10 April 2000
Physics Letters A 268 Ž2000. 158–160
www.elsevier.nlrlocaterphysleta
Entanglement measures and the Hilbert–Schmidt distance
Masanao Ozawa)
School of Informatics and Sciences, Nagoya UniÕersity, Chikusa-ku, Nagoya 464-8601, Japan
Received 24 November 1999; received in revised form 14 February 2000; accepted 24 February 2000
Communicated by P.R. Holland
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. q 2000 Elsevier Science B.V. All
rights reserved.
PACS: 03.67.-a
Keywords: Entanglement; Completely positive maps; Operations; Hilbert–Schmidt norm
As classical information arises from probability
correlation between two random variables, quantum
information arises from entanglement w1,2x. 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. w3x 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 Žto 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
)
Tel.: q81-52-789-4728; fax: q81-52-721-2686.
E-mail address: mozawa@math.human.nagoya-u.ac.jp
ŽM. Ozawa..
quantum relative entropy and the Bures metric have
the CP nonexpansive property w3x, and it has been
conjectured that so does the Hilbert–Schmidt distance w4x.
In the interesting Letter w5x, 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 w4x. 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 H s H1 m H2 be the Hilbert space of a
quantum system consisting of two subsystems with
Hilbert spaces H1 and H2 . We assume that H1 and
H2 have the same finite dimension. We shall consider the notion of entanglement with respect to the
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 1 7 1 - 7
M. Ozawar Physics Letters A 268 (2000) 158–160
above two subsystems. Let T be the set of density
operators on H . The set D 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
w3,4x:
ŽE1. EŽ s . s 0 for all s g D.
ŽE2. For any family of bounded operators Vi 4 of
the form Vi s A i m Bi such that Ý i Vi † Vi s I,
Ža. EŽÝ i Vi s Vi † . F EŽ s .,
Žb. Ý iTrw Vi si Vi † x EŽ Vi si Vi †rTrw Vi si Vi † x. F EŽ s ..
Condition ŽE1. ensures that disentangled states
have a zero value of entanglement. Condition ŽE2.
ensures that the amount of entanglement does not
increase totally or in average by so-called purification procedures. Note that ŽE2-a. implies the following condition:
ŽE3. EŽ s . s EŽU1 m U2 s U1† m U2† . for all unitary operators Ui on Hi for i s 1,2.
Condition ŽE3. ensures that a local change of
basis has no effect on the amount of entanglement.
Vedral et. al. w3x proposed the following general
construction of the measure of entanglement E. Let
T = T R be a function satisfying the followD:T
ing conditions:
ŽD1. DŽ s , r . G 0 and DŽ s , s . s 0 for any s , r
g T.
ŽD2. DŽQs ,Qr . F DŽ s , r . for any s , r g T and
for any completely positive trace preserving map Q
on the space of operators on H .
Condition ŽD1. ensures that D has some properties of ‘distance’. Condition ŽD2. ensures that the
‘distance’ does not increase by any nonselective
operations. Then, it is shown that the ‘distance’
EŽ s . of a state s to the set D of disentangled
states defined by
™
E Ž s . s inf r g D D Ž s , r .
Ž 1.
satisfies conditions ŽE1. and ŽE2-a.. It is shown that
the quantum relative entropy and the Bures metric
satisfy ŽD1. and ŽD2. w3x, and it is conjectured that
the Hilbert–Schmidt distance is a reasonable candidate of a ‘distance’ to generate an entanglement
measure w4x. Here, the Hilbert–Schmidt distance is
defined by
D HS Ž s , r .
s 5 s y r 5 2HS s Tr
Ž syr .
2
159
for all s , r g T , which satisfies ŽD1. since 5 s y
r 5 HS is a true metric.
Recently, Witte and Trucks w5x claimed that the
Hilbert–Schmidt distance also satisfies ŽD2. and that
the prospective measure of entanglement, EHS , defined by
E HS Ž s . s inf r g D D HS Ž s , r .
satisfies ŽE1. and ŽE2-a..
