We show that the process of entanglement distillation is irreversible by
showing that the entanglement cost of a bound entangled state is finite. Such
irreversibility remains even if extra pure entanglement is loaned to assist the
distillation process.Comment: RevTex, 3 pages, no figures Result on indistillability of PPT states
  under pure entanglement catalytic LOCC adde

A. Uhlmann

C. H. Bennett

D. Jonathan

D. P. DiVincenzo

E. M. Rains

G. Vidal

J. I. Cirac

M. A. Nielsen

M. Horodecki

P. Horodecki

P. W. Shor

R. Josza

S. Popescu

W. K. Wootters

English

arXiv

Crossref

Irreversibility in Asymptotic Manipulations of Entanglement

We show that the process of entanglement distillation is irreversible by showing that the entanglement cost of a bound entangled state is finite. Such irreversibility remains even if extra pure entanglement is loaned to assist the distillation process

Vidal, G.

Cirac, J. I.

University of Queensland eSpace

VOLUME 86, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 JUNE 2001Irreversibility in Asymptotic Manipulations of Entanglement
G. Vidal and J. I. Cirac
Institut für Theoretische Physik, Universität Innsbruck, A-6020 Innsbruck, Austria
(Received 8 February 2001)
We show that the process of entanglement distillation is irreversible by showing that the entanglement
cost of a bound entangled state is finite. Such irreversibility remains even if extra pure entanglement is
loaned to assist the distillation process.
DOI: 10.1103/PhysRevLett.86.5803 PACS numbers: 03.67.Hk, 03.65.Ud, 03.65.CaThe appearance of irreversibility in physical processes
can be regarded as one of the most fundamental and stud-
ied problems in the history of physics. In the context of
quantum information, it was pointed out that an irreversible
loss of entanglement might be present in the process of en-
tanglement distillation [1,2]. That is, the amount of pure
entanglement that can be distilled out of N copies of some
state r might be strictly smaller than the one needed to
create those copies if only local operations and classical
communication (LOCC) are allowed and in the asymp-
totic limit (N ! `). Although when r represents a pure
state this process is known to be reversible [3], it is gen-
erally believed that for mixed states this is not the case
[4]. This last statement has not been proved so far [5,6].
In this Letter we prove it, i.e., we show that, by means of
an example, the process of entanglement distillation is in-
herently irreversible. We will also extend this result to a
broader context set by catalytic local operations and clas-
sical communication (LOCCc) [7], where pure entangle-
ment —to be subsequently returned —is loaned to assist
the distillation process.
Perhaps, the strongest indication that we have so far
of the irreversibility of entanglement distillation is given
by the existence of so-called bound entangled states [9].
Those are states from which no entanglement can be dis-
tilled but for which, in order to create a single copy, en-
tanglement is required. Notice that, in spite of being very
suggestive, this indication is not conclusive. It does not
rule out the possibility that, in order to create a larger num-
ber of copies, the amount of entanglement needed per copy
vanishes in the asymptotic limit. Although this seems un-
likely, it has not been disproved so far. On the other hand, it
is also not clear yet whether bound entangled states can be
activated, and ultimately distilled, with the help of some
borrowed pure entanglement. This would still leave an
open door for some form of catalytic reversibility.
