Entanglement measures and the Hilbert-Schmidt distance
Masanao Ozawa
In order to construct a measure of entanglement on the basis of a ``distance'' between two states, it is one of desirable properties that the ``distance'' is nonincreasing under every completely positive trace preserving map. Contrary to a recent claim, this letter shows that the Hilbert-Schmidt distance does not have this property.
mo000214.tex

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% TITLE:
  \title{\bf \Title}
  \author{\sc \Author \\ 
  \small\em  School of Informatics and Sciences,
  \small\em  Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan}
  \date{}
%---------------------------------------------------------------------
% BEGIN DOCUMENT :                                                    
  \begin{document}
% TITLE AND ABSTRACT:
\maketitle
\begin{abstract}
In order to construct a measure of entanglement on the basis of
a ``distance'' between two states, it is one of desirable properties 
that the ``distance'' is nonincreasing under every completely positive 
trace preserving map.
Contrary to a recent claim, this letter shows that the Hilbert-Schmidt 
distance does not have this property.
\end{abstract}

{\em PACS:} 03.67-a

{\em Keywords:} entanglement; completely positive maps; operations;
Hilbert-Schmidt norm
\bigskip


%---------------------------------------------------------------------
% TEXT:

\setcounter{equation}{0}

As classical information arises from probability correlation
between two random variables, quantum information arises from
entanglement \cite{Wer89,BDSW96}.  
Motivated by the finding of an entangled state 
which does not violate Bell's inequality, the problem of 
quantifying entanglement has received an increasing interest
recently.  

Vedral et.\ al.\ \cite{VPRK97} proposed three necessary conditions
that any measure of entanglement has to satisfy and showed that if 
a ``distance'' between two states has the property that it is 
nonincreasing under every completely positive trace preserving 
map (to be referred to as the CP nonexpansive property), 
the ``distance'' of a state to the set of disentangled 
states satisfies their conditions.
It has been shown that the quantum relative entropy and the Bures metric 
have the CP nonexpansive property \cite{VPRK97}, 
and it has been conjectured that so does the Hilbert-Schmidt 
distance \cite{VP98}.

In the interesting Letter \cite{WT99}, Witte and Trucks claimed
that the Hilbert-Schmidt distance really has the CP nonexpansive
property and conjectured that the distance generates a measure
of entanglement satisfying even the stronger condition posed 
later by Vedral and Plenio \cite{VP98}.  
However, it can be readily seen that their suggested proof includes 
a serious gap.  
In this Letter, it will be shown that, contrary to 
their claim, the Hilbert-Schmidt distance does not have the 
CP nonexpansive property by presenting a counterexample.

Let $\cH=\cH_{1}\otimes\cH_{2}$ be the Hilbert space of 
a quantum system consisting of two subsystems with Hilbert
spaces $\cH_{1}$ and $\cH_{2}$.
We assume that $\cH_{1}$ and $\cH_{2}$ have the same finite 
dimension.
We shall consider the notion of entanglement with respect to
the above two subsystems.
Let $\cT$ be the set of density operators on $\cH$.
The set $\cD$ of disentangled states is the set of all
convex combinations of pure tensor product states.
There are several requirements that every measure of entanglement,
$E$, should satisfy \cite{VPRK97,VP98}:

(E1) $E(\si)=0$ for all $\si\in\cD$.

(E2) For any family of bounded operators $\{V_{i}\}$ of the form
$V_{i}=A_{i}\otimes B_{i}$ such that $\sum_{i}V_{i}^{\da}V_{i}=I$,

(a) $E(\sum_{i}V_{i}\si V_{i}^{\da})\le E(\si)$,

(b) $\sum_{i}\Tr[V_{i}\si_{i}V_{i}^{\da}]
E(V_{i}\si_{i}V_{i}^{\da}/\Tr[V_{i}\si_{i}V_{i}^{\da}])
\le E(\si)$.

Condition (E1) ensures that disentangled states have a zero value
of entanglement. Condition (E2) ensures
that the amount of entanglement does not increase totally or
in average by so-called purification procedures. 
Note that (E2-a) implies the following condition:

(E3) $E(\si)=E(U_{1}\otimes U_{2} \si U_{1}^{\da}\otimes U_{2}^{\da})$
for all unitary operators $U_{i}$ on $\cH_{i}$ for $i=1,2$.

