Mixed-state entanglement and distillation: is there a ``bound'' entanglement in nature?

4 Pages·0.12 MB·English

by  Michal Horodecki

Mixed-state entanglement and distillation: is there a “bound” entanglement in
nature?
Micha l Horodecki
∗
Institute of Theoretical Physics and Astrophysics
University of Gdan´sk, 80–952 Gdan´sk, Poland
Pawe l Horodecki
∗∗
Faculty of Applied Physics and Mathematics
Technical University of Gdan´sk, 80–952 Gdan´sk, Poland
Ryszard Horodecki
∗∗∗
Institute of Theoretical Physics and Astrophysics
8 University of Gdan´sk, 80–952 Gdan´sk, Poland
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an ensemble described by some mixed state by means of

n Itisshownthatifamixedstatecanbedistilledtothesin-
local quantum operations and classical communication
a glet form, it must violate partial transposition criterion [A.
[10].
J Peres, Phys. Rev. Lett. 76, 1413 (1996)]. It implies that
The process can be described as follows : the two ob-
9 there are two qualitatively different types of entanglement: “
2 free” entanglement which is distillable, and “bound” entan- servers, Alice and Bob, each have N quantum systems
coming from entangled pairs prepared in a given state
glementwhichcannotbebroughttothesingletformusefulfor
1 quantum communication purposes. Possible physical mean- ρ. Each one can perform local operations with her/his
v
ing of the result is discussed. N particles, and exchange classical information with the
9
other one. The question is whether they can in this way
6
0 Pacs Numbers: 03.65.Bz obtain a pair of entangled qubits (the rest of the quan-
1 tum systems being discarded). They need not succeed
0 every time, but at least they know when they have been
8 SincethefamousEinstein,PodolskyandRosen[1]and
successful. If they managed to do this, one says that
9 Schr¨odinger [2] papers quantum entanglement still re-
they have distilled some amount of pure entanglement
/ mains one of the most striking implications of quantum
h fromthe state̺. Subsequently, the distilledsingletpairs
p formalism. In recent years, a great effort was made to
can be used e.g. for reliable transmission of quantum
- understand a role of entanglement in nature and funda-
t information via teleportation [5].
n mental applications were found in the field of quantum
Recently, it has been shown [12] that any inseparable
a information theory [3–6]. The most familiar example of
u pure entangledstate is the singletstate [7] of two spin-1 two-qubit state [13] represents the entanglement which,
q 2 however small, can be distilled to a singlet form. The
particles
v: result was obtained by use of the necessary [14] and suf-
i 1 ficient [15] condition of separability for two-qubit states,
X Ψ = ( ), (1)
− √2 |↑↓i−|↓↑i local filtering [17,16] and Bennett et al. distillation pro-
r tocol [10]i.
a whichcannotbe reducedto directproductby anytrans-
In this context it seems very natural to make the fol-
formationof the bases pertaining to eachone ofthe par-
lowing conjecture:
ticles.
Conjecture - Any inseparable state can be distilled to
Inpractice,duetodecoherenceeffects,weusuallydeal
the singlet form.
with mixed states [8]. A mixed state of quantum system
Surprisingly enough, this conjecture is wrong. In the
consistingoftwosubsystemsissupposedtorepresenten-
present Letter we will show that there are inseparable
tanglement if it is inseparable [9] i.e. cannot be written
statesthat cannot be distilled. Morespecifically, wefirst
in the form
show that any state which can be distilled must violate
̺= p ̺A ̺B, p 0, p =1. (2) Peres separability criterion [14]. Then the result fol-
X i i ⊗ i i ≥ X i
i i lows from the fact [18] that there are inseparable states
that satisfy the criterion. It shows that there are two
were ̺A and ̺B are states for the two subsystems. How-
i i qualitatively different types of entanglement. The first,
ever, to use the entanglement for quantum information
“free”entanglement, canbe distilled to the singletform.