In this Letter we will show that the bound entangled
state with positive partial transposition (PPT) introduced
in Ref. [10] has a nonvanishing entanglement cost in the
asymptotic limit. We will also show that more pure en-
tanglement cannot be distilled from PPT states than just
the amount that may have been used in order to assist the
distillation process. In this way, the irreversibility of the
asymptotic manipulation of entanglement in the context of0031-90070186(25)5803(4)$15.00LOCC— and also in that of LOCCc —will immediately
follow.
Let us formulate more precisely the problem. We con-
sider two parties located in spatially separated regions and
possessing N copies of the state r. Let us consider a trans-
formation r≠N ! N which fulfills
lim
N!`
DN , jC Cj≠M ! 0 , (1)
for some integer M depending on N , where jC 
j0, 1 2 j1, 0p2 is the two-qubit singlet state and D is
a properly chosen distance measure [11]. The entangle-
ment of distillation EDr is defined as the maximal
asymptotic ratio MN with respect to all possible trans-
formations which consist of LOCC [12]. On the other
hand, let us assume now that the parties possess M two-
qubit singlet states and they are able to transform them
into the state rM fulfilling
lim
M!`
DrM ,r≠N  ! 0 , (2)
for some integer N . The entanglement cost ECr is de-
fined as the minimal asymptotic ratio MN also with re-
spect to all LOCC [13]. The distillation process of a state
r is irreversible if EDr , ECr.
Let us consider a density operator r acting on HA ≠ HB
and let us call P the projector onto the range of r. Then,
we have that the entanglement cost of r can be bounded
below as follows:
Theorem 1.— If e, fjP≠N je, f # aN for all normal-
ized product vectors je, f [ HA≠N ≠ HB≠N , then
ECr $ 2 log2a.
Proof.—We will use the results of Ref. [13], where it is
shown that
ECr  lim
N!`
Ef r≠N 
N
, (3)
where the limit exists. The entanglement of formation Ef
can be determined by considering decompositions of the
form [1,14]
r≠N 
X
i
pijCi Cij , (4)
and minimizing the quantity
P
i piECi with respect to
all possible decompositions. Here, E denotes the entropy© 2001 The American Physical Society 5803
VOLUME 86, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 JUNE 2001of entanglement [3]. By writing the Schmidt decomposi-
tion jCi 
P
k ci,kjei,k , fi,k, we see that
jci,kj2 # ei,k , fi,kjP≠N jei,k , fi,k # aN , (5)
where the first inequality is a consequence of the fact that
jCi Cij # P≠N , since all the vectors jCi must be in
the range of r≠N . Given the fact that
P
k jci,kj2  1,
we obtain ECi $ 2N log2a for all i and thereforeP
i piECi $ 2N log2a for all decompositions. 
Let us consider the bound entangled state rb intro-
duced in Ref. [10], where HA  HB  C3. It is defined
as rb  Pb4, where Pb is a projector operator onto the
orthogonal complement to the subspace spanned by the
following vectors:
j0 ≠ j0 1 j1 ,
j0 1 j1 ≠ j2 ,
j2 ≠ j1 1 j2 ,
j1 1 j2 ≠ j0 ,
j0 2 j1 1 j2 ≠ j0 2 j1 1 j2 .
This state has a positive partial transposition, and there-
fore it is not distillable [EDrb  0] [9]. Later we will
elaborate on this result. Our goal for now is to show that
ECrb . 0.
We begin by proving the following two properties of the
operator Pb :
' 1 Pb 
X
k
jak , bk ak , bkj , (6a)
a1  sup
je,ffi0
e, fjPbje, f  , 1 . (6b)
Equation (6a) indicates that the operator '1 Pb is sepa-
rable. This can be proven by showing that the projec-
tor operators P1  ' 2 ja0,a0 a0, a0j and P2  Pb 1
ja0,a0 a0, a0j are both separable, where ja0  j0 2
j1 1 j2p3. By choosing ja1 and ja2 in such a way
that jak2k0 forms an orthonormal basis, one immedi-
