BC}$
described by the matrix
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{UBC}
U_{BC}=\left(\begin{array}{ccc}
\openone & 0 & 0\\
0 & \frac{1}{\sqrt{2}}\openone &  \frac{1}{\sqrt{2}}\openone\\
0 & \frac{1}{\sqrt{2}}\openone&  -\frac{1}{\sqrt{2}}\openone\\
\end{array}\right).
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The resulting CM reads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{gamma3}
\gamma_{3}=U_{BC}\gamma_{2}U_{BC}^{T}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
and, remarkably, exhibits entanglement between modes $A$ and $B$.
Moreover, mode $C$ remains separable from the subsystem $(AB)$.

First, we will verify entanglement between mode $A$ and $B$. For
this purpose we express the CM $\gamma_{3,AB}$ of the two modes in
the block form
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gamma3AB}
\gamma_{3,AB}=\left(\begin{array}{cc}
{\cal A} & {\cal C} \\
{\cal C}^{T} & {\cal B} \\
\end{array}\right)
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
with the blocks given by the following matrices:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{calABC}
{\cal A}=\left(\begin{array}{cc}
a_{+}+(g_1^{2}+1)x & (g_{0}+g_{1})x \\
(g_{0}+g_{1})x & a_{-}+(g_0^{2}+1)x \\
\end{array}\right),\nonumber\\
{\cal B}=\left(\begin{array}{cc}
b_{+}+(h_1^{2}+\frac{1}{2})x & \frac{h_{0}-h_{1}}{\sqrt{2}}x \\
\frac{h_{0}-h_{1}}{\sqrt{2}}x  & b_{-}+(h_0^{2}+\frac{1}{2})x \\
\end{array}\right),\nonumber\\
{\cal C}=\left(\begin{array}{cc}
c_{+}+(g_1h_1-\frac{1}{\sqrt{2}})x & -(h_{0}+\frac{g_{1}}{\sqrt{2}})x \\
(h_{1}-\frac{g_{0}}{\sqrt{2}})x  & c_{-}-(g_{0}h_{0}+\frac{1}{\sqrt{2}})x \\
\end{array}\right),
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $a_{\pm}=(e^{\pm 2d}a+1)/2$, $b_{\pm}=[e^{\pm
2d}(3a\mp2\sqrt{2}c)+1]/4$, $c_{\pm}=[e^{\pm
2d}(a\mp\sqrt{2}c)-1]/2\sqrt{2}$,
$g_{j}=1+\sin(\phi+j\frac{\pi}{2})/\sqrt{2}$, and
$h_{j}=[\sqrt{2}-(-1)^{j}]\sin(\phi+j\frac{\pi}{2})/2+(-1)^{j}/\sqrt{2}$,
$j=0,1$. The entanglement between modes $A$ and $B$ can be verified
by calculating the least symplectic eigenvalue of the matrix
$\gamma_{3,AB}^{(T_{B})}$. The eigenvalue can be computed
analytically as a root of a biquadratic equation and attains the
form \cite{Vidal_02}:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{ABsymplectic}
\nu=\sqrt{\frac{\kappa-\sqrt{\kappa^2-4\mbox{det}(\gamma_{3,AB})}}{2}},
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $\kappa=\mbox{det}({\cal A})+\mbox{det}({\cal
B})-2\mbox{det}({\cal C})$ and the $2\times2$ matrices ${\cal A}$,
${\cal B}$ and ${\cal C}$ are defined in Eq.~(\ref{calABC}).
Hence, restricting ourselves to the case when $e^{2r_A}=3/2$,
$e^{2r_{B}}=2$ and $x=1.041>x_{\rm th}$, one can obtain by direct
calculation using Eqs.~(\ref{calABC}) and (\ref{ABsymplectic}) a
lengthy but exact expression for $\nu$ that is approximately equal
to $\nu\approx0.9571<1$ which is a clear evidence of the
entanglement between modes $A$ and $B$.

Finally we will show that like in the qubit protocol the auxiliary
mode $C$ is separable from the two-mode subsystem $(AB)$. To show
this we will use once again the criterion based on symplectic
invariants \cite{Serafini_06}. By calculating the characteristic
polynomial of the matrix $\Omega\gamma_3^{(T_C)}$ one finds
three symplectic invariants $J_{1}$, $J_{2}$ and
$J_{3}=\mbox{det}(\gamma_{3}^{(T_{C})})=I_{3}$ that yield, like in
the step 2, the following analytical expression for the quantity
$\Sigma_{3}=J_3-J_2+J_1-1$:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{Sigma3}
\Sigma_{3}=x(wx+z),
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where the parameters $w$ and $z$ are given in the Appendix. Substituting
$e^{2r_{A}}=3/2$, $e^{2r_{B}}=2$ and $x=1.041$ into the
Eq.~(\ref{Sigma3}) we arrive at $\Sigma_{3}\approx0.3957>0$ thus
finally verifying the separability of the mode $C$ from the modes
$(AB)$.

