Rafa{\l} Demkowicz-Dobrza\'nski\(^{1}\)
Maciej Lewenstein\(^{2,3}\)
%\footnote{Also at Instituci\'o Catalana de Recerca i Estudis Avan{\c c}ats.}
Aditi Sen(De)\(^{3,4}\)
Ujjwal Sen\(^{3,4}\)
Dagmar Bru\ss\(^{5}\)
%We solve the problem of the optimal local cloning of local orbits of pure entangled two-qubit states , by using semidefinite programming %and analytic methods. We solve the problem of the optimal cloning of pure entangled two-qubit states with a fixed degree of entanglement using local operations and classical communication. %with a fixed degree of entanglement. %Our method relies on semidefinite programming, and subsequently leads to %an analytical formula for the fidelity. We study the question whether We show, that amazingly, classical communication between the parties can improve the fidelity of local cloning %, %and find that this is the case if and only if the initial entanglement is higher than a certain critical value. It is completely useless for weakly entangled states. We also show that bound entangled states with positive partial transpose are not useful as a resource to improve the best local cloning fidelity.
clonent_1Feb2006.tex
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\begin{document}

%\title{%''Phase transition'' in the optimal LOCC cloning of two-qubit entangled states
%Classical communication can improve local cloning of entangled states
%Optimal local cloning of entangled states may benefit from communication

%\title{Classical communication improves local cloning of entangled quantum states after a critical amount of shared entanglement}

%\title{Uselessness of classical communication for local cloning of weakly entangled states}

\title{Usefulness of classical communication for local cloning of entangled states}

\author{Rafa{\l} Demkowicz-Dobrza\'nski\(^{1}\), Maciej Lewenstein\(^{2,3}\),
%\footnote{Also at Instituci\'o Catalana de Recerca i Estudis Avan{\c c}ats.}, 
Aditi Sen(De)\(^{3,4}\), Ujjwal Sen\(^{3,4}\), Dagmar Bru\ss\(^{5}\)}

\affiliation{\(^{1}\)Center for Theoretical Physics, Polish Academy of Sciences
Aleja Lotnik{\'o}w 32/44, 02-668 Warszawa, Poland\\
\(^{2}\)ICREA and ICFO-Institut de Ci\`encies Fot\`oniques, E-08860  Castelldefels (Barcelona), Spain\\
\(^{3}\)Institut f\"ur Theoretische Physik, Universit\"at Hannover, D-30167 Hannover,
Germany\\
\(^{4}\)ICFO-Institut de Ci\`encies Fot\`oniques, E-08860  Castelldefels (Barcelona), Spain\\
\(^{5}\)Institut f\"ur Theoretische Physik III, Heinrich-Heine-Universit\"at
D\"usseldorf, 40225 D\"usseldorf, Germany}


%\author{Rafa{\l} Demkowicz-Dobrza\'nski}
%\affiliation{Center for Theoretical Physics, Polish Academy of Sciences\\
%Aleja Lotnik{\'o}w 32/44, 02-668 Warszawa, Poland}
%\author{Maciej Lewenstein}
%\affiliation{ICFO-Institut de Ci{\� e}ncies Fot{\� o}niques, Jordi Girona 29, Edifici Nexus II, E-08034 Barcelona, Spain \\
%Institut f\"ur Theoretische Physik, Universit\"at Hannover, D-30167 Hannover,
%Germany}
%\author{Aditi Sen(De)}
%\affiliation{ICFO-Institut de Ci{\� e}ncies Fot{\� o}niques, Jordi Girona 29, Edifici Nexus II, E-08034 Barcelona, Spain \\
%Institut f\"ur Theoretische Physik, Universit\"at Hannover, D-30167 Hannover,
%Germany}
%\author{Ujjwal Sen}
%\affiliation{ICFO-Institut de Ci{\� e}ncies Fot{\� o}niques, Jordi Girona 29, Edifici Nexus II, E-08034 Barcelona, Spain \\ 
%Institut f\"ur Theoretische Physik, Universit\"at Hannover, D-30167 Hannover,
%Germany}
%\author{Dagmar Bru\ss}
%\affiliation{Institut f\"ur Theoretische Physik III, Heinrich-Heine-Universit\"at
%D\"usseldorf, 40225 D\"usseldorf, Germany}
%\date{\today}

\begin{abstract}
%We solve the problem of the optimal local cloning of local orbits of pure entangled two-qubit states , by using semidefinite programming
%and analytic methods.
We solve the problem of the optimal cloning of pure entangled two-qubit states with a fixed degree of entanglement 
using local operations and classical communication.
%with a fixed degree of entanglement.
%Our method relies on semidefinite programming, and subsequently leads to
%an analytical formula for the fidelity. We study the question whether
We show, that amazingly, classical communication between the parties can improve the fidelity of local cloning
%,
%and find that this is the case 
\emph{if and only if} the initial entanglement is higher
than a certain \emph{critical} value. It is completely useless for weakly entangled states. 
We also show that bound entangled states with positive partial transpose 
are not useful as a resource to improve the best local cloning fidelity. 

\end{abstract}

\maketitle

\section{Introduction}

Quantum  information processing (QIP) and engineering in quantum mechanical systems has always been regarded 
on two levels of complexity. First, one can consider quantum nature of single systems with regard to global operations. 
The second, and usually more complex, level is to consider composite systems with regard to local operations. 
The examples of such duality  are numerous: Quantumness \cite{pakhi} vs. entanglement \cite{lagaan}, 
global vs. local accessible information \cite{sundar, royal-bengal}, global vs. local dense coding 
\cite{chandrabindu, bisarga}, etc. In this paper, we discuss for the first time the duality between global and local cloning 
and find optimal local cloner for pure states of two qubits. 



%In quantum information processing (QIP), one usually deals with two qualitatively different types of scenarios.
%%%%%%%%%%%%%: Single systems and composite ones. 
%One may deal with \emph{single} systems, and understand the  laws that govern QIP in such systems. The subject of quantum communication, however, 
%needs an understanding of the laws governing \emph{composite} systems also.  




%Among the laws that Nature imposes on QIP in composite systems, 
The impossibility of perfect cloning \cite{wootters} is a fundamental law
that Nature imposes on QIP.
%The no-cloning theorem \cite{wootters} is a fundamental law that Nature 
%imposes on quantum information processing. 
Although perfect quantum
cloning is forbidden, 
approximate quantum cloning is possible \cite{buzek1996, cloning, werner1998, cloninf}.
%, and can set bounds on the
%information gain of the adversary in quantum cryptography.
%Limits on approximate covariant cloning  have been found
%in e.g. \cite{buzek1996, cloning, werner1998} for finite-dimensional quantum states, and in e.g. \cite{cloninf} for
%infinite dimensional ones. 
Only recently, the question of cloning states with a fixed degree of entanglement 
%pure 
has been addressed. 
%It was shown that
%it is impossible to find a map that clones maximally entangled states perfectly while
%being separability preserving \cite{lamoureux2004}. 
The optimal operation cloning maximally entangled states of two qubits  was given in \cite{lamoureux2004},
%all degrees
%of entanglement with equal fidelity (Cerf). 
while the more general case of non-maximally entangled states was solved in \cite{novotny2004}.
% when considering {\em global} transformations.
% This approach is equivalent to cloning of a given subset
%of quantum states in higher dimensions, where the  subset is defined via
%the fixed degree of entanglement.

A conceptually different approach,
which may shed new light on the nature of entanglement, 
is {\em local} cloning of 
%entanglement
entangled states: The state is then distributed 
between two parties that may only act locally on their respective
subsystems, with or without 
%exchange of 
classical communication.
The impossibility of locally cloning  a {\em known} entangled state has
been pointed out and used to define an entanglement measure \cite{sens}. 
%Cloning of unknown states
%with local operations and classical communication (LOCC) has
%only been studied for orthogonal entangled states \cite{martin}.
Cloning of orthogonal entangled states with local operations and classical communication (LOCC), 
under the assumption of sharing an entangled blank state has been studied in \cite{martin}.