ately sees that the range of P1 is spanned by the mutu-
ally orthogonal product vectors jak1 , ak2, where k1, k2 
0, 1, 2, except for k1  k2  0, and therefore P1 is sepa-
rable. Analogously, the range of P2 is spanned by the
following mutually orthogonal product vectors
j0 ≠ j0 2 j1 ,
j0 2 j1 ≠ j2 ,
j2 ≠ j1 2 j2 ,
j1 2 j2 ≠ j0 ,
j1 ≠ j1 .
On the other hand, Eq. (6b) is a direct consequence of the
fact that the range of Pb contains no product vectors and
that e, fjPbje, f  is a continuous function of je, f  de-
fined on a compact set so that it reaches its supremum [15].
We will now show that, for any normalized product
vector jeN , fN  where jeN , jfN  [ C3≠N ,5804eN , fN jP≠Nb jeN , fN  , bN , (7)
where b  1 1 a12 , 1. Then, the above theorem
readily implies that the entanglement cost EC of the bound
entangled state rb  Pb4 (and of any mixed state with
the same support Pb) is finite. We will use induction over
the number of copies N to show that Eq. (7) holds. For
N  1 it is true because of Eq. (6b) and a1 , b. Now,
let us assume that it is true for a given N . Then, for any
product vector jeN11, fN11 [ C3≠N11 ≠ C3≠N11 we
have
eN11, fN11j ' 1 Pb ≠∑
' 2
1
bN
P≠Nb
∏
jeN11, fN11 $ 0 . (8)
The reason is that by using (6a) and defining jeNk  
ak j eN11 and j fNk   bkjfN11 we can write the left-
hand side of this equation as
X
k
eNk , f
N
k j
∑
' 2
1
bN
P≠Nb
∏
jeNk , fNk  , (9)
where all the terms in the sum are positive according to the
induction hypothesis (7). Now, we can write
'1 Pb ≠
∑
'2
1
bN
P≠Nb
∏
# '1 Pb ≠ '2 2
bN
P≠N11b .
(10)
By substituting this expression in Eq. (8), we arrive at
eN11, fN11jP≠N11b jeN11, fN11 #
bN
1
2 1 1 e
N11, fN11jPb ≠ 'jeN11, fN11 #bN11,
(11)
as we wanted to prove.
The very same techniques can be applied to obtain lower
bounds for the entanglement cost EC also for more general
mixed states. Notice that the relevant ingredients we have
used are Eqs. (6a) and (6b), and that both conditions are
only concerned with the support of the state rb . Therefore,
for any projector P satisfying Eqs. (6a) and (6b), we get
a nontrivial bound for the entanglement cost of any state
supported on it [16].
We move now to consider the distillability of the PPT
state rb and the extension of the irreversibility result to cat-
alytic (i.e., LOCCc-based) distillation. In [9] it was shown
that inseparable states sb with PPT cannot be distilled into
two-qubit singlet states jC using LOCC. The original
proof relies on the fact that singlet states have a negative
partial transposition (NPT), and LOCC cannot transform
a PPT state into a NPT state. Notice, however, that if the
parties initially share, in addition to the N copies of the
state sb , L two-qubit singlet states jC, then the original
argument cannot be applied, because s≠Nb ≠ jC Cj≠L is
a NPT state for any L $ 1. And, then, maybe the trans-
formation
s≠Nb ≠ jC Cj≠L ! jC Cj≠M1L (12)
VOLUME 86, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 JUNE 2001is asymptotically possible with some finite ratio MN , in
what would be a LOCCc distillation. Thus, our previous
results for the state rb do not yet exclude the possibility
that in the large N limit the equivalence
r≠Nb ≠ jC Cj≠L 	 jC Cj≠M1L (12)
under LOCC holds, i.e., the distillation of rb can be turned
into a reversible process using entanglement catalysis.
The following general result on bound entanglement
readily implies that rb is not distillable even with LOCCc,
thereby providing an example of asymptotic irreversibil-
ity also in a broader sense than that of LOCC. While the
proof we present here is original, the theorem also follows
