“Squashed entanglement”: An additive entanglement measure

“Squashed entanglement”: An additive entanglement measureMatthias Christandl and Andreas Winter Citation: J. Math. Phys. 45, 829 (2004); doi: 10.1063/1.1643788 View online: http://dx.doi.org/10.1063/1.1643788 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v45/i3 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

Downloaded 21 Aug 2013 to 128.112.200.107. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions

http://jmp.aip.org/?ver=pdfcov

http://oasc12039.247realmedia.com/RealMedia/ads/click_lx.ads/www.aip.org/pt/adcenter/pdfcover_test/L-37/726139175/x01/AIP-PT/HC_JMPCoverPg_0713/chp_books_banner1640x440.jpg/6c527a6a7131454a5049734141754f37?x

http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Matthias Christandl&possible1zone=author&alias=&displayid=AIP&ver=pdfcov

http://jmp.aip.org/search?sortby=newestdate&q=&searchzone=2&searchtype=searchin&faceted=faceted&key=AIP_ALL&possible1=Andreas Winter&possible1zone=author&alias=&displayid=AIP&ver=pdfcov


http://link.aip.org/link/doi/10.1063/1.1643788?ver=pdfcov

http://jmp.aip.org/resource/1/JMAPAQ/v45/i3?ver=pdfcov

http://www.aip.org/?ver=pdfcov


http://jmp.aip.org/about/about_the_journal?ver=pdfcov

http://jmp.aip.org/features/most_downloaded?ver=pdfcov

http://jmp.aip.org/authors?ver=pdfcov

‘‘Squashed entanglement’’: An additive entanglementmeasure

Matthias Christandla)

Center for Quantum Computation, Department of Applied Mathematics and TheoreticalPhysics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA,United Kingdom

Andreas Winterb)

School of Mathematics, University of Bristol,University Walk, Bristol BS8 1TW, United Kingdom

~Received 24 November 2003; accepted 25 November 2003!

In this paper, we present a new entanglement monotone for bipartite quantumstates. Its definition is inspired by the so-called intrinsic information of classicalcryptography and is given by the halved minimum quantum conditional mutualinformation over all tripartite state extensions. We derive certain properties of thenew measure which we call ‘‘squashed entanglement’’: it is a lower bound onentanglement of formation and an upper bound on distillable entanglement. Fur-thermore, it is convex, additive on tensor products, and superadditive in general.Continuity in the state is the only property of our entanglement measure which wecannot provide a proof for. We present some evidence, however, that our quantityhas this property, the strongest indication being a conjectured Fannes-type inequal-ity for the conditional von Neumann entropy. This inequality is proved in theclassical case. ©2004 American Institute of Physics.@DOI: 10.1063/1.1643788#

I. INTRODUCTION

Ever since Bennettet al.1,2 introduced the entanglement measures ofdistillable entanglementand entanglement of formationin order to measure the amount of nonclassical correlation in abipartite quantum state, there has been an interest in an axiomatic approach to entanglementmeasures. One natural axiom is LOCC-monotonicity, which means that an entanglement measureshould not increase underLocal Operations and Classical Communication. Furthermore, everyentanglement measure should vanish on the set of separable quantum states; it should be convex,additive, and a continuous function in the state. Though several entanglement measures have beenproposed, it turns out to be difficult to find measures that satisfy all of the above axioms. Oneunresolved question is whether or not entanglement of formation is additive. This is an importantquestion and has recently been connected to many other additivity problems in quantum informa-tion theory.24 Other examples are distillable entanglement, which shows evidence of being neitheradditive nor convex,25 and relative entropy of entanglement,28 which can be proven to benonadditive.30

In this paper we present a functional called ‘‘squashed entanglement’’ which has many ofthese desirable properties: it is convex, additive on tensor products and superadditive in general. Itis upper bound by entanglement cost, lower bound by distillable entanglement, and we are able topresent some evidence of continuity.