processing, we must have it in pure singlet form. The
The second type of entanglement is not distillable and
procedureofconvertingmixedstateentanglementto the
is considered here in analogy with thermodynamics as a
singlet form is called distillation [10]. It amounts to ex-
“bound” entanglementwhich cannotbe used to perform
traction of pairs [11] of particles in singlet state from
1
a useful “informational work” like reliable transmission Note that the operators A and B act into two-
i0 i0
of quantum data via teleportation. dimensional space C2, hence they can be written in the
Now, let us first shortly describe the Peres criterion. form
A state ̺ satisfies the criterion, if all eigenvalues of its
partial transposition ̺TB are nonnegative (i.e. if ̺TB is Ai0 =|0ihψA|+|1ihφA|, Bi0 =|0ihψB|+|1ihφB|, (7)
a positive operator). Here the partial transposition ̺TB
where 1 and 0 constituteorthonormalbasisinC2 and
absassoisciiasteddefiwnietdhbaynathrbeitmraartyrixpreoldemucetnotsrtihnotnhoirsmbaalseisi:⊗fj ψA,φA| ∈i HA⊗|Ni, ψB,φB ∈ HB⊗N are arbitrary (possi-
bly unnormalized) vectors. Let us now consider two-
̺TmBµ,nν ≡hem⊗fµ|̺TB|en⊗fνi=̺mν,nµ. (3) dimensional projectors PA and PB which project onto
the spaces spanned by ψ ,φ and ψ ,φ respectively.
A A B B
Clearly, the matrix ̺TB depends on the basis, but its Then we have
eigenvalues do not. Thus given a state, one can check
whether it violates the criterion performing the partial ̺i0 =Ai0 ⊗Bi0(cid:0)PA⊗PB̺⊗NPA⊗PB(cid:1)A†i0 ⊗Bi†0. (8)
transpositioninanarbitraryproductbasis. Inparticular,
Now, since a product action cannot convert separable
it implies that ̺ violates the criterion if and only if any
N-fold tensor product ̺ N =̺ ... ̺ does [14]. state into inseparable one, we obtain that also the state
⊗
⊗ ⊗
Peres showed that the crite|rion{Nzmus}t be satisfied by ̺′ =PA⊗PB̺⊗NPA⊗PB (9)
anyseparablestate[14]. Ithasbeenalsoshown[15]that is inseparable. Let us write this state in basis f
i
for two-qubit (and qubit-trit) states the criterion is also |gki,i=1,2,...,dimHA⊗N,k =1,2,...,dimHB⊗N wit|hfiou⊗r
sufficient condition for separability. This does not hold vectors f1 , f2 (g1 , g2 ) spanning the subspaces de-
| i | i | i | i
for higher dimensions. The explicit examples of insepa- fined by projectors P , P . The only nonzero matrix
A B
rable mixtures satisfying criterion were constructed [18]. elements are due to products of those vectors and they
Now we are in position to present the main result of define a 4 4 matrix M2q which can be thought as two-
×
this Letter. Suppose Alice and Bob have a large number qubit state. The operation of partial transposition on
N of pairs each in a state ̺ acting on the Hilbert space ̺ affects only those elements (as the remaining ones are
′
H =HA⊗HB. Then the joint state of N pairs is given equaltozero). IfM2q werepositiveafterpartialtranspo-
by̺⊗N. Supposenowthatthestate̺isdistillable. This sition,then,due tothe sufficiencyofthe partialtranspo-
means,thatAliceandBobareabletoobtainpuresinglet sitiontest fortwo-qubitcase[15],M2q wouldrepresenta
two-qubit pairs for N tending to infinity. This however separable two-qubit state. Hence, if embedded into the
implies, that for some finite N, they are able to obtain whole space N, it would still remain separable. Con-
⊗
H
an inseparable two-qubit state ̺˜2q. The most general sequently, the state ̺′ would be separable, which is the
operation producing a two-qubit pair they can perform contradiction. Thus partial transposition of M2q must
over the initial amount of N pairs can be written in the be negative. Now, since M2q is formed by all nonzero
following form [19] elementsof̺,thenweobtainthatalsothestate̺ must
′ ′
violatethePerescriterion,i. e. ̺TB musthaveanegative
1 ′
̺˜2q = M XAi⊗Bi̺⊗NA†i ⊗Bi†, (4) eigenvalue. Now let ψ be the eigenvector corresponding
i to the eigenvalue. As the vector belongs to the subspace
where M =TrPiAi⊗Bi̺⊗NA†i ⊗Bi† is the normaliza- Hψ2q(̺itNfo)lTlBowψs tahraetetqhuealm.aHtreinxceelwemeeonbttsaihnψ|̺′TB|ψi and