from results originally derived in [17].
Theorem 2.— Given N copies of a PPT state sb and K
copies of some other state s, the number of singlets that
can be asymptotically distilled from them are, at most, the
number of singlets required to create s≠K :
EDs≠Nb ≠ s≠K  # KECs , (14)
which in particular means that
EDsb ≠ s # ECs, EDsb  0 . (15)
Proof.—Consider the upper bound on distillability [18]
given by the logarithmic negativity EN r  log2jjrTB jj,
where TB stands for partial transposition and jjAjj1 
tr
p
AyA is the trace norm of A. EN is an additive function
which vanishes for PPT states and is 1 for singlet states.
Therefore, for general N ,L we have
EN s≠Nb ≠ jC Cj≠L  L , (16)
which implies that, at most, L singlet states can be dis-
tilled from s≠Nb ≠ jC Cj≠L. Setting L  KECs, and
observing that such a number of singlets is sufficient to
create s≠K locally, so that no more singlets can be dis-
tilled from s≠Nb ≠ s≠K than from s≠Nb ≠ jC Cj≠L, we
obtain Eq. (14). 
Thus, an optimal LOCC transformation of the form of
Eq. (12) has M  0, and pure entanglement does not help
at distilling PPT states for LOCCc transformations [19].
On the other hand, it is easy to see that the entanglement
cost of creating a mixed and a pure state is additive, so
that [20]
ECs≠Nb ≠ jC Cj≠L  NECsb 1 L . (17)
Then, when considering both the upper bound on distill-
ability of Eq. (16) and the entanglement cost of Eq. (17),
both applied to the PPT state rb for which we have proved
that ECrb . 0, we readily conclude that the asymptotic
LOCCc manipulation of entanglement is also irreversible.
In summary, we have shown that the asymptotic en-
tanglement cost EC for locally preparing a given bound
entangled state rb is finite. Since no pure-state entangle-
ment can be distilled from the state rb even in the
asymptotic limit [i.e., EDrb  0], this result implies
that the asymptotic interconversion, by means of LOCC,between pure- and mixed-state entanglement is in general
not a reversible process. We have finally proved that
such an irreversibility also occurs for asymptotic LOCCc
transformations.
The authors acknowledge discussions with Jens Eisert,
Paweł Horodecki, Maciej Lewenstein, and Martin B. Ple-
nio. This work was supported by the Austrian Science
Foundation under the SFB “Control and Measurement of
Coherent Quantum Systems” (Project 11), the European
Community under the TMR network ERB-FMRX-CT96-
0087, project EQUIP (Contract No. IST-1999-11053), and
Contract No. HPMF-CT-1999-00200, the European Sci-
ence Foundation, and the Institute for Quantum Informa-
tion GmbH. We thank the Erwin Schrödinger Institute for
hosting us during the quantum information program, where
some of the ideas for this work were developed.
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D. Rohrlich, Phys. Rev. A 56, R3319 (1997).
[4] For a single copy N  1 or even for a finite number of
copies, the process of entanglement distillation is irrever-
sible, even for pure states. See H.-K. Lo and S. Popescu,
quant-ph/9707038; M. A. Nielsen, Phys. Rev. Lett. 83, 436
(1999); G. Vidal ibid. 83, 1046 (1999).
[5] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.
Lett. 84, 4260 (2000). This paper shows that for a family
of states the process of distillation is irreversible. However,
the proof of this statement is based on the results of E. M.
Rains [Phys. Rev. A 60, 179 (1999)] which were corrected
in an erratum [E. M. Rains, Phys. Rev. A 63, 019902(E)
(2001) (See also K. G. H. Vollbrecht and R. F. Werner,
quant-ph/0010095.) As a consequence, the arguments lead-