The remaining sections are organized as follows: in Sec. II we will define squashed entangle-ment and prove its most important properties. In Sec. III we will explain its analogy to a quantitycalled intrinsic information, known from classical cryptography. This constitutes the motivationfor our definition.

a!Electronic mail: [email protected]!Electronic mail: [email protected]

JOURNAL OF MATHEMATICAL PHYSICS VOLUME 45, NUMBER 3 MARCH 2004

8290022-2488/2004/45(3)/829/12/$22.00 © 2004 American Institute of Physics


http://dx.doi.org/10.1063/1.1643788

The only property that we could not find proof for is continuity. A detailed discussion of thisproblem follows in Sec. IV, where we show that squashed entanglement is continuous on theinterior of the set of states and where we provide evidence in favor of continuity in general. Thisevidence is based on a Fannes-type inequality for the conditional von Neumann entropy. It isconjectured in general and is true in the classical case, which we will prove in the Appendix.

II. SQUASHED ENTANGLEMENT

In this paper all Hilbert spaces are assumed to be finite dimensional, even though some of thefollowing definitions and statements make sense also in infinite dimension.

Definition 1:Let rAB be a quantum state on a bipartite Hilbert spaceH5HA^ HB . We definethe squashed entanglementof rAB by

Esq~rAB!ª inf$ 12 I ~A;BuE!:rABE extension ofrAB%.

The infimum is taken over all extensions ofrAB, i.e., over all quantum statesrABE with rAB

5Tr ErABE. I (A;BuE)ªS(AE)1S(BE)2S(ABE)2S(E) is the quantum conditional mutual in-formation of rABE.3 rA stands for the restriction of the staterABE to subsystemA, and S(A)5S(rA) is the von Neumann entropy of the underlying state, if it is clear from the context. If not,we emphasize the state in subscript,S(A)r . Note that the dimension ofE is a priori unbounded.

Tucci26 has previously defined the same functional~without the factor12) in connection with

his investigations into the relationship between quantum conditional mutual information and en-tanglement measures, in particular entanglement of formation.

Our name for this functional comes from the idea that the right choice of a conditioningsystem reduces the quantum mutual information betweenA and B, thus ‘‘squashing out’’ thenonquantum correlations. See Sec. III for a similar idea in classical cryptography, which motivatedthe above definition.

Example 2:Let rAB5uc&^cuAB be a pure state. All extensions ofrAB are of the formrABE

5rAB^ rE; therefore

12 I ~A;BuE!5S~rA!5E~ uc&),

which impliesEsq(uc&^cu)5E(uc&).Proposition 3: Esq is an entanglement monotone,i.e., it does not increase under local quan-

tum operations and classical communication (LOCC) and it is convex.Proof: According to Ref. 29 it suffices to verify thatEsq satisfies the following two criteria:

~1! For any quantum staterAB and any unilocal quantum instrument (Ek)—theEk are completelypositive maps and their sum is trace preserving6—performed on either subsystem,

Esq~rAB!>(k

pkEsq~ rkAB!,

where

pk5Tr Ek~rAB! and rkAB5

1

pkEk~rAB!.

~2! Esq is convex, i.e., for all 0<l<1,

Esq~lrAB1~12l!sAB!<lEsq~rAB!1~12l!Esq~sAB!.

In order to prove 1, we modify the proof of theorem 11.15 in Ref. 19 for our purpose. Bysymmetry we may assume that the instrument (Ek) acts unilocally onA. Now, attach two ancillasystemsA8 andA9 in statesu0&A8 and u0&A9 to the systemABE ~i!. To implement the quantumoperation

830 J. Math. Phys., Vol. 45, No. 3, March 2004 M. Christandl and A. Winter


rABE→ rAA8BEª(

k~Ek^ idE!~rABE! ^ uk&^kuA8,

with (uk&A8)k being an orthonormal basis onA8, we perform~ii ! a unitary transformationU onAA8A9 followed by ~iii ! tracing out the systemA9. Here, i denotes the systemi P$A,B,AB% afterthe unitary evolutionU. Then, for any extension ofrAB,

I ~A;BuE!5~ i!

I ~AA8A9;BuE!

5~ ii !

I ~AA8A9;BuE!

>~ iii !

I ~AA8;BuE!

5~ iv!

I ~A8;BuE!1I ~A;BuEA8!

>~v!

(k

pkI ~A;BuE!rk

>~vi!

(k

2pkEsq~rk!.