tion factor and Ai and Bi map the large Hilbert spaces h | ⊗ | i
HA⊗,NB intoC2. Forconvenience,wewilluseunnormalized ψ (̺⊗N)TB ψ <0. (10)
states,asthepropertyofseparabilityaswellassatisfying h | | i
the Peres criterion do not depend on the positive factor. Thusthestate̺ N violatesthepartialtranspositioncri-
⊗
Then, for unnormalized states, we omit the condition terion. However, as it was mentioned, this implies that
Pipi = 1 in the definition of separability (2). Conse- also ̺ does. All the above considerationcan be formally
quently let summarised as follows. If the output state of this action
appears to have negative partial transposition, then the
̺2q =XAi⊗Bi̺⊗NA†i ⊗Bi† (5) basiccomponent̺ofinputstate̺⊗N musthavehadalso
i negative partial transposition . This means nothing but
and that any action of type (4) on ̺ (including collecting N
pairs) preserves positivity of partial transposition. This
̺i =Ai⊗Bi̺⊗NA†i ⊗Bi†. (6) result can be generalized [20]: any action of the form
Since ̺2q is inseparable then at least for some i = i0 M1 PiAi ⊗Bi̺⊗NA†i ⊗Bi† producing an arbitrary two-
the state ̺ must be inseparable. Indeed, by summing component system (not necessarily 2 2 one) preserves
i0 ×
separable states we cannot get inseparable one. positivity of partial transposition.
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Thus we showed that if a state ̺ is distillable, it can be always converted via distillation protocol to the
must violate the Peres separability criterion. It is an “active” singlet form.
important result as it implies that there are insepara- To complete the analogy, one could consider the
ble states which cannot be distilled! Indeed, quite re- asymptotic number of singlets which are needed to pro-
centlyoneofus[18]constructedinseparablestateswhich duce a given mixed state as internal entanglement E
int
do not violate the criterion. Some of those peculiar (thecounterpartofinternalenergy)[21]. Thenthebound
states are density matrices for two spin-1 particles (the entanglementcanbequantitativelydefinedbythefollow-
two-trit case). Using the standard basis for this case ( ing equation
1 1 , 1 2 , 1 3 , 2 1 , 2 2 , and so on ... ) those
m| ia|triic|esi|cain|bie|wiri|ttie|niin| tih|eiform: Eint =Efree+Ebound. (12)
a 0 0 0 a 0 0 0 a In particular, for pure states we have E = E and
int free
 0 a 0 0 0 0 0 0 0  E = 0. Indeed, pure states can be converted in a
bound
 0 0 a 0 0 0 0 0 0  “lossless”wayintoactivesingletform[16]. Inthepresent
 0 0 0 a 0 0 0 0 0  letter we showed that there exist inseparable states hav-
̺a = 1 a 0 0 0 a 0 0 0 a , (11) ing reciprocal properties. Namely for the states of type
8a+1 0 0 0 0 0 a 0 0 0  (11) we have Eint =Ebound and Efree =0.
 0 0 0 0 0 0 1+2a 0 √12−a2  Now the question arises: is it that Eint = Ebound = 0
 0 0 0 0 0 0 0 a 0  or = 0? Both cases are curious. In the first case, we
  6
a 0 0 0 a 0 √1 a2 0 1+a  would have inseparable states which can be produced
2− 2
from asymptotically zero number of singlet pairs. This
with 0 < a < 1. It has been shown [18] by means of would imply, in turn, that entanglement of formation is
independent separability criterion that those states are not additive state function [23], as by the very defini-
inseparable despite they have positive partial transposi- tion it does not vanish for any inseparable states. In the
tion. However, as we have shown above, that the den- second case, we would have curious states which absorb
sity matrices with positive partial transposition cannot entanglement in an irreversible way. To produce such
be distilled to the singlet form. Consequently, any state states, one needs some amount of entanglement. But
of the form (11) cannot be distilled. oncethestateswereproduced,thereisnowaytorecover
It is remarkable, that the question whether a state is any,howeverlittle,pieceoftheinitialentanglement. The
distillable or not has been reduced to the one whether latter is entirely lost.