ing to the statement that the process of distillation is irre-
versible are invalid. Thus, this question remains open.
[6] P. W. Shor, J. A. Smolin, and B. M. Terhal, Phys. Rev. Lett.
86, 2681 (2001). In this paper it is argued that if certain
conjecture on indistillability of some Werner states is true,
then the entanglement cost EC of some other PPT en-
tanglement is nonzero. This result relies, however, on an
unproved conjecture.
[7] Entanglement assisted LOCC (ELOCC) transformations
were studied by D. Jonathan and M. B. Plenio, [Phys. Rev.
Lett. 83, 3566 (1999)] in the context of transformations
of a finite number of copies of a pure state, where they
showed that entanglement catalysis is possible. Here we
will adopt the nomenclature suggested in [8] and will write
LOCCc instead of ELOCC.
[8] C. H. Bennett, S. Popescu, D. Rohrlich, J. A. Smolin, and
A. V. Thapliyal, Phys. Rev. A 63, 012307 (2001), and will
write LOCCc instead of ELOCC.
[9] M. Horodecki, P. Horodecki, and R. Horodecki, Phys. Rev.
Lett. 80, 5239 (1998); P. Horodecki, Phys. Lett. A 232,
333 (1997).5805
VOLUME 86, NUMBER 25 P H Y S I C A L R E V I E W L E T T E R S 18 JUNE 2001[10] C. H. Bennett, D. P. DiVincenzo, T. Mor, P. W. Shor, J. A.
Smolin, and B. M. Terhal, Phys. Rev. Lett. 82, 5385 (1999).
[11] The distance D used is the Bures distance given by
Dr,r0  2
p
1 2 Fr,r0, where Fr,r0 
trr12r0r1212 is the Uhlmann fidelity [A. Uhlmann,
Rep. Math. Phys. 9, 273 (1976). For justification of the
use of this distance, see R. Josza, J. Mod. Opt. 41, 2315
(1994).
[12] C. H. Bennett, G. Brassard, S. Popescu, B. Schumacher,
J. A. Smolin, and W. K. Wootters, Phys. Rev. Lett. 76, 722
(1996); E. M. Rains, Phys. Rev. A 60, 173 (1999).
[13] P. M. Hayden, M. Horodecki, and B. Terhal, quant-ph/
0008134; M. A. Nielsen, Phys. Rev. A 61, 064301 (2000).
[14] W. K. Wootters, Phys. Rev. Lett. 80, 2245 (1998).
[15] B. M. Terhal, quant-ph/9810091. In this paper an upper
bound to a1 is given. Numerically, we find a1 , 0.98.
[16] Consider, as an alternative— i.e., not using Eq. (6a)—
example of how to get bounds for EC , the projector P
spanned by the vectors
q
2
3 j0, 2 1
q
1
3 j1, 0 ,
q
2
3 j2, 0 1
q
1
3 j1, 2 ,
of a C3 ≠ C3 system. For any product vector je, f [
C3 ≠ C3 we have that
e, fjPje, f  # ejtrBPje  ej 23 je  23 ,5806and, similarly, for any product vector jeN , fN , where
jeN , jfN  [ C3≠N ,
eN , fN jP≠N jeN , fN #  23 N .
Then the above theorem implies that the entanglement
cost EC of any state r supported on P is at least
2 log2
2
3 .
[17] See propositon 3 in section VI of D. P. DiVincenzo,
T. Mor, P. W. Shor, J. A. Smolin, and B. M. Terhal,
quant-ph/9908070. See also D. P. DiVincenzo, P. W. Shor,
J. A. Smolin, B. M. Terhal, and A. V. Thapliyal, Phys.
Rev. A 61, 062312 (2000).
[18] This bound has been derived by R. F. Werner [see G. Vidal
and R. F. Werner quant-ph/0102117] and by the Horodecki
family in [5].
[19] Let us marginally comment on a weaker result that also
follows from Eq. (16). Namely, that PPT states cannot
be distilled in the so-called LOCCq [8] setting, when an
asymptotically vanishing amount (per copy to be distilled)
of pure entanglement is used in the distillation process. In
a sense, this provides robustness to the original proof [9]
of indistillability of PPT states.
[20] Since EC is defined in an asymptotic sense, it is also an
additive for identical copies of a state, and thus ECs≠Nb  
NECsb.


Irreversibility in asymptotic manipulations of entanglement

Irreversibility in asymptotic manipulations of entanglement

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