The justification of these steps is as follows: attaching auxiliary pure systems does not change theentropy of a system, step~i!. The unitary evolution affects only the systemsAA8A9 and thereforedoes not affect the quantum conditional mutual information in step~ii !. To show that discardingquantum systems cannot increase the quantum conditional mutual information

I ~AA8;BuE!<I ~AA8A9;BuE!

we expand it into

S~AA8E!1S~BE!2S~AA8BE!2S~E!<S~AA8A9E!1S~BE!2S~AA8A9BE!2S~E!,

which is equivalent to

S~AA8E!2S~AA8BE!<S~AA8A9E!2S~AA8A9BE!,

the strong subadditivity;17 this shows step~iii !, and for step~iv! we use thechain rule,

I ~XY;ZuU !5I ~X;ZuU !1I ~Y;ZuUX!.

For step ~v!, note that the first term,I (A8;BuE), is non-negative and that the second term,

I (A;BuEA8), is identical to the expression in the next line. Finally, we have~vi! sincerkABE is a

valid extension ofrk . As the original extension ofrAB was arbitrary, the claim follows.To prove convexity, property 2, consider any extensionsrABE andsABE of the statesrAB and

sAB, respectively. It is clear that we can assume, without loss of generality, that the extensions aredefined on identical systemsE. Combined,rABE andsABE form an extension

tABEE8ªlrABE

^ u0&^0uE81~12l!sABE^ u1&^1uE8

of the statetAB5lrAB1(12l)sAB. The convexity of squashed entanglement then follows fromthe observation

lI ~A;BuE!r1~12l!I ~A;BuE!s5I ~A;BuEE8!t>2Esq~tAB!.

Proposition 4: Esq is superadditive in general, and additive on tensor products, i.e.,

Esq~rAA8BB8!>Esq~rAB!1Esq~rA8B8!

is true for every density operatorrAA8BB8 on HA^ HA8^ HB^ HB8 , rAB5Tr A8B8rAA8BB8, and

rA8B85Tr ABrAA8BB8,

Esq~rAA8BB8!5Esq~rAB!1Esq~rA8B8!

831J. Math. Phys., Vol. 45, No. 3, March 2004 ‘‘Squashed entanglement’’


for rAA8BB85rAB^ rA8B8.

Proof: We start with superadditivity and assume thatrAA8BB8E on HA^ HA8^ HB^ HB8^ HE is an extension ofrAA8BB8, i.e., rAA8BB85Tr ErAA8BB8E. Then,

I ~AA8;BB8uE!5I ~A;BB8uE!1I ~A8;BB8uEA!

5I ~A;BuE!1I ~A;B8uEB!1I ~A8;B8uEA!1I ~A8;BuEAB8!

>I ~A;BuE!1I ~A8;B8uEA!>2Esq~rAB!12Esq~rA8B8!.

The first inequality is due to strong subadditivity of the von Neumann entropy. Note thatE is anextension for systemAB and EA extends systemA8B8. Hence, the last inequality is true sincesquashed entanglement is defined via the infimum over all extensions of the respective states. Thecalculation is independent of the choice of the extension, which proves superadditivity.

A special case of the above is superadditivity on product statesrAA8BB8ªrAB

^ rA8B8. Toconclude thatEsq is indeed additive on tensor products, it therefore suffices to prove subadditivityon tensor products.

Let rABE on HA^ HB^ HE be an extension ofrAB and letrA8B8E8 on HA8^ HB8^ HE8 be anextension forrA8B8. It is evident thatrABE

^ rA8B8E8 is a valid extension forrAA8BB85rAB

^ rA8B8, hence

This inequality holds for arbitrary extensions ofrAB andrA8B8. We therefore conclude thatEsq issubadditive on tensor products. h

Proposition 5: Esq is upper bounded byentanglement of formation1,2

Esq~rAB!<EF~rAB!.

Proof: Let $pk ,uCk&% be a pure state ensemble forrAB,

(k

pkuCk&^CkuAB5rAB.

The purity of the ensemble implies

(k

pkS~A!Ck5 1

2(k

pkI ~A;B!Ck.

Consider the following extensionrABE of rAB:

rABEª(

kpkuCk&^CkuAB

^ uk&^kuE.

It is elementary to compute

(k

pkS~A!Ck5 1

2(k

pkI ~A;B!Ck5 1

2 I ~A;BuE!.

Thus, it is clear that entanglement of formation can be regarded as an infimum over a certain classof extensions ofrAB. Squashed entanglement is an infimum overall extensions ofrAB, evaluatedon the same quantity12I (A;BuE) and therefore smaller or equal to entanglement of formation.h

Corollary 6: Esq is upper bounded byentanglement cost,



Esq~rAB!<EC~rAB!.