It is interesting to find what is the maximum entanglement that
can be distributed with our protocol. Numerical analysis reveals
that the maximum is attained if $e^{2r_{A}}=1.8$, $e^{2r_{B}}=2.38$ and $x\approx404.9539$.
The initial symplectic eigenvalue of the matrix $\gamma_{AB}^{(T_B)}$ is then given by
$\nu_{\rm in}=a-c=e^{-2r}\approx0.8697$ which yields the logarithmic negativity
$E_{{\cal N},{\rm in}}=-\log_{2}\nu_{\rm in}\approx0.2015$ ebits. In this case the symplectic eigenvalue
(\ref{ABsymplectic}) of the matrix $\gamma_{3,AB}^{(T_{B})}$ reads $\nu\approx0.9299$ which gives
$E_{\cal N}\approx0.1048$ ebits. The amount of entanglement can be further slightly
increased by performing on the ancilla $C$ homodyne detection of a suitably rotated quadrature
and properly displacing modes $A$ and $B$ according to the measurement outcome. Then we
obtain the optimal symplectic eigenvalue $\nu_{\rm opt}\approx0.9166$ and the corresponding
logarithmic negativity $E_{{\cal N},{\rm opt}}\approx0.1257$ ebits.
The proposed protocol can be improved even further. By taking $y=0.69\times\left(1,1\right)^{T}$
instead of as in Eq.~(\ref{y}) and optimizing the scheme with respect to initial squeezing
one finds the distributed entanglement is maximized for $e^{2r_{A}}=1.74$, $e^{2r_{B}}=2.46$ and $x=2$.
In this case we get $\nu_{\rm in}\approx0.8410$ which yields the initial logarithmic negativity of
$E_{{\cal N},{\rm in}}\approx0.2498$ ebits. The lower symplectic eigenvalue (\ref{ABsymplectic}) of the
partially transposed final entangled state shared by Alice and Bob then reads as
$\nu\approx0.9224$ and the corresponding logarithmic negativity is $E_{\cal N}\approx0.1165$ ebits. Like in
the previous case, the eigenvalue can be further reduced by the homodyne detection of a proper quadrature on
mode $C$ which gives $\nu_{\rm opt}\approx0.8823$ that leads to the highest logarithmic negativity
of $E_{{\cal N},{\rm opt}}\approx0.1806$ ebits that we found for the proposed protocol.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The protocol in Fig.~\ref{fig2} reveals that the original LOCC protocol \cite{Mista_08} can
be effectively replaced by another protocol that is no more LOCC and in which the origin
of the entanglement distributed by separable ancilla is easy to understand. Namely, it
shows that the initial fully separable state with CM (\ref{gamma1}) can be created from
the pure entangled state with CM (\ref{gammaAB}) and the vacuum mode $C$ by performing local
random correlated Gaussian displacements on them. Since the displacements of mode $C$
are correlated with displacements of modes $A$ and $B$, they can be compensated partially
by mixing the two modes sequentially on beam splitters $BS_{AC}$ and $BS_{BC}$ with
auxiliary mode $C$ thereby partially recreating the original entanglement between modes $A$ and $B$.
Moreover, provided that the distribution of displacements is chosen suitably mode $C$ need
not to get entangled with the pair of modes $A$ and $B$ at any stage of the protocol.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
appendix*
\section{}


The parameters $u$ and $v$ in Eq.~(\ref{Sigma1}) read
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\small
\begin{widetext}
\begin{eqnarray}\label{uv}
u&=&e^{2r}\left[7+\cos^{2}(2\phi)\right]\cosh(2d)-e^{4r}\left[1+\cos^{2}(2\phi)\right]+
2\sqrt{2}e^{r}\left[e^{d}\sinh(d+r)\sin(\phi)-e^{-d}\sinh(d-r)\cos(\phi)\right]\sin(2\phi)\nonumber\\
&&+2\left[2\cosh(2d)+e^{-2r}-3e^{2r}\right]\sinh(2d)\cos(2\phi)-2\cosh(4d)-4e^{2r}\sinh(2r)\sin(2\phi)-4\cosh(2r)\cosh(2d)
\label{u},\\
v&=&2e^{2r}\cosh(2d)-\cosh(4d)-8\sinh^{2}(2r)-e^{4r}+2\left[\cosh(2d)-e^{2r}\right]\sinh(2d)\cos(2\phi)\nonumber\\
&&+4\sqrt{2}e^{r}\left[e^{d}\sinh(d-r)\sin(\phi)-e^{-d}\sinh(d+r)\cos(\phi)\right]\sinh(2r)\label{v}.