This paper addresses the problem of finding the optimal LOCC operation for cloning 
an unknown  entangled state, with a fixed degree of entanglement.
% (belonging to the same local orbit).
In all previous papers touching upon this problem \cite{broadcasting1997,lamoureux2004}, classical communication was 
%neglected  
not considered,
and 
%it was implicitly assumed that the best local cloning amounts to applying 
only 
%two 
independent Bu{\v z}ek-Hillery cloners 
were applied
to each subsystem.
Here we will show that 
%such assumptions were not legitimate.
classical communication produces very interesting effects.


The plan of the paper is as follows. 
We restrict ourselves  to $1\to2$ cloning (optimal production of two copies from one copy) of two-qubit pure entangled states,
%We look for an LOCC operation, 
such that all states with the same degree of entanglement are cloned equally well,
namely with the same fidelity. The degree of entanglement is explicitly given in Sect. \ref{sec-dui}.
%We will optimize the fidelity in both the 
%LOCC case and the case without communication.
In the three succeeding sections, we find the optimal local  protocol for cloning two-qubit states with a given amount of entanglement, and 
show that 
only 
when the degree of entanglement is higher than a certain ``critical"  threshold, 
classical communication improves 
the cloning fidelity. In particular, this critical threshold, is obtained in Sect. \ref{subsec-tin-ka}. 
%is significantly better than the optimal fidelity obtained with a local protocol without classical communication. 
(For other examples of usefulness of 
%of phenomena where 
% local protocols with 
classical communication, 
%excels over those without communication, include 
see Refs. \cite{jomey-doi, bisarga}.) 
This is, to our knowledge, the first quantum protocol for which a critical entanglement threshold for usefulness
of classical communication is demonstrated. 
%To our knowledge, a critical value of entanglement for local cloning of entanglement states was hitherto not known. 
In Sect. \ref{sec-chhoi},
we  show that bound entangled states with positive partial transpose (PPT) \cite{HHH_bound} are not useful as a resource for local cloning. 
We discuss our results in Sect. \ref{sec-Mithhun}.

\section{Degree of entanglement}
\label{sec-dui}

The degree of entanglement of a bipartite state
can be quantified with the help of the Schmidt decomposition. Every two-qubit state
can be written as
%\begin{equation}
\[\ket{\Phi}=U_A \otimes U_B \left(\alpha \ket{00} + \sqrt{1-\alpha^2} \ket{11}\right),\] 
%\nonumber
%\end{equation}
where $U_A$, $U_B$ are single qubit 
unitary operations, 
%unitaries,
and $\alpha\in (0,1/\sqrt{2})$ is a 
Schmidt coefficient determining the degree of entanglement of the state \cite{Bennett-partial}.
%For $\alpha=0$ the state is a product state, while for $\alpha=1/\sqrt{2}$ the
% state is maximally entangled. 
We look for the cloning transformation that clones all states with a given $\alpha$ 
with the same fidelity.


\section{Setting up the stage: The constraints on a local cloner}


The cloning transformation  is a linear and
completely positive (CP)
 map $\mathcal{E}$
 that takes a two-qubit input state 
\[\rho_{\textrm{in}}=\ket{\Phi}\bra{\Phi} \in \mathcal{L}(\mathcal{H}_A \otimes \mathcal{H}_B)\]
to a four-qubit state 
\[\rho_{\textrm{out}}\in \mathcal{L}(\mathcal{H}_{1A} \otimes \mathcal{H}_{1B} \otimes \mathcal{H}_{2A}\otimes \mathcal{H}_{2B}).\]
(\(\mathcal{L}(\mathcal{H})\) denotes the set of all states on a Hilbert space \(\mathcal{H}\).)
The reduced density matrices of the two clones are
%respectively 
 $\rho_{1(2)}=\textrm{Tr}_{2A,2B(1A,1B)}(\rho_{\textrm{out}})$, where
$\textrm{Tr}_{2A,2B(1A,1B)}$ denotes tracing over $\mathcal{H}_{2A}\otimes \mathcal{H}_{2B}$ ($\mathcal{H}_{1A} \otimes \mathcal{H}_{1B}$).
In the following, we investigate symmetric cloning, i.e. the situation when $\rho_1 = \rho_2$. 
As a 
figure of merit for optimality, we adopt 
here
the so-called local fidelity
\[F= \bra{\Phi} \rho_{1(2)} \ket{\Phi},\]
 quantifying the similarity of each of the clones to the input.
% state.
%This is the figure of merit that is regarded as more appropriate, as the (impossible) exact cloning demands that 
%every clone should be indistinguishable from the input (see e.g. \cite{werner1998})
This is the figure of merit which judges each clone independently, and 
is more in the spirit of original formulation of the problem of cloning than the global fidelity (see e.g. \cite{werner1998}).

A fact which is often exploited in investigations on optimal cloning is, that 
 the optimum can always be reached by a covariant operation 
\cite{werner1998,novotny2004}. This is also true in our case, and so 
%In our case 
we impose covariance 
with respect to local unitary operations:  
%\begin{equation}
%\label{eq:CPcov}
\[\mathcal{E}(U_A \otimes U_B \rho_{\textrm{in}} U_A^\dagger \otimes U_B^\dagger)
=(U_A \otimes U_B)^{\otimes 2} \mathcal{E}(\rho_{\textrm{in}}) (U_A^\dagger \otimes U_B^\dagger)^{\otimes 2}.\]
%\nonumber
%\end{equation}
In general, with every CP map $\mathcal{E}: \mathcal{L}(\mathcal{H}) \to \mathcal{L}(\mathcal{K})$, 
one can associate a positive operator $P_\mathcal{E} \in \mathcal{L}(\mathcal{K}\otimes\mathcal{H})$, 
%with the help of 
via the 
Jamio{\l}kowski isomorphism \cite{jamiolkowski1972}. The covariance condition of the cloning map $\mathcal{E}$ 
 can be  written using the corresponding positive operator 
$P_\mathcal{E}\in \mathcal{L}(\mathcal{H}_{1A} \otimes \mathcal{H}_{1B} 
\otimes \mathcal{H}_{2A}\otimes \mathcal{H}_{2B} \otimes \mathcal{H}_A \otimes \mathcal{H}_B)$ 
as follows \cite{ariano2001}:
%\begin{equation}
\[[P_\mathcal{E},U_A \otimes U_B \otimes U_A \otimes U_B \otimes U_A^* \otimes U_B^*]=0.\]
%\end{equation}
For convenience, let us introduce an operator $\tilde{P}_\mathcal{E}$ which is the operator $P_\mathcal{E}$ but with 
different ordering of subspaces, namely:
$\tilde{P}_\mathcal{E} \in \mathcal{L}(\mathcal{H}_{1A} \otimes \mathcal{H}_{2A} \otimes \mathcal{H}_{A}\otimes 
\mathcal{H}_{1B} \otimes \mathcal{H}_{2B} \otimes \mathcal{H}_B)$. The covariance condition for $\tilde{P}_\mathcal{E}$ reads
%\begin{equation}
%\label{eq:comPt}
\[[\tilde{P}_\mathcal{E},U_A \otimes U_A \otimes U_A^* \otimes U_B \otimes U_B \otimes U_B^*]=0.\]
%\end{equation}
In order to write the most general form of $\tilde{P}_\mathcal{E}$ satisfying the above condition, let us first write the most general
form of a hermitian operator $A$, satisfying 
%the condition
%To write the general form of $\tilde{P}_\mathcal{E}$ satisfying  (\ref{eq:comPt}), we first write the general
%form of a hermitian operator $A$ satisfying
%\begin{equation}
%\label{eq:comA}
\[[A,U_A \otimes U_A \otimes U_A^*]=0.\]
%\end{equation}
The space $\mathcal{H}_{1A} \otimes \mathcal{H}_{2A} \otimes \mathcal{H}_{A}$ can be decomposed into invariant subspaces under the action of 
$U_A \otimes U_A \otimes U_A^*$. There are two two-dimensional invariant subspaces and one four dimensional subspace \cite{was_a_footnote1}:
%\begin{equation}
\(\mathcal{H}_{1A} \otimes \mathcal{H}_{2A} \otimes \mathcal{H}_{A} = \mathcal{M}_1 \oplus  \mathcal{M}_2 \oplus  \mathcal{M}_3\).
%\end{equation}
In what follows, $\mathcal{M}_1$ is the space spanned by the vectors $(\ket{011}-\ket{101})/\sqrt{2}$ and $(\ket{010}-\ket{100})/\sqrt{2}$,
$\mathcal{M}_2$ is spanned by  $(2\ket{000}+\ket{011}+\ket{101})/\sqrt{6}$ and $(\ket{010}+\ket{100}+2\ket{111})/\sqrt{6}$, while $\mathcal{M}_3$ 
%is
%spanned 
by the remaining four orthogonal vectors.
Let $T_i$ be a projection operator on the subspace $\mathcal{M}_i$. Let $T_{12}$ ($T_{21}$) be an isomorphism from 
%the subspace
 $\mathcal{M}_1$ to $\mathcal{M}_2$ ($\mathcal{M}_2$ to $\mathcal{M}_1$). For convenience we also introduce Hermitian operators 
$T_4 = T_{12}+T_{21}$ and $T_5=iT_{12} - iT_{21}$. 
The subspaces $\mathcal{M}_1$ and  $\mathcal{M}_2$ support equivalent two-dimensional irreducible representations of $\textrm{SU(2)}$, while 
 $\mathcal{M}_3$ supports a four-dimensional irreducible representation.
Using Schur's lemma, one can show that the most general form of a Hermitian operator $A$, satisfying
covariance condition, 
%Eq.(\ref{eq:comA}) 
has the form \cite{ariano2001}
%\begin{equation}
\[A=\sum_{i=1}^5 a_i T_i,\] 
%\end{equation}
where $a_i$ are arbitrary real parameters.
Hence, the most general form of a positive operator $\tilde{P}_\mathcal{E}$, satisfying covariance condition,
%Eq.(\ref{eq:comPt}),
 %when no local communication is allowed, is a product operator
%and reads
%\begin{equation}
%\tilde{P}_{\mathcal{E},LO} = \sum_{i}^5 a_{i} T_i \otimes  \sum_{j}^5 b_{j}T_j,
%\end{equation} 
%Here the 10 parameters
%$a_{i},b_{j}$ are real and 
%such that the operator $\tilde{P}_{\mathcal{E},LO}$ is positive.
%For LOCC operations there is more freedom, and the cloning operator
reads 
%\begin{equation}
\[\tilde{P}_{\mathcal{E}} = \sum_{i,j=1}^5 a_{ij} T_i \otimes T_j,\]
%\end{equation} 
where we now have 25 real parameters $a_{ij}$, which have to be chosen 
such that the operator $\tilde{P}_{\mathcal{E}}$ is positive. 
%CP maps satisfying Eq.(\ref{eq:CPcov}) can thus be parametrized with
%$25$ real parameters. 
Additional constraints 
%on the
%parameters %$a_{ij}$ 
come from the fact that 
the CP map corresponding to the positive operator is trace preserving. This is the case provided that
%\begin{equation}
\(\textrm{Tr}_{1A,1B,2A,2B}{P_\mathcal{E}} = \openone \in \mathcal{L}(\mathcal{H}_A \otimes \mathcal{H}_B)\),
%\end{equation}
which imposes one linear equality for the $a_{ij}$:
%\begin{equation}
%\label{eq:trace}
\[a_{11} + a_{12} + 2a_{13} + a_{21}+  a_{22} + 2a_{23} + 2a_{31} + 2a_{32} +  4a_{33} = 1.\]
%\nonumber
%\end{equation}
There is also a number of linear constraints coming from the symmetry condition on the two clones, $\rho_{1}=\rho_{2}$. 
%And last but not least 
Lastly, there is the condition that the operation should be LOCC.
%the operation should be either LO, or  LOCC. 