there is a two-qubit entanglement in a collection of N A natural problem which arises in the context of the
pairs for some N. Thus the latter condition is the nec- presentedresultis: whatisthe physicalreasonforwhich
essary and sufficient condition for any given state to be the partial transposition is connected with distillability?
distilled. Indeed, as shown above, if a state ̺ is distill- Ourconjectureisthatitistimewhichlinksintimatelythe
ablethenthereexisttwo-dimensionalprojectionsP and two things. Indeed, transposition can be interpreted as
A
P so that the state ̺ given by eq. (9) is inseparable. theoperationoftime-reversal[24]. Alsointhecontextof
B ′
Conversely, if the latter condition is satisfied then ̺ can distillation,thereappearedtheproblemoftime. Namely,
be distilled by projecting ̺ N locally by means of P distillationis inherently connectedwith the quantumer-
⊗ A
and P and then applying the protocol proposed in [12] ror correction for quantum noisy channel supplemented
B
which is able to distill any two-qubit inseparable state. by two-way classical channel [6]. The quantum capacity
Thereisanopenquestion,whethertheconditionimplies of such channels can be strictly larger than without the
satisfying Peres criterion. Then the latter would acquire classicalchannel. However,thepricewemustpayisthat
thephysicalsense: itwouldbeequivalenttodistillability. the error correction with two-way classical communica-
Let us now discuss shortly possible physical meaning tion cannot be used to store the quantum information
of our result. As a matter of fact, we have revealed a in noisy environment [6] because one cannot send signal
kind of entanglement which cannot be used for send- backward in time. Needless to say deeper investigation
ing reliably quantum information via teleportation. Us- oftheconnectionamongthedistillation,partialtranspo-
ing an analogy with thermodynamics [22], we can con- sitionandtimereversalseemstobemorethandesirable.
sider entanglementasa counterpartof energy,andsend- Finally,itisperhapsworthtomentionaboutthecircle
ing of quantum information as a kind of “informational describedbystoryofthenonlocalityofmixedstates,be-
work”. Consequently we can consider “free entangle- ginningwiththeworkofWerner[9]. Thelattersuggested
ment” (E ) which can be distilled, and “bound en- that there are curious inseparable states which do not
free
tanglement” (E ). In particular, the free entangle- exhibit nonlocal correlations. Then Popescu[25] showed
bound
mentisnaturallyidentifiedwithdistillableentanglement that there is a subtle kind of pure quantum correlations
D as the latter says us how much qubits can we reliably whichisexhibitedbyWernermixtures. Thedistillability
teleport via the mixed state. This kind of entanglement of all two-qubit states [12] proved that all they are also
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nonlocal. One could suspect that the story will end by [21] One of the important problems is finding whether the
showingthatallinseparablestatescanbedistilled,hence so defined internal entanglement is equal to the entan-
theyarenonlocal. Hereweshowedthatitisnottrue. So, glement of formation [23]. The latter is defined [6] by
one is now faced with the problem similar to the initial E(̺) = inf ipiS(̺i), where S(̺) = −Tr̺log2̺. The
P
one i.e. are the inseparable states with positive partial infimumistakenoverallensemblesofpurestatesψi sat-
titrafonlslpowossittihoantntohnelopcraolb?leNmowce,ritnaivnileywcaonfnthotebaebosvoelvreedsublyt sisefeynintgh̺ait=EiTnrtB(̺|ψ)i=ihψliim|,NanN1dE̺(=̺⊗PN)ip(iw|ψeicihoψuild|.Istacyatnhbaet
Eint is “additification” of E).
means of distillation concept.
[22] S.PopescuandD.Rohrlich,Phys.Rev.56,3219(1997);
We would like to thank Asher Peres for helpful com-
M. Horodecki and R. Horodecki, Are there basic laws
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∗ E-mail address: [email protected]
time reversal and canonical decomposition, Report No.
∗∗ E-mail address:[email protected]
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2
system is usually referred to as qubit. Similarly, a three
level (spin-1) system is called trit.
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[20] A. Peres, private communication. It follows from the
simple fact that for any operators A,B,C,D we have
(A⊗B̺C ⊗D)TB = (A⊗DT̺TBC ⊗BT) where T is
usual transposition.
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