\end{eqnarray}
%\end{widetext}
\normalsize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The parameters $w$ and $z$ in Eq.~(\ref{Sigma3}) read
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\small
%\begin{widetext}
\begin{eqnarray}\label{wz}
w&=&\frac{1}{8}\left\{e^{2r}\left[25\cosh(2d)-14\sqrt{2}\sinh(2d)\right]-9e^{4r}+8\sqrt{2}\sinh(4d)-12\cosh(4d)-4e^{-2r}\cosh(2d)
+\left\{e^{2r}\left[16\sqrt{2}\cosh(2d)\right.
\right.\right.\nonumber\\
&&\left.\left.\left.-28\sinh(2d)\right]-8\sqrt{2}e^{4r}-8\sqrt{2}\cosh(4d)+12\sinh(4d)+4e^{-2r}\sinh(2d)\right\}\cos(2\phi)
+\left\{e^{2r}\left[3\cosh(2d)-2\sqrt{2}\sinh(2d)\right]\right.\right.\nonumber\\
&&\left.\left.-3e^{4r}\right\}\cos(4\phi)+\left\{2\left[\sqrt{2}+2+2\left(\sqrt{2}-2\right)e^{4r}-\left(3\sqrt{2}-2\right)e^{2(d+r)}\right]
\sin\phi+2\left[\sqrt{2}-2+2\left(\sqrt{2}+2\right)e^{4r}\right.\right.\right.\nonumber\\
&&\left.\left.\left.-\left(3\sqrt{2}+2\right)e^{2(r-d)}\right]\cos\phi+4\left(e^{4r}-1\right)\right\}\sin(2\phi)\right\}\label{w},\\
z&=&\frac{1}{4}\left\{4-2e^{-4r}-5e^{4r}+6e^{2r}\cosh(2d)-4\sqrt{2}e^{2r}\sinh(2d)-3\cosh(4d)+2\sqrt{2}\sinh(4d)+\left(2-\sqrt{2}\right)
\left[2e^{2d}\sinh(2r)+1\right.\right.\nonumber\\
&&\left.\left.-e^{4r}\right]\sin\phi-\left(2+\sqrt{2}\right)\left[2e^{-2d}\sinh(2r)+1-e^{4r}\right]\cos\phi+\left\{
2e^{2r}\left[2\sqrt{2}\cosh(2d)-3\sinh(2d)-\sqrt{2}e^{2r}\right]\right.\right.\nonumber\\
&&\left.\left.-2\sqrt{2}\cosh(4d)+3\sinh(4d)\right\}\cos(2\phi)\right\}\label{z}.
\end{eqnarray}
\end{widetext}
\normalsize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
whereas for $\sin(2\phi)=0$ we have
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{xsep2}
x_{0}=\frac{\sinh(2r)}{\delta}=e^{-2d}\sinh(2r),
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The schemes in Fig.~\ref{fig1} are within the reach of the current technology. They involve
interference of two squeezed beams on a balanced beam splitter followed by another interference
of one of the output beams on another balanced beam splitter which was demonstrated in a number
experiments, e.g. \cite{Furusawa_98}. The improved protocol requires two squeezed beams with 3dB squeezing
$\langle(\Delta p_{A})^2\rangle=\langle(\Delta x_{B})^2\rangle=e^{-2t}=1/2$
which can be also relatively easily prepared in the contemporary quantum-optical laboratory. Numerical analysis
further confirms that the protocol is robust against phase noise excess in antisqueezed quadratures of
squeezed beams $A$ and $C$ and works also with weakly unbalanced beam splitters.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!]
\centerline{\psfig{width=6.5cm,angle=0,file=fig3.eps}}
\caption{Improved scheme for distribution of
continuous-variable entanglement by separable Gaussian states.
Step 1: preparation of a fully separable Gaussian state of
three modes $A$, $B$ and $C$. Step 2 entangles mode $A$ with a pair
of modes $(BC)$. Step 3 entangles mode $A$ with mode $B$. Mode $C$
remains separable from the pair of modes $(AB)$ in all steps. $BS_{{AC},{BC}}$
denote balanced beam splitters, $D_{A},D_{B}$, $D_{C}$ are local
displacements distributed according to the Gaussian distribution with correlation matrix $xQ$.