It should be pointed out first that their suggested
proof of condition ŽD2. for D HS is not justified. Let
f be a convex function on Ž0,`. and let f Ž0. s 0. Let
F be a trace preserving positive map on the space of
operators such that 5F 5 F 1. Then, Lindblad’s theorem w6x asserts that for every positive operator A we
have
Tr f Ž F A . F Tr f Ž A . ,
Ž 2.
where f Ž 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
D HS Ž Qs ,Qr . F D HS Ž s , r .
Ž 3.
by regarding D HS as a convex function on Tq Ž H .
[ Tq Ž H . for all positive mappings Q . However, it
is not clear at all how D HS and Q satisfy the
assumptions of Lindblad’s theorem.
Now, we shall show a counterexample to the
claim that D HS satisfies condition ŽD2.. Let A and
B be 4 = 4 matrices defined by
0
1
As
0
0
0
0
0
0
0
0
0
1
0
0
,
0
0
0
0
Bs
0
0
0
1
0
0
0
0
0
0
0
0
.
0
1
0
0
Then we have
1
0
A As
0
0
†
0
0
0
0
0
0
1
0
0
0
.
0
0
0
It follows that A†A q B †B s I4 and hence
Qs s A s A† q Bs B † ,
M. Ozawar Physics Letters A 268 (2000) 158–160
160
where s is arbitrary, defines a completely positive
trace preserving map. Let s and r be density
matrices defined by
and hence
1r2
0
ss
0
0
We conclude therefore
0
0
rs 0
0
0
1r2
0
0
0
0
0
0
0
0
0
0
0
0
1r2
0
0
0
,
0
0
0
0
D HS Ž Qs ,Qr . F D HS Ž s , r .
0
0
1r4
0
and hence
D HS Ž s , r . s Tr Ž s y r .
2
s 2.
From the above counterexample, we conclude that
the inequality
Then we have
0
1r4
0
0
2
D HS Ž Qs ,Qr . ) D HS Ž s , r . .
0
0
0 .
1r2
1r4
0
2
Ž syr . s
0
0
D HS Ž Qs ,Qr . s Tr Ž Qs y Qr .
s 1.
0
0
0
1r4
0
is not generally true for completely positive trace
preserving maps Q . Therefore, it is still quite open
whether EHS is a good candidate for an entanglement
measure or not.
In order to obtain a tight bound for D HS ŽQs ,Qr .,
we take advantage of Kadison’s inequality w7x: If F
is a positive map, then we have
2
F Ž A . F 5F 5 F Ž A 2 .
Ž 4.
for all Hermitian A. Applying the above inequality
to the positive trace preserving map F s Q and
A s s y r , we have
2
2
Ž Qs y Qr . F 5 Q 5 Q Ž s y r . .
On the other hand, we have
0
0
A s A† s
0
0
0
1r2
0
0
0
0
0
0
0
0
,
0
0
By taking the trace of the both sides we obtain the
following conclusion: For any trace preserÕing positiÕe map Q and any states s and r , we haÕe
0
0
Bs B † s
0
0
0
1r2
0
0
0
0
0
0
0
0
,
0
0
The previous example shows that the bound can be
attained with 5 Q 5 s 2.
0
0
†
Ar A s
0
0
0
0
0
0
0
0
0
0
0
0
,
0
1r2
0
0
Br B † s
0
0
0
0
0
0
0
0
0
0
0
0
.
0
1r2
0
0
0
0
D HS Ž Qs ,Qr . F 5 Q 5 D HS Ž s , r . .
Acknowledgements
I thank V. Vedral and M. Murao for calling my
attention to the present problem.
References
It follows that
0
0
Ž Qs y Qr . s
0
0
2
0
1
0
0
0
0
0
0
Ž 5.
0
0
0
1
0
w1x R.F. Werner, Phys. Rev. A 40 Ž1989. 4277.
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w3x V. Vedral, M.B. Plenio, M.A. Rippin, P.L. Knight, Phys. Rev.
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w5x C. Witte, M. Trucks, Phys. Lett. A 257 Ž1999. 14.
w6x G. Lindblad, Commun. Math. Phys. 39 Ž1974. 111.
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