The symmetry condition together with the LOCC constraint
make the problem much more difficult than the problem of optimal global cloning
with the global fidelity $F_g= \bra{\Phi}\bra{\Phi} \rho_{\textrm{out}} \ket{\Phi}\ket{\Phi}$
 as a figure of merit,
 as considered in \cite{novotny2004}, where the optimal cloning operation was found analytically \cite{guchhiey-lyang}. 
%Consequently, we will not attempt to find the optimal cloning procedure analytically, but we will rather resort to numerics.
%Consequently, w
We first deal with the problem numerically, and  this leads us to an analytical solution.


\section{The optimal local cloner and the threshold for usefulness of classical communication}

\subsection{Finding the optimal local cloner}

Disregarding for the moment the LOCC constraint, 
the search for the optimal cloning operation can be written as a semidefinite program. 
In a semidefinite program, one tries to maximize a linear function of variables, while keeping certain matrices (that depend in a linear way 
on these variables) positive semidefinite.
%(variables can enter only linearly). 
One can also impose additional linear equalities on the variables. This is 
%exactly 
just 
the kind of problem we are dealing with.
The fidelity we want to maximize is linear in the parameters,
%$a_{ij}$, 
we have linear equalities (trace preservation, symmetry), and we have to keep the operator $\tilde{P}_\mathcal{E}$ positive 
semidefinite. 

%The LO case (without communication) can be solved in a straightforward way.
Imposing the LOCC constraint is in general not easy, so we first impose a weaker constraint, namely the positivity of the partial transpose  of 
$\tilde{P}_\mathcal{E}$ with respect to the subsystem $B$ ($\mathcal{H}_{1B} \otimes \mathcal{H}_{2B} \otimes \mathcal{H}_B$) (see \cite{Lewen-Jamiol}). 
The
PPT condition is a weaker condition, because 
if an operation is LOCC it is separable, and if it is separable it satisfies PPT.
(The opposite implications do not hold \cite{HHH_bound, nlwe}.)
The PPT condition has the advantage that it can be easily incorporated into a semidefinite program. Fortunately, 
no stronger constraint than PPT will be needed, because in our case imposing the PPT constraint 
will yield an operation which will be proven later to be LOCC.

We have performed the numerics in Matlab using a package for semidefinite programing called SeDuMi. In Fig. \ref{fig:fidglobPPT}
the results of optimization are presented.
 \begin{figure}[t]
\begin{center}
%\includegraphics[height=0.15\textheight,width=0.4\textwidth]{fidglobPPT.eps}
\includegraphics[width=0.5\textwidth]{fidglobPPT.eps}
\caption{Fidelities of three different operations performing cloning of entangled states, with a given degree of entanglement $\alpha$. 
%The
%solid curve represents the fidelity of the optimal cloning operation 
%%%%%when no restrictions on locality of operation are imposed. 
%without any locality restriction.
%The LOCC-curve is for the optimal cloning operation that 
%can be realized with the help of local operations and classical communication. The $\textrm{BH}^{\otimes 2}$-curve
%corresponds to an independent 
%application of two Bu{\v z}ek-Hillery cloners, 
%to each qubit from the entangled pair. 
%The split of two curves, corresponding to LOCC and $\textrm{BH}^{\otimes 2}$, appears
%at \(\alpha=\alpha_0 \approx .3357\).
%%%%%=\sqrt{1/2-\sqrt{15}/10}.
%%%%% For $\alpha>\alpha_0$ the optimal LOCC operation using one bit of 
%%%%%classical communication 
%%%%%outperforms $\textrm{BH^{\otimes 2}$.
}
 \label{fig:fidglobPPT}
\end{center}
\end{figure}

\begin{figure}[t]
\begin{center}
%\includegraphics[height=0.15\textheight,width=0.4\textwidth]{paramPPT.eps}
\includegraphics[width=0.5\textwidth]{paramPPT.eps}
\caption{Parameters describing the optimal LOCC operation that clones entangled states, with given degree of entanglement $\alpha$.
%For $\alpha \leq \alpha_0 $, there is only one non-zero parameter viz. $a_{22}=1$, and the optimal operation is $\textrm{BH}^{\otimes 2}$.
%For $\alpha>\alpha_0$ there are five non-zero parameters, resulting in an LOCC operation that necessarily 
%requires one bit of classical communication.
}
 \label{fig:paramPPT}
\end{center}
\end{figure}
The solid curve represents the optimal cloning when we do not impose a PPT constraint on
the optimization. For this operation, most
 $a_{ij}$ are zero, except for  \cite{was_a_footnote2}
\begin{eqnarray}
a_{11} &=& \frac{1}{2} - \frac{4(1-\alpha^2+\alpha^4)}{c(\alpha)},\nonumber\\
a_{22} &=& 1-a_{11}, \nonumber\\
a_{44} &=& \frac{\sqrt{a_{11}a_{22}}}{2}, \nonumber \\ 
a_{55} &=& -a_{44}, \nonumber
\end{eqnarray}
 where 
\[c(\alpha)=\sqrt{73+16\alpha^2(1-\alpha^2)(1+40\alpha^2-40\alpha^4)}.\]  
%
%
%Qualitatively this is a very similar transformation to the one obtained in \cite{novotny2004}, where the global fidelity was maximized. 
%The same  $a_{ij}$ are zero, and the remaining four behave similarly with the change of $\alpha$. Additionally, the
The fidelity dependence on $\alpha$ has a very similar character when optimizing local fidelity 
or global fidelity \cite{novotny2004}, with a minimum 
at 
\[\alpha=\alpha_0=\sqrt{\frac{1}{2} - \frac{\sqrt{15}}{10}} \approx .3357.\]
We found for the local fidelity
\[F=\frac{16+(1-4\alpha^2)^2-8\alpha^4+c(\alpha)}{36}.\]
For maximally entangled states ($\alpha=\sqrt{2}/2$), the 
fidelity equals $F=(5+\sqrt{13})/12$, which agrees with the result obtained in \cite{lamoureux2004}.
For product states ($\alpha=0$), the fidelity equals $F=(17+\sqrt{73})/36$. 