See text for details} \label{fig3}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[!]
\centerline{\psfig{width=6.5cm,angle=0,file=fig1.eps}}
\caption{LOCC scheme of the protocol for distribution of
continuous-variable entanglement by separable Gaussian states.
Step 1: preparation of a fully separable Gaussian state of
three modes $A$, $B$ and $C$. Step 2 entangles mode $A$ with a pair
of modes $(BC)$. Step 3 entangles mode $A$ with mode $B$. Mode $C$
remains separable from the pair of modes $(AB)$ in all steps. $BS_{{AC},{BC}}$
denote balanced beam splitters, $D_{A},D_{B}$, $D_{C}$ are local
displacements distributed according to the Gaussian distribution with correlation matrix $Q$.
See text for details} \label{fig1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The entire three-mode state is then described by
the CM of the form:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{gammaABC}
\tilde{\gamma}_{ABC}=\left(\begin{array}{ccc}
\cosh(2s)\openone & 0 & \sinh(2s)\sigma_{3} \\
0 & \openone & 0 \\
\sinh(2s)\sigma_{3} & 0 & \cosh(2s)\openone\\
\end{array}\right).
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In the step 1 Alice and Bob prepare a manifestly separable state
$\rho_{abc}$ of three two-level systems (qubits) $a$, $b$ and $c$
of the form:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{qstep1}
\rho_{abc}\!\!=\!\!\frac{1}{6}(\sum_{k=0}^{3}|\Psi_{k},\Psi_{-k},0\rangle\langle\Psi_{k},\Psi_{-k},0|+
\sum_{i=0}^{1}|i,i,1\rangle\langle i,i,1|),\nonumber\\
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $|0\rangle$ and $|1\rangle$ are basis states and
$|\Psi_{k}\rangle=(1/\sqrt{2})(|0\rangle+e^{ik\pi/2}|1\rangle)$.
For our purposes it is important to note that, the reduced state
$\tilde{\rho}_{ab}=\mathrm{Tr}_{c}(\rho_{abc})$ of the subsystem
$ab$ reads
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{qstep2}
\tilde{\rho}_{ab}=\frac{1}{3}|\Phi_{+}\rangle_{ab}\langle\Phi_{+}|+\frac{1}{6}\openone_{ab},
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $|\Phi_{+}\rangle=(1/\sqrt{2})(|00\rangle+|11\rangle)$ is a
maximally entangled state and $\openone_{ab}$ is a two-qubit
identity matrix. The state (\ref{qstep2}) is the two-qubit Werner
state \cite{Werner_89} lying exactly on the boundary of separable
and entangled states (only Werner states with $p>1/3$ are
entangled). In the step 2 Alice performs a controlled NOT (CNOT)
operation on her qubit $a$ (control qubit) and the ancillary qubit
$c$ (target qubit) thereby entangling qubit $a$ with the qubits
$bc$ . The resulting state is however invariant under the exchange
of qubits $b$ and $c$ and since it is obviously separable with
respect to splitting $b-ac$ it has to be separable also with
respect to splitting $c-ab$. In the step 3, Bob applies a CNOT
operation on his qubit $b$ (control qubit) and the ancilla $c$
(target qubit) thus creating the state
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{eqnarray}\label{qstep3}
\tau_{abc}=\frac{1}{3}|\Phi_{+}\rangle_{ab}\langle\Phi_{+}|\otimes|0\rangle_{c}\langle0|+
\frac{2}{3}\openone_{ab}\otimes|1\rangle_{c}\langle1|.
\end{eqnarray}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Obviously, the qubit $c$ is separable from the two qubits $ab$ but
there is still no entanglement between qubits $a$ and $b$ because
the reduced state of the two qubits is again the separable Werner
state (\ref{qstep2}). Therefore, the protocol has to be finalized
by the step 4, when Bob either extracts with probability $1/3$ a
maximally entangled state $|\Phi_{+}\rangle_{ab}$ by measuring the
qubit $c$ in the computational basis or alternatively he can
extract a non-maximally entangled state by applying a suitable
trace-preserving operation on qubits $b$ and $c$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Importantly, the reduced state
$\tilde{\rho}_{ab}=\mathrm{Tr}_{c}(\rho_{abc})$ of the subsystem
$ab$ is the separable Werner state \cite{Werner_89} lying exactly
on the boundary of separable and entangled states.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
for the matrices (\ref{NtildeN})
$\mbox{det}N>0,\mbox{det}\tilde{N}>0$ holds
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\begin{equation}\label{criterion2}
%s_{i}\geq1\quad.