Condition (E3) ensures that a local change of basis
has no effect on the amount of entanglement.

Vedral et.~al. \cite{VPRK97} proposed the following general construction
of the measure of entanglement $E$. Let $D:\cT\times \cT\to \R$ be
a function satisfying the following conditions:

(D1) $D(\si,\rh)\ge 0$ and $D(\si,\si)=0$ for any $\si,\rh\in\cT$.

(D2) $D(\Th\si,\Th\rh)\le D(\si,\rh)$
for any $\si,\rh\in\cT$ and for any completely positive trace
preserving map $\Th$ on the space of operators on $\cH$.

Condition (D1) ensures that $D$ has some properties of ``distance''.
Condition (D2) ensures that the ``distance'' does not increase
by any nonselective operations.
Then, it is shown that the ``distance'' $E(\si)$ of a state
$\si$ to the set $\cD$ of disentangled states defined by
\beql{1115a}
E(\si)=\inf_{\rh\in\cD}D(\si,\rh)
\eeq
satisfies conditions (E1) and (E2-a).
It is shown that the quantum relative entropy and the Bures metric 
satisfy (D1) and (D2) \cite{VPRK97}, and it is conjectured that 
the Hilbert-Schmidt distance is a reasonable candidate of a
``distance'' to generate an entanglement measure \cite{VP98}.
Here, the Hilbert-Schmidt distance is defined by
$$
D_{HS}(\si,\rh)=\|\si-\rh\|^{2}_{HS}=\Tr[(\si-\rh)^{2}]
$$
for all $\si,\rh\in\cT$, which satisfies (D1) since 
$\|\si-\rh\|_{HS}$ is a true metric.

Recently, Witte and Trucks \cite{WT99} 
claimed that the Hilbert-Schmidt distance also satisfies (D2) and
that the prospective measure of entanglement, $E_{HS}$, defined by
$$
E_{HS}(\si)=\inf_{\rh\in\cD}D_{HS}(\si,\rh)
$$
satisfies (E1) and (E2-a).

It should be pointed out first that their suggested proof 
of condition (D2) for $D_{HS}$ is not justified. 
Let $f$ be a convex function on $(0,\infty)$ and let $f(0)=0$.
Let $\Ph$ be a trace preserving positive map on the space of 
operators such that $\|\Ph\|\le 1$.
Then, Lindblad's theorem  \cite{Lin74} asserts that for every 
positive operator $A$ we have
\beql{1117b}
\Tr[f(\Ph A)]\le \Tr[f(A)],
\eeq
where $f(A)$ is defined as usual through the spectral resolution 
of $A$.  
It is suggested that with the help of the above
theorem it can be shown that 
\beql{1117a}
D_{HS}(\Th\si,\Th\rh)\le D_{HS}(\si,\rh)
\eeq
by regarding $D_{HS}$ as a convex function on 
$\cT_{+}(\cH)\oplus\cT_{+}(\cH)$ for all positive mappings $\Th$.
However, it is not clear at all how $D_{HS}$ and $\Th$ satisfy the 
assumptions of Lindblad's theorem.