The dashed-dotted curve in Fig. \ref{fig:fidglobPPT} represents the fidelity of cloning when 
two optimal $1\to 2$ Bu{\v z}ek-Hillery cloners \cite{buzek1996} are applied independently to each qubit from the entangled pair
 ($\textrm{BH}^{\otimes 2}$). This is actually the optimal local cloning operation when no communication is allowed.
This can be derived writing the operator $P_{\mathcal{E}}$ as a product of two covariant local operators: $P_{A}\otimes P_{B}$,
and optimizing the fidelity, keeping in mind the trace and symmetry constraints.
The only non-zero parameter of the optimal transformation is $a_{22}=1$, and the corresponding fidelity:
\[F_{\textrm{BH}^{\otimes 2}}=\frac{25-16\alpha^2+16\alpha^4}{36}.\] 


\subsection{The critical threshold of entanglement above which classical communication helps in local cloning}
\label{subsec-tin-ka}


The crux of the paper is the dotted curve in Fig. \ref{fig:fidglobPPT} which represents the fidelity of the optimal LOCC cloning.
For $\alpha$ smaller than a critical value $\alpha_0$,
the optimal LOCC operation is just $\textrm{BH}^{\otimes 2}$.
Namely, for weakly entangled states, classical communication between the parties does not help to improve the
fidelity of local cloning.
%In this region the dotted line coincides with the dashed-dotted line representing the %fidelity of cloning with the help of $\textrm{BH}^{\otimes 2}$ operation.
Beyond the threshold, i.e.
 for $\alpha>\alpha_0$, $\textrm{BH}^{\otimes 2}$ ceases to be the optimal
 LOCC cloning protocol.
The point of the transition 
%from the 
 is indicated by an arrow. Notice that this point
 is exactly the point at which the unrestricted global cloning 
has the minimum.
%\footnote{This can signify that the set of entangled states with $\alpha=\alpha_0$ (local orbit), may have some interesting geometrical properties, for example
%it could be a local orbit with the biggest volume with respect to some natural measure induced by the Haar measure of $SU(4)$ groups, among all two-qubit local orbits}.} 
For $\alpha>\alpha_0$, the corresponding non-zero parameters are given by 
\begin{eqnarray}
a_{11} &=& \frac{(1-10\alpha^2+10\alpha^4)^2}{4(1+8\alpha^2-8\alpha^4)^2}, \nonumber \\
a_{22} &=& (1-\sqrt{a_{11}})^2, \nonumber \\ 
a_{12} &=& a_{21}=a_{44}=\sqrt{a_{11}a_{22}}. \nonumber
\end{eqnarray}
The optimal LOCC fidelity for $\alpha>\alpha_0$  reads 
\[F_{\textrm{LOCC}}=\frac{3+8\alpha^2(1-\alpha^2)(2+\alpha^2-\alpha^4)}{4(1+8\alpha^2-8\alpha^4)}.\]
The optimal LOCC operation outperforms $\textrm{BH}^{\otimes 2}$ most visibly in the case of cloning of maximally entangled states, where
the optimal LOCC yields the fidelity of cloning $F=5/8=0.625$, whereas 
the $\textrm{BH}^{\otimes 2}$ operation can achieve only $F=7/12 \approx 0.583$.

In Fig. \ref{fig:paramPPT}, we present the dependence on $\alpha$, of 
the non-zero $a_{ij}$ of the optimal LOCC transformation.
For $\alpha\leq \alpha_0$, the only non-zero parameter is $a_{22}=1$, and the optimal operation is $\textrm{BH}^{\otimes 2}$. 
For $\alpha>\alpha_0$, there are five non-zero $a_{ij}$'s describing the optimal transformation, of which three are identical.
%Non-differentiable behaviour of parameters $a_{ij}$ at $\alpha=\alpha_0$ 
%prompts us to call this transition  a ''phase transition''.
The non-differentiable behaviour of $a_{ij}$ at $\alpha=\alpha_0$
 is quite a surprising effect, as it indicates that there is a ``critical" 
threshold $\alpha_0$ for the degree of entanglement of two-qubit states,
below which  classical communication does not help in enhancing the fidelity of cloning, while above the threshold, classical 
communication does improve the cloning fidelity.

\section{Explicit protocol for implementing the optimal local cloner}

We proceed now to give the promised proof that the optimal LOCC cloning transformation that we  discussed is indeed LOCC.
Recall that we have obtained this transformation  numerically, imposing only the
PPT constraint, so in principle it could be a nonlocal operation.
%As mentioned before the operation has five non zero parameters: 
%$a_{11}$, $a_{22}$, $a_{44}=a_{12}=a_{21}$. 
%Additionally the optimal operation has the property that
% $a_{44}=\sqrt{a_{11} a_{22}}$.
%Trace preservation condition (\ref{eq:trace}) imposes a constraint: 
%$a_{11}+a_{22}+2a_{44}=1$, which taking into account the previous 
%equality can be written as
%$\sqrt{a_{11}}+\sqrt{a_{22}}=1$. 