%\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
$(B-i\Omega_2)^{-1}$ and $(B-i\Omega_2)^{-1}$ Before going further
let us note that the entanglement properties of studied Gaussian
states we are interested in are determined solely by their
covariance matrix. Since the phase-space displacements $d$ do not
change the covariance matrix they do not affect the entanglement
and in what follows we assume $d=0$ for simplicity.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{equation}\label{eigenvalue}
\lambda_{1,2}=e^{-2r}\cosh(2d)\mp\sqrt{1-e^{-4r}+e^{-4r}\cosh^{2}(2d)},
\lambda_{3,4}=e^{2r}\cosh(2d)\mp\sqrt{1+e^{4r}\sinh^{2}(2d)},\\
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The symplectic eigenvalues of the matrix $\gamma_{1}^{T_B}$ can be
calculated along the same lines and they coincide with those of
the matrix $\gamma_{1}^{T_B}$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Apparently, the radius of the circle ${\cal C}$ increases with
increasing $x$ while the position of the point $m_{\cal E}$ is
independent on $x$ which indicates that for a sufficiently large
$x$ the point will lie inside the circle and the CM $\gamma_1$
will be fully separable.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%COMPLETE PART FROM THE SHORT PAPER%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix
\begin{center}
\centerline{\bf APPENDIX}
\end{center}
The coefficients $u$ and $v$ in Eq.~(\ref{quadratic}) are given
by:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\small
\begin{widetext}
\begin{eqnarray}
u&=&e^{2r}\left[7+\cos^{2}(2\phi)\right]\cosh(2d)-e^{4r}\left[1+\cos^{2}(2\phi)\right]+
2\sqrt{2}e^{r}\left[e^{d}\sinh(d+r)\sin(\phi)-e^{-d}\sinh(d-r)\cos(\phi)\right]\sin(2\phi)\nonumber\\
&&+2\left[2\cosh(2d)+e^{-2r}-3e^{2r}\right]\sinh(2d)\cos(2\phi)-2\cosh(4d)-4e^{2r}\sinh(2r)\sin(2\phi)-4\cosh(2r)\cosh(2d)
\label{u},\\
v&=&2e^{2r}\cosh(2d)-\cosh(4d)-8\sinh^{2}(2r)-e^{4r}+2\left[\cosh(2d)-e^{2r}\right]\sinh(2d)\cos(2\phi)\nonumber\\
&&+4\sqrt{2}e^{r}\left[e^{d}\sinh(d-r)\sin(\phi)-e^{-d}\sinh(d+r)\cos(\phi)\right]\sinh(2r)\label{v}.
\end{eqnarray}
\end{widetext}
\normalsize
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Using the formalism of Wigner functions, the resulting state of modes $A$
and $C$ before the beam splitter which however already involves all the displacements is %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
\begin{equation}\label{WAC}
W_{AC}(r_{A},r_{B},r_{C})=\int\tilde{\cal{Q}}(\bar{\xi}_{A},\bar{\xi}_{B},\bar{\xi}_{C})W_{A}(r_{A}-\bar{\xi}_{A})
W_{C}(r_{C}-\bar{\xi}_{C}-G\bar{\xi}_{B})d\bar{\xi}_{A}d\bar{\xi}_{B}d\bar{\xi}_{C},
\end{equation}
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
where $W_{A}(r_{A})$ and $W_{C}(r_{C})$ are the Wigner functions of the initial pure squeezed states of
modes $A$ and $C$ with $CMs$ given in Eq.~(\ref{gammaACgammaB}) and $\tilde{\cal{Q}}$ is the following
classical Gaussian distribution of displacements with correlation matrix (\ref{tildeQ})
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{widetext}
\begin{equation}\label{tildecalQ}
\tilde{\cal{Q}}(\bar{x}_{A},\bar{p}_{A},\bar{x}_{B},\bar{p}_{B},\bar{x}_{C},\bar{p}_{C})=\frac{1}{4\pi x}\delta(\bar{x}_{A})\delta(\bar{p}_{A}+\frac{\bar{p}_{B}}{\sqrt{2}})\delta(\bar{x}_{C}-\frac{\bar{x}_{B}}{\sqrt{2}})\delta(\bar{p}_{C})
e^{-\frac{1}{4x}\left(\bar{x}_{B}^2+\bar{p}_{B}^2\right)}.
\end{equation}
\end{widetext}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

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