Now, we shall show a counterexample to the claim that $D_{HS}$
satisfies condition (D2).
Let $A$ and $B$ be $4\times4$
matrices defined by
$$
A=\left(\begin{array}{cccc}
0&0&0&0\\
1&0&0&0\\
0&0&0&0\\
0&0&1&0\end{array}
\right),
\quad
B=\left(\begin{array}{cccc}
0&0&0&0\\
0&1&0&0\\
0&0&0&0\\
0&0&0&1\end{array}
\right).
$$
Then we have
$$
A^{\dagger}A
=\left(\begin{array}{cccc}
1&0&0&0\\
0&0&0&0\\
0&0&1&0\\
0&0&0&0\end{array}
\right).
$$
It follows that $A^{\dagger}A+B^{\dagger}B=I_{4}$ and hence 
$$
\Th\si=A\si A^{\dagger}+B\si B^{\dagger},
$$
where $\si$ is arbitrary, 
defines a completely positive trace preserving map.
Let $\si$ and $\rh$ be density matrices defined by
$$
\si=\left(\begin{array}{cccc}
1/2&0&0&0\\
0&1/2&0&0\\
0&0&0&0\\
0&0&0&0\end{array}
\right),
\quad
\rh=\left(\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
0&0&1/2&0\\
0&0&0&1/2\end{array}
\right).
$$
Then we have
$$
(\si-\rh)^{2}
=\left(\begin{array}{cccc}
1/4&0&0&0\\
0&1/4&0&0\\
0&0&1/4&0\\
0&0&0&1/4\end{array}
\right)
$$
and hence
$$
D_{HS}(\si,\rh)={\rm Tr}[(\si-\rh)^{2}]=1.
$$
On the other hand, we have
$$
A\si A^{\dagger}
=\left(\begin{array}{cccc}
0&0&0&0\\
0&1/2&0&0\\
0&0&0&0\\
0&0&0&0\end{array}
\right),\quad
B\si B^{\dagger}
=\left(\begin{array}{cccc}
0&0&0&0\\
0&1/2&0&0\\
0&0&0&0\\
0&0&0&0\end{array}
\right),
$$
$$
A\rh A^{\dagger}
=\left(\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
0&0&0&0\\
0&0&0&1/2\end{array}
\right),\quad
B\rh B^{\dagger}
=\left(\begin{array}{cccc}
0&0&0&0\\
0&0&0&0\\
0&0&0&0\\
0&0&0&1/2\end{array}
\right).
$$
It follows that
$$
(\Th\si-\Th\rh)^{2}
=\left(\begin{array}{cccc}
0&0&0&0\\
0&1&0&0\\
0&0&0&0\\
0&0&0&1\end{array}
\right)
$$
and hence
$$
D_{HS}(\Th\si,\Th\rh)={\rm Tr}[(\Th\si-\Th\rh)^{2}]=2.
$$
We conclude therefore
$$
D_{HS}(\Th\si,\Th\rh)>D_{HS}(\si,\rh).
$$

From the above counterexample, we conclude that the inequality
$$
D_{HS}(\Th\si,\Th\rh)\le D_{HS}(\si,\rh)
$$
is not generally true for completely positive trace preserving 
maps $\Th$.  Therefore, it is still quite open 
whether $E_{HS}$ is a good candidate for an entanglement measure 
or not.

In order to obtain a tight bound for $D_{HS}(\Th\si,\Th\rh)$,
we take advantage of Kadison's inequality \cite{Kad52}: 
If $\Ph$ is a positive map, then we have
\beql{0210a}
\Ph(A)^{2}\le\|\Ph\|\,\Ph(A^{2})
\eeq
for all Hermitian $A$.  Applying the above inequality to the
positive trace preserving map $\Ph=\Th$ and $A=\si-\rh$, 
we have 
$$
(\Th\si-\Th\rh)^{2}\le\|\Th\|\,\Th[(\si-\rh)^{2}].
$$
By taking the trace of the both sides we obtain the following
conclusion:  {\em For any trace preserving positive map $\Th$ 
and any states $\si$ and $\rh$, we have}
\beq
D_{HS}(\Th\si,\Th\rh)\le\|\Th\|D_{HS}(\si,\rh).
\eeq
The previous example shows that the bound can be attained 
with $\|\Th\|=2$. 
\bigskip

{\bf Acknowledgements}

I thank V. Vedral and M. Murao for calling
my attention to the present problem.

%---------------------------------------------------------------------
% BIBLIOGRAPHY:                                                    
\begin{thebibliography}{1}\itemsep=0in

\bibitem{Wer89}
R.~F. Werner,
\newblock {Phys.\ Rev.\ A} {40} (1989) 4277.

\bibitem{BDSW96}
C.~H. Bennet, D.~P. DiVincenzo, J.~A. Smolin, and W.~K. Wootters,
\newblock {Phys.\ Rev.\ A} {54} (1996) 3824.

\bibitem{VPRK97}
V. Vedral, M.~B. Plenio, M.~A. Rippin, and P.~L. Knight,
\newblock {Phys.\ Rev.\ Lett.} {78} (1997) 2275.

\bibitem{VP98}
V. Vedral and M.~B. Plenio,
\newblock {Phys.\ Rev.\ A} {57} (1998) 1619.

\bibitem{WT99}
C. Witte and M. Trucks,
\newblock {Phys.\ Lett.\ A} {257} (1999) 14.

\bibitem{Lin74}
G. Lindblad,
\newblock {Commun.\ Math.\ Phys.} {39} (1974) 111.


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\end{thebibliography}
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