Having the explicit form of the positive operator $P_\mathcal{E}$ 
that corresponds to this operation, we will first show that it is a separable operator, 
%and we found the explicit decomposition of $P_\mathcal{E}$ into convex sum 
%of eight product states. 
by expressing the corresponding 
 CP map $\mathcal{E}$ in terms of {\em local} Kraus operators
\(K_i\)
%.
%We find that the CP map
%can be written with the help of
%eight Kraus operators $K_i$ 
as 
%\begin{equation}
%  \label{eq:9}
 \[ \mathcal{E}(\rho_{\textrm{in}}) = \sum_{i=1}^8 K_i \rho_{\textrm{in}} K_i^\dagger,\quad \sum_i K_i^\dagger K_i = \openone,\]
%\end{equation}
where $K_i$ act from a two qubit input space into the tensor product of two qubit spaces of the two clones:
\begin{equation}
 % \label{eq:10}
  K_i:\ \underbrace{\mathcal{H}_A \otimes \mathcal{H}_B}_{\textrm{input space}} 
\mapsto \underbrace{\mathcal{H}_{1A} \otimes \mathcal{H}_{1B}}_{\textrm{first clone}}\otimes
\underbrace{ \mathcal{H}_{2A} \otimes \mathcal{H}_{2B}}_{\textrm{second clone}}.
\nonumber 
\end{equation}
Additionally, the 
%Kraus operators 
$K_i$ can be written as products of local Kraus operators acting on the $A$ or $B$ subsystem as 
%\begin{equation}
 % \label{eq:12}
 \[ K_i = K_{i,A} \otimes K_{i,B},\] 
%\end{equation}
where
%\begin{equation}
 % \label{eq:13}
%\begin{array}{rrcl}
\[ K_{i,s}:  \mathcal{H}_s \mapsto \mathcal{H}_{1s} \otimes \mathcal{H}_{2s}, \quad s = A,B.\]
%
% \( K_{i,A}:  \mathcal{H}_A \mapsto \mathcal{H}_{1A} \otimes \mathcal{H}_{2A}\),
% \( K_{i,B}:  \mathcal{H}_B  \mapsto \mathcal{H}_{1B} \otimes \mathcal{H}_{2B}\). 
%\end{array}
%\end{equation}
In order to write down the $K_i$ operators in a simple manner it will prove useful to introduce four matrices $M_i$:
%Let
%\[M_{1,2} = (w_0, c_{1,2}), \quad M_{3,4} = (c_{2,1}, w^0),\] 
%where 
%\begin{eqnarray}
%w_0 &=& (w,0,0,0)^T, \nonumber \\
%w^0 &=& (0,0,0,w)^T, \nonumber \\
%c_{1,2} &=& (0,w/2 \mp v,w/2 \pm v,0)^T,  \nonumber
%\end{eqnarray}
\begin{equation}
\begin{array}{rclrcl}
M_{1}&=&\left(\begin{array}{cc}w & 0 \\ 0 & \frac{1}{2}w-v \\ 0 &\frac{1}{2}w+v \\ 0 & 0  \end{array}\right),& 
M_{2}&=&\left(\begin{array}{cc}w & 0 \\ 0 & \frac{1}{2}w+v \\ 0 &\frac{1}{2}w-v \\ 0 & 0  \end{array}\right), \\ \\
\end{array}
\nonumber
\end{equation}
\begin{equation}
\begin{array}{rclrcl}
M_{3}&=&\left(\begin{array}{cc}0 & 0 \\ \frac{1}{2}w+v & 0 \\\frac{1}{2}w-v &0\\ 0 & w  \end{array}\right), &
M_{4}&=&\left(\begin{array}{cc}0 & 0 \\ \frac{1}{2}w-v & 0 \\\frac{1}{2}w+v &0\\ 0 & w  \end{array}\right), 
\end{array}
\nonumber
\end{equation}
where 
%with
%$w,v$ are
%expressed with help of parameters $a_{11},a_{22}$:
%\begin{equation}
\[w= \frac{a_{22}^{1/4}}{\sqrt{3}}, \quad v=\frac{a_{11}^{1/4}}{2}.\]
%\end{equation}
%The Kraus operators $K_{i}$ can 
%now 
%be written in a compact form as 
Then
%\(K_1=\sqrt{2} M_1 \otimes M_1\),  
%\(K_2=\sqrt{2} M_1 \otimes M_3\),
%\(K_3=\sqrt{2} M_3 \otimes M_1\),
%\(K_4=\sqrt{2} M_3 \otimes M_3\),
%\(K_5=\sqrt{2} M_2 \otimes M_2\),  
% \(K_6=\sqrt{2} M_2 \otimes M_4\), 
%\end{array}
%\nonumber
%\end{equation}
%\begin{equation}
%\begin{array}{lcl}
%,&&  
%\(K_7=\sqrt{2} M_4 \otimes M_2\),
%\(K_8=\sqrt{2} M_4 \otimes M_4\), 
\begin{eqnarray}
%\begin{equation}
%\begin{array}{lcl}
K_1=\sqrt{2} M_1 \otimes M_1, &&  
K_5=\sqrt{2} M_2 \otimes M_2,  \nonumber \\
K_2=\sqrt{2} M_1 \otimes M_3,&&  
K_6=\sqrt{2} M_2 \otimes M_4,  \nonumber \\
%\end{array}
%\nonumber
%\end{equation}
%\begin{equation}
%\begin{array}{lcl}
K_3=\sqrt{2} M_3 \otimes M_1,&&  
K_7=\sqrt{2} M_4 \otimes M_2, \nonumber \\
K_4=\sqrt{2} M_3 \otimes M_3,&&  
K_8=\sqrt{2} M_4 \otimes M_4, \nonumber 
%\end{array}
%\nonumber
%\end{equation}
\end{eqnarray}
where the prefactor $\sqrt{2}$ is needed 
%to fulfill the 
for 
trace preservation.
% condition.
The LOCC protocol which realizes the above operation requires
 only one-way communication of a single classical bit and can be implemented
 as follows.
First, a local POVM (positive operator valued measurement) 
%, using the four 
%Kraus operators 
%$M_i$ 
%(for which we have \(\sum_{i=1}^{4} M_i^\dagger M_i = \openone\)),
is performed on $A$:
\[\ket{\Phi}\bra{\Phi} \to \sum_{i=1}^4 M_i \otimes \openone \ket{\Phi}\bra{\Phi} M^\dagger_i \otimes \openone\]
(\(\sum_{i=1}^{4} M_i^\dagger M_i = \openone\)).
Next, a (one-bit) classical one-way communication
% from 
%subsystem 
%A to B
% is carried 
%through, 
%out,
%informing 
informs
B whether the result \(i\) of the POVM on 
%the subsystem 
$A$ was in $\{1,3\}$, or
$\{2,4\}$. In the first (second) case, \(B\) is operated on by 
%an operation on the subsystem $B$ is performed 
using the Kraus operators
$\sqrt{2}M_{1}, \sqrt{2}M_{3}$ ($\sqrt{2}M_{2}, \sqrt{2}M_{4}$), completing the protocol.  
The equality  $M_{1}^\dagger M_{1} + M_{3}^\dagger M_{3} = \openone / 2$ (and similarly for $M_{2}, M_{4}$), 
guarantee
%which 
%prove 
that the 
%above 
protocol is indeed a deterministic LOCC.

%To ease the future experimental implementation of the protocol, 
%We now give the explicit realization of the above mentioned local POVMs. 
%In order to perform the local operation on subsystem A, described by four operators $M_i$, one 
To perform the POVM at her side, Alice
%needs to 
uses a four dimensional local ancilla ($\mathcal{H}_{aA}$),
and
%First, one 
performs first a unitary on $\mathcal{H}_{1A} \otimes \mathcal{H}_{2A} \otimes \mathcal{H}_{aA}$, 
which acts as
%which acting on two relevant input states 
%(initially blank qubit and ancilla are always in the state $\ket{0a_1}$) yields:
%\begin{widetext}
\begin{eqnarray}
%\begin{array}{rcl}
U_A \ket{00a_1} = w\ket{00}\left(\ket{a_1}+\ket{a_2}\right)+ \ket{\underbar{01}^+}\ket{a_3} +
\ket{\underbar{01}^-}\ket{a_4}, 
\nonumber \\
U_A \ket{10a_1} =\ket{\underbar{01}^-}\ket{a_1} +
\ket{\underbar{01}^+}\ket{a_2} + w\ket{11}\left(\ket{a_3}+\ket{a_4}\right), 
\nonumber 
\end{eqnarray}
%\end{equation}
where 
\[\ket{\underbar{01}^\pm} = (\frac{1}{2}w \pm v)\ket{01}+(\frac{1}{2}w \mp v)\ket{10},\]
%\begin{equation}
%\begin{array}{rcl}
%U_A \ket{00a_1} &=& w\ket{00}\left(\ket{a_1}+\ket{a_2}\right)+ \left[ c^+\ket{01}+c^-\ket{10}\right]\ket{a_3} +
%\left[ c^-\ket{01}+c^+\ket{10}\right]\ket{a_4} \\
%\\
%U_A \ket{10a_1} &=&\left[ c^-\ket{01}+c^+\ket{10}\right]\ket{a_1} +
%\left[ c^+\ket{01}+c^-\ket{10}\right]\ket{a_2} + w\ket{11}\left(\ket{a_3}+\ket{a_4}\right),
%\end{array}
%\end{equation}
%\begin{equation}
%\begin{array}{rcl}
%U_A \ket{00a_1} &=& w\ket{00}\left(\ket{a_1}+\ket{a_2}\right)+ \left[ (\frac{1}{2}w+v)\ket{01}+(\frac{1}{2}w-v)\ket{10}\right]\ket{a_3} +
%\left[ (\frac{1}{2}w-v)\ket{01}+(\frac{1}{2}w+v)\ket{10}\right]\ket{a_4} \\
%\\
%U_A \ket{10a_1} &=&\left[ (\frac{1}{2}w-v)\ket{01}+(\frac{1}{2}w+v)\ket{10}\right]\ket{a_1} +
%\left[ (\frac{1}{2}w+v)\ket{01}+(\frac{1}{2}w-v)\ket{10}\right]\ket{a_2} + w\ket{11}\left(\ket{a_3}+\ket{a_4}\right),
%\end{array}
%\end{equation}
and $\ket{a_i}$ are 
%four 
orthogonal,
% states of ancilla.
and measures on \(\{|a_i\rangle\}\).
%
%Subsequently, the measurement of the state of ancilla is performed, resulting in one of four possible outcomes corresponding to different $M_i$.
%
%
Bob requires only a two-dimensional ancilla ($\mathcal{H}_{aB}$),
and uses the unitaries given by
%
%
%The local operation performed on subsytem $B$, after classical communication took place, requires only the use of a 
%two dimensional ancilla ($\mathcal{H}_{aB}$),
%and can be realized with the help of the unitary operations given by
\begin{eqnarray}
%\begin{array}{rcl}
U_B^{\pm} \ket{00a_1} = \sqrt{2}w\ket{00a_1}+ \sqrt{2} \ket{\underbar{01}^\pm} \ket{a_2}, 
\nonumber \\
U_B^{\pm} \ket{10a_1} =\sqrt{2} \ket{\underbar{01}^\mp}  \ket{a_1} + \sqrt{2}w\ket{11a_2}, \nonumber
\end{eqnarray}
%\end{equation}
%\end{widetext}
where the upper (lower) sign
 corresponds to the use of 
%Kraus operators 
$\sqrt{2}M_{1}, \sqrt{2}M_{3}$ 
% while the lower sign to 
($\sqrt{2}M_{2}, \sqrt{2}M_{4}$).

\section{Bound entangled states with positive partial transpose are not useful as additional resources in local cloning}
\label{sec-chhoi}

It is interesting to note 
%Second,  we remark 
that even though we have imposed a 
PPT %-preservation 
constraint on the cloning operation, which is weaker than an LOCC constraint, we have found an LOCC 
operation to be the optimal one.
This means that there are no 
PPT %-preserving entangled 
operations that helps to improve the fidelity over 
 LOCC.
%for cloning two qubit entangled states.
In particular, this implies that if apart from the ability to perform LOCC,
 we were given some PPT bound entangled states \cite{HHH_bound}
as additional resources, this would not help in the local cloning process
(cf., e.g. \cite{activation}).
%(*what to cite for connection between operations and use of entangled states?*).
This behaviour is different  from, e.g., the recent observations
regarding the convertibility properties of pure quantum states, where
  PPT operations can
significantly outperfom LOCC operations \cite{martin2}.
% We have thus showed another example 
%of uselessness of bound entangled states, this time in the case of 
%cloning of entangled two-qubit states.


\section{Discussion}
\label{sec-Mithhun}


A local cloning machine has several potential applications. In particular, in secret sharing \cite{Marek} 
(see also \cite{byapok}), a boss requires to 
send the states 
\[\frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle), \quad \frac{1}{\sqrt{2}}(|00\rangle \pm i|11\rangle)\] to 
two subordinates.
%, as considered in \cite{Marek}, requires 
Our local cloning machine gives bounds on the security in  a local  eavesdropping on this protocol.
%, by using 
%our local cloning machine. 
Moreover, 
%we conjecture that 
for a different secret sharing protocol (which we conjecture to be more secure \cite{darun-paper} than the one in \cite{Marek}), 
that uses the three quartets 
\begin{eqnarray}
\left\{\frac{1}{\sqrt{2}}(|\phi_1\rangle \pm |\phi_2\rangle),  \frac{1}{\sqrt{2}}(|\phi_3\rangle \pm |\phi_4\rangle)\right\}, \nonumber \\ 
\left\{\frac{1}{\sqrt{2}}(|\phi_1\rangle \pm |\phi_3\rangle),  \frac{1}{\sqrt{2}}(|\phi_2\rangle \pm |\phi_4\rangle)\right\}, \nonumber \\
\left\{\frac{1}{\sqrt{2}}(|\phi_1\rangle \pm |\phi_4\rangle),  \frac{1}{\sqrt{2}}(|\phi_2\rangle \pm |\phi_4\rangle)\right\}, \nonumber 
\end{eqnarray}
our local cloning machine gives the 
optimal local attack, when considering
cloning-based strategies.
%cloning-based optimal local attack. 
Here, 
\begin{eqnarray}
|\phi_1\rangle = \frac{1}{\sqrt{2}}(|00\rangle + |11\rangle), && 
|\phi_2\rangle = \frac{-i}{\sqrt{2}}(|00\rangle - |11\rangle), \nonumber \\
|\phi_3\rangle = \frac{1}{\sqrt{2}}(|01\rangle - |10\rangle), &&
|\phi_4\rangle = \frac{-i}{\sqrt{2}}(|01\rangle + |10\rangle). \nonumber
\end{eqnarray}
%
%\((|00\rangle \pm |11\rangle)/\sqrt{2}\), \((|++\rangle \pm |--\rangle)/\sqrt{2}\), \((|+y,+y\rangle \pm |-y,-y\rangle)/\sqrt{2}\),
%where \(|\pm\rangle = (|0\rangle \pm |1\rangle)/\sqrt{2}\), \(|\pm y\rangle = (|0\rangle \pm i|1\rangle)/\sqrt{2}\), 
%our local cloning machine would give the cloning-based optimal local attack.
Note that a  local attack is the realistic one for secret sharing, and, to our knowledge,
has not yet been considered \cite{darun-paper}. 


%our local cloning machine gives the optimal local attack when considering
%cloning-based strategies

In conclusion, we have found the optimal fidelity for local cloning of
pure two-qubit states, when the degree of entanglement is fixed. For
weakly entangled states, the optimal transformation is shown to  consist
of two independent optimal single-qubit cloners, a la Bu{\v z}ek-Hillery.
For states with entanglement that is higher than a certain threshold, 
communication between the parties improves the fidelity considerably.
We have thus pointed out the existence of a critical entanglement value
for the usefulness of classical communication in local cloning.
Moreover, non-local operations that are separable or 
%that preserve the positivity of partial transpose, 
PPT 
 do not lead to any advantage beyond local operations and classical communications. This, in particular, implies 
 that bound entangled states with positive partial transpose are not useful 
 as an additional resource for local cloning. 



\begin{acknowledgments}


%We thank Toni Ac{\'\i}n for discussions, and 
We acknowledge discussions with Toni Ac{\'\i}n, and support from the DFG (SFB 407, SPP 1078, 432 POL), 
the AvH
%Humboldt 
Foundation, the EC Program 
%Contract No. IST-2002-38877 
QUPRODIS, the ESF Program QUDEDIS, EU IP SCALA and SECOQC,
and the Polish Ministry of Scientific Research and Information Technology under the (solicited) grant No.
PBZ-Min-008/P03/03.


\end{acknowledgments}

%\bibliography{clonent}



%\footnotesize{\(^*\) Also at Instituci\'o Catalana de Recerca i Estudis Avan{\c c}ats.}


\begin{thebibliography}{99}

%\bibitem[{*}] aInstituci\'o Catalana de Recerca i Estudis Avan{\c c}ats.


\bibitem{pakhi} Th. Richter and W. Vogel,
Phys. Rev. Lett. \textbf{89}, 283601 (2002). 

\bibitem{lagaan} M.A. Nielsen and I.C. Chuang, \emph{Quantum Computation and Quantum Information} (Cambridge University Press, Cambridge, 2000).


\bibitem{sundar} A.S. Holevo, Probl. Pereda. Inf. \textbf{9}, 3 1973 [Probl. Inf.
Transm. \textbf{9}, 110 (1973)]; R. Josza, D. Robb, and W. Wotters, Phys. Rev. A, \textbf{49}, 668 (1994).


\bibitem{royal-bengal} P. Badzi{\c{a}}g, M. Horodecki, A. Sen(De), and
U. Sen, Phys. Rev. Lett. \textbf{91}, 117901 (2003); 
A. Sen(De), U. Sen, and M. Lewenstein, quant-ph/0505137. 



\bibitem{chandrabindu} C.H. Bennett and S.J. Wiesner, 
Phys. Rev. Lett. \textbf{69}, 2881 (1992). 

\bibitem{bisarga} D. Bru\ss, G.M. D'Ariano, M. Lewenstein, C. Macchiavello, A. Sen(De), and U. Sen,
Phys. Rev. Lett. \textbf{93}, 210501 (2004).


\bibitem{wootters} W.K. Wootters and W.H. Zurek, Nature (London) \textbf{299}, 802 (1982); D. Dieks, Phys. Lett. \textbf{92A}, 271 (1982).

 \bibitem{buzek1996}  
V. Bu{\v z}ek and M. Hillery,
Phys. Rev. A \textbf{54}, 1844 (1996). 


\bibitem{cloning} 
 D. Bru{\ss}, 
D.P. DiVincenzo, A. Ekert, C.A. Fuchs, C. Macchiavello, and J.A. Smolin,
% \emph{et al.}, 
Phys. Rev. A \textbf{57}, 2368 (1998); R. Werner, \emph{ibid.} {\bf 58}, 1827 (1998).

%put some cloning papers if space allows

\bibitem{werner1998} M. Keyl and R.F. Werner,
%Comments: 16 Pages, REVTeX
J. Math. Phys. \textbf{40}, 3283 (1999).

%R. Werner, Phys. Rev. A {\bf 58}, 1827 (1998).

\bibitem{cloninf} N.J. Cerf, A. Ipe, and X. Rottenberg,
Phys. Rev. Lett. \textbf{85}, 1754 (2000).
%cloning in infinite dimensions (Cerf)

\bibitem{lamoureux2004}
        L.-P. Lamoureux, P. Navez, J.
                  Fiur\'{a}\v{s}ek, and N. J. Cerf,
%       title={Cloning of entanglement of a pair of quantum bits},
        Phys. Rev. A
        \textbf{69}, 040301 (R), 
        (2004);
        E. Karpov, P. Navez, and N. J. Cerf, Phys. Rev. A \textbf{72}, 042314 (2005).


\bibitem{novotny2004}
J. Novotn\'{y}, G. Alber, and I. Jex, Phys. Rev. A \textbf{71}, 042332 (2005).
%title={Optimal copying of entangled two-qubit states},
%quant-ph/0411105.

\bibitem{sens}  M. Horodecki, A. Sen(De), and U. Sen,
Phys. Rev. A \textbf{70}, 052326 (2004).

\bibitem{martin}  S. Ghosh, G. Kar, and A. Roy,
Phys. Rev. A \textbf{69}, 052312 (2004); F. Anselmi, A. Chefles, and M. B. Plenio,
New. J. Phys. \textbf{6}, 164 (2004).

\bibitem{broadcasting1997}
V. Bu{\v z}ek, V. Vedral, M.B. Plenio, P.L. Knight, M. Hillery, Phys. Rev A \textbf{55}, 3327 (1997).


\bibitem{jomey-doi} C.H. Bennett,
% \emph{et al.}, 
G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W.K. Wootters,
Phys. Rev. Lett. \textbf{70}, 1895 (1993); C.H. Bennett, 
G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters,
\emph{ibid.} \textbf{76}, 722 (1996); C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters,
Phys. Rev. A \textbf{54}, 3824 (1996); J. Walgate, A.J. Short, L. Hardy, and V. Vedral,
Phys. Rev. Lett. \textbf{85}, 4972 (2000). 

\bibitem{HHH_bound} P. Horodecki, Phys. Lett. A \textbf{232}, 333 (1997); M. Horodecki, P. Horodecki, and R. Horodecki,
Phys. Rev. Lett. \textbf{80}, 5239 (1998).



\bibitem{Bennett-partial} C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher,
Phys. Rev. A \textbf{53}, 2046 (1996).


%\bibitem{werner1998} R. Werner, Phys. Rev. A {\bf 58}, 1827 (1998).

\bibitem{jamiolkowski1972}
 A. Jamio{\l}kowski,
Rep. Math. Phys.
 \textbf{3},
 275
 (1972).

\bibitem{ariano2001}  G.M. D'Ariano and P. Lo Presti, 
Phys. Rev. A \textbf{64}, 042308 (2001).
%quant-ph/0101100.

\bibitem{was_a_footnote1} 
%One can think of it as adding three spins 
%$1/2$ and looking for subspaces with fixed total angular momentum, the only difference is that the third representation of 
%$\textrm{SU}(2)$ is complex conjugated,
%which amounts to a slight change in the
%Clebsch-Gordan coefficients, namely the 
%basis in the third subspace $\mathcal{H}_A$ should be renamed: $\ket{0}\to \ket{1}$, $\ket{1}\to -\ket{0}$.
%%%
This is like adding three spins 
$1/2$ and looking for subspaces with fixed total angular momentum. The difference is that the third representation of 
$\textrm{SU}(2)$ is complex conjugated,
which 
requires 
%amounts to a slight change in the
%Clebsch-Gordan coefficients, namely the 
%basis in the third subspace $\mathcal{H}_A$ should be renamed: 
the renaming 
$\ket{0}\to \ket{1}$, $\ket{1}\to -\ket{0}$ in the third subspace $\mathcal{H}_A$.


\bibitem{guchhiey-lyang} 
For the global fidelity \(F_g\), classical communication turns out to be useless, and 
two independent Bu{\v z}ek-Hillery cloners  \cite{buzek1996} are optimal. 
%However, 
%the \emph{global} fidelity \(F_g\) is not the appropriate  figure of merit. 


\bibitem{Lewen-Jamiol} J.I. Cirac, W. D{\" u}r, B. Kraus, and M. Lewenstein,
Phys. Rev. Lett. \textbf{86}, 544 (2001).





\bibitem{nlwe} C.H. Bennett, 
D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, and W.K. Wootters,
Phys. Rev. A \textbf{59}, 1070 (1999).


\bibitem{was_a_footnote2} 
Although we did numerics, we were able to obtain analytical results, 
as the numerical calculations indicated which \(a_{ij}\) can be put to zero, and with the rest, we could deal symbolically.

\bibitem{activation} P. Horodecki, M. Horodecki, and R. Horodecki,
Phys. Rev. Lett.  \textbf{82}, 1056 (1999);  P.W. Shor, J.A. Smolin, and B.M. Terhal,
\emph{ibid.} \textbf{86}, 2681 (2001);
T. Eggeling, K.G.H. Vollbrecht, R.F. Werner, and M.M. Wolf,
\emph{ibid.} \textbf{87}, 257902 (2001).


\bibitem{martin2} S. Ishizaka, Phys. Rev. Lett. \textbf{93}, 190501 (2004);
 S. Ishizaka and M. B. Plenio, quant-ph/0412193.


\bibitem{Marek} M. \.Zukowski, A. Zeilinger, M. Horne, and H. Weinfurter, Acta Phys. Pol. A \textbf{93}, 187 (1998);
M. Hillery, V. Bu{\v z}ek, and A. Berthiaume, Phys. Rev. A \textbf{59}, 1829 (1999).


\bibitem{byapok}  V. Scarani and  N. Gisin, Phys. Rev. Lett. \textbf{87},  117901 (2001);
 V. Scarani and  N. Gisin, Phys. Rev. A \textbf{65}, 012311 (2002);
  A. Sen(De), U. Sen, and M. \.Zukowski,
Phys. Rev. A \textbf{68}, 032309 (2003).


\bibitem{darun-paper} M. Lewenstein \emph{et al.}, in preparation.


%\bibitem{bennett1996}
 %       C. H. Bennett,  D. P. DiVincenzo, J. A. Smolin,
  %                and W. K.  Wootters,
%       title={Mixed-state entanglement and quantum error correction},
   %     Phys. Rev. A
    %    54, 3824
     %   (1996).

\end{thebibliography}

\end{document}


quant-ph0502103.extracted_bib
\begin{thebibliography}{99}

%\bibitem[{*}] aInstituci\'o Catalana de Recerca i Estudis Avan{\c c}ats.


\bibitem{pakhi} Th. Richter and W. Vogel,
Phys. Rev. Lett. \textbf{89}, 283601 (2002). 

\bibitem{lagaan} M.A. Nielsen and I.C. Chuang, \emph{Quantum Computation and Quantum Information} (Cambridge University Press, Cambridge, 2000).


\bibitem{sundar} A.S. Holevo, Probl. Pereda. Inf. \textbf{9}, 3 1973 [Probl. Inf.
Transm. \textbf{9}, 110 (1973)]; R. Josza, D. Robb, and W. Wotters, Phys. Rev. A, \textbf{49}, 668 (1994).


\bibitem{royal-bengal} P. Badzi{\c{a}}g, M. Horodecki, A. Sen(De), and
U. Sen, Phys. Rev. Lett. \textbf{91}, 117901 (2003); 
A. Sen(De), U. Sen, and M. Lewenstein, quant-ph/0505137. 



\bibitem{chandrabindu} C.H. Bennett and S.J. Wiesner, 
Phys. Rev. Lett. \textbf{69}, 2881 (1992). 

\bibitem{bisarga} D. Bru\ss, G.M. D'Ariano, M. Lewenstein, C. Macchiavello, A. Sen(De), and U. Sen,
Phys. Rev. Lett. \textbf{93}, 210501 (2004).


\bibitem{wootters} W.K. Wootters and W.H. Zurek, Nature (London) \textbf{299}, 802 (1982); D. Dieks, Phys. Lett. \textbf{92A}, 271 (1982).

 \bibitem{buzek1996}  
V. Bu{\v z}ek and M. Hillery,
Phys. Rev. A \textbf{54}, 1844 (1996). 


\bibitem{cloning} 
 D. Bru{\ss}, 
D.P. DiVincenzo, A. Ekert, C.A. Fuchs, C. Macchiavello, and J.A. Smolin,
% \emph{et al.}, 
Phys. Rev. A \textbf{57}, 2368 (1998); R. Werner, \emph{ibid.} {\bf 58}, 1827 (1998).

%put some cloning papers if space allows

\bibitem{werner1998} M. Keyl and R.F. Werner,
%Comments: 16 Pages, REVTeX
J. Math. Phys. \textbf{40}, 3283 (1999).

%R. Werner, Phys. Rev. A {\bf 58}, 1827 (1998).

\bibitem{cloninf} N.J. Cerf, A. Ipe, and X. Rottenberg,
Phys. Rev. Lett. \textbf{85}, 1754 (2000).
%cloning in infinite dimensions (Cerf)

\bibitem{lamoureux2004}
        L.-P. Lamoureux, P. Navez, J.
                  Fiur\'{a}\v{s}ek, and N. J. Cerf,
%       title={Cloning of entanglement of a pair of quantum bits},
        Phys. Rev. A
        \textbf{69}, 040301 (R), 
        (2004);
        E. Karpov, P. Navez, and N. J. Cerf, Phys. Rev. A \textbf{72}, 042314 (2005).


\bibitem{novotny2004}
J. Novotn\'{y}, G. Alber, and I. Jex, Phys. Rev. A \textbf{71}, 042332 (2005).
%title={Optimal copying of entangled two-qubit states},
%quant-ph/0411105.

\bibitem{sens}  M. Horodecki, A. Sen(De), and U. Sen,
Phys. Rev. A \textbf{70}, 052326 (2004).

\bibitem{martin}  S. Ghosh, G. Kar, and A. Roy,
Phys. Rev. A \textbf{69}, 052312 (2004); F. Anselmi, A. Chefles, and M. B. Plenio,
New. J. Phys. \textbf{6}, 164 (2004).

\bibitem{broadcasting1997}
V. Bu{\v z}ek, V. Vedral, M.B. Plenio, P.L. Knight, M. Hillery, Phys. Rev A \textbf{55}, 3327 (1997).


\bibitem{jomey-doi} C.H. Bennett,
% \emph{et al.}, 
G. Brassard, C. Crepeau, R. Jozsa, A. Peres, and W.K. Wootters,
Phys. Rev. Lett. \textbf{70}, 1895 (1993); C.H. Bennett, 
G. Brassard, S. Popescu, B. Schumacher, J.A. Smolin, W.K. Wootters,
\emph{ibid.} \textbf{76}, 722 (1996); C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, and W.K. Wootters,
Phys. Rev. A \textbf{54}, 3824 (1996); J. Walgate, A.J. Short, L. Hardy, and V. Vedral,
Phys. Rev. Lett. \textbf{85}, 4972 (2000). 

\bibitem{HHH_bound} P. Horodecki, Phys. Lett. A \textbf{232}, 333 (1997); M. Horodecki, P. Horodecki, and R. Horodecki,
Phys. Rev. Lett. \textbf{80}, 5239 (1998).



\bibitem{Bennett-partial} C.H. Bennett, H.J. Bernstein, S. Popescu, and B. Schumacher,
Phys. Rev. A \textbf{53}, 2046 (1996).


%\bibitem{werner1998} R. Werner, Phys. Rev. A {\bf 58}, 1827 (1998).

\bibitem{jamiolkowski1972}
 A. Jamio{\l}kowski,
Rep. Math. Phys.
 \textbf{3},
 275
 (1972).

\bibitem{ariano2001}  G.M. D'Ariano and P. Lo Presti, 
Phys. Rev. A \textbf{64}, 042308 (2001).
%quant-ph/0101100.

\bibitem{was_a_footnote1} 
%One can think of it as adding three spins 
%$1/2$ and looking for subspaces with fixed total angular momentum, the only difference is that the third representation of 
%$\textrm{SU}(2)$ is complex conjugated,
%which amounts to a slight change in the
%Clebsch-Gordan coefficients, namely the 
%basis in the third subspace $\mathcal{H}_A$ should be renamed: $\ket{0}\to \ket{1}$, $\ket{1}\to -\ket{0}$.
%%%
This is like adding three spins 
$1/2$ and looking for subspaces with fixed total angular momentum. The difference is that the third representation of 
$\textrm{SU}(2)$ is complex conjugated,
which 
requires 
%amounts to a slight change in the
%Clebsch-Gordan coefficients, namely the 
%basis in the third subspace $\mathcal{H}_A$ should be renamed: 
the renaming 
$\ket{0}\to \ket{1}$, $\ket{1}\to -\ket{0}$ in the third subspace $\mathcal{H}_A$.


\bibitem{guchhiey-lyang} 
For the global fidelity \(F_g\), classical communication turns out to be useless, and 
two independent Bu{\v z}ek-Hillery cloners  \cite{buzek1996} are optimal. 
%However, 
%the \emph{global} fidelity \(F_g\) is not the appropriate  figure of merit. 


\bibitem{Lewen-Jamiol} J.I. Cirac, W. D{\" u}r, B. Kraus, and M. Lewenstein,
Phys. Rev. Lett. \textbf{86}, 544 (2001).





\bibitem{nlwe} C.H. Bennett, 
D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, and W.K. Wootters,
Phys. Rev. A \textbf{59}, 1070 (1999).


\bibitem{was_a_footnote2} 
Although we did numerics, we were able to obtain analytical results, 
as the numerical calculations indicated which \(a_{ij}\) can be put to zero, and with the rest, we could deal symbolically.

\bibitem{activation} P. Horodecki, M. Horodecki, and R. Horodecki,
Phys. Rev. Lett.  \textbf{82}, 1056 (1999);  P.W. Shor, J.A. Smolin, and B.M. Terhal,
\emph{ibid.} \textbf{86}, 2681 (2001);
T. Eggeling, K.G.H. Vollbrecht, R.F. Werner, and M.M. Wolf,
\emph{ibid.} \textbf{87}, 257902 (2001).


\bibitem{martin2} S. Ishizaka, Phys. Rev. Lett. \textbf{93}, 190501 (2004);
 S. Ishizaka and M. B. Plenio, quant-ph/0412193.


\bibitem{Marek} M. \.Zukowski, A. Zeilinger, M. Horne, and H. Weinfurter, Acta Phys. Pol. A \textbf{93}, 187 (1998);
M. Hillery, V. Bu{\v z}ek, and A. Berthiaume, Phys. Rev. A \textbf{59}, 1829 (1999).


\bibitem{byapok}  V. Scarani and  N. Gisin, Phys. Rev. Lett. \textbf{87},  117901 (2001);
 V. Scarani and  N. Gisin, Phys. Rev. A \textbf{65}, 012311 (2002);
  A. Sen(De), U. Sen, and M. \.Zukowski,
Phys. Rev. A \textbf{68}, 032309 (2003).


\bibitem{darun-paper} M. Lewenstein \emph{et al.}, in preparation.


%\bibitem{bennett1996}
 %       C. H. Bennett,  D. P. DiVincenzo, J. A. Smolin,
  %                and W. K.  Wootters,
%       title={Mixed-state entanglement and quantum error correction},
   %     Phys. Rev. A
    %    54, 3824
     %   (1996).

\end{thebibliography}