Proof: Entanglement cost is equal to the regularized entanglement of formation,12

EC~rAB!5 limn→`

1

nEF~~rAB! ^ n!.

This, together with proposition 5, and the additivity of the squashed entanglement~proposition 4!implies

EC~rAB!5 limn→`

1

nEF~~rAB! ^ n!> lim

n→`

1

nEsq~~rAB! ^ n!5Esq~rAB!.

h

Theorem 7: Squashed entanglement vanishes for every separable density matrixrAB, i.e.,

rABseparable⇒Esq~rAB!50.

Conversely, if there exists a finite extensionrABE of rAB with vanishing quantum conditionalmutual information, thenrAB is separable, i.e.,

I ~A;BuE!50 and dim H E,`⇒rAB separable.

Proof: Every separablerAB can be written as a convex combination of separable pure states,

rAB5(i

pi uc i&^c i uA^ uf i&^f i uB.

The quantum conditional mutual information of the extension

rABEª(

ipi uc i&^c i uA^ uf i&^f i uB^ u i &^ i uE,

with orthonormal states (u i &E), is zero. Squashed entanglement thus vanishes on the set of sepa-rable states.

To prove the second assertion assume that there exists an extensionrABE of rAB withI (A;BuE)50 and dimHE,`. Now, a recently obtained result13 on the structure of such statesrABE applies: there it was shown that in this case the systemE can be written as a direct sumof tensor products, such that with a suitable basis transformationE→EE8E9 the state can berewritten

rABE5(i

pir iAE8^ r i

E9B^ u i &^ i uE.

ThenrAB5( i pi(Tr E8r iAE8) ^ (Tr E9r i

E9B) is separable.Remark 8: The minimization in squashed entanglement ranges over extensions ofrAB with a

priori unbounded size. Esq(r)50 is thus possible, even if any finite extension has strictly positivequantum conditional mutual information. Therefore, without a bound on the dimension of theextending system, the second part of theorem 7 does not suffice to conclude that Esq(r

AB) impliesseparability ofrAB. A different approach to this question could be provided by a possible approxi-mate version of the main result of Ref. 13: if there is an extensionrABE with small quantumconditional mutual information, then therAB is close to a separable state. For further discussionon this question, see Secs. III and IV.



Note that the strict positivity of squashed entanglement for entangled states would, via cor-ollary 6, imply strict positivity of entanglement cost for all entangled states. This is not yet proven,but conjectured as a consequence of the additivity conjecture of entanglement of formation.

Example 9:It is worth noting that in generalEsq is strictly smaller thanEF andEC : considerthe totally antisymmetric statesAB in a two-qutrit system

sAB5 13 ~ uI&Îu1uII &ÎI u1uIII &ÎII u!,

with

uI&51

&~ u2&Au3&B2u3&Au2&B),

uII &51

&~ u3&Au1&B2u1&Au3&B),

uIII &51

&~ u1&Au2&B2u2&Au1&B).

On the one hand, it is known from Ref. 31 thatEF(sAB)5EC(sAB)51, though, on the other hand,we may consider the trivial extension,

Esq~sAB!< 12 I ~A;B!5 1

2 log 3'0.792.

The best known upper bound onED for this state, theRains bound,21 gives the only slightly

smaller value log53'0.737. It remains open if there exist states for which squashed entanglementis smaller than the Rains bound.

Proposition 10: Esq is lower bounded bydistillable entanglement1,2

ED~rAB!<Esq~rAB!.

Proof: Consider any entanglement distillation protocol by LOCC, takingn copies of the state(rAB) ^ n to a statesAB such that

isAB2us&^suABi1<d, ~1!

with us& being a maximally entangled state of Schmidt ranks. We may assume without loss ofgenerality that the support ofsA andsB is contained in thes-dimensional support of TrBus&^suand TrAus&^su, respectively. Using propositions 4 and 3, we have

nEsq~rAB!5Esq~~rAB! ^ n!>Esq~sAB!, ~2!

so that it is only necessary to estimateEsq(sAB) vs Esq(us&^suAB)5 logs ~see example 2!. For this,

let sABE be an arbitrary extension ofsAB and consider a purification of it,uC&PHABEE8 . Chainrule and monotonicity of the quantum mutual information allow us to estimate

I ~A;BuE!5I ~AE;B!2I ~E;B!>I ~A;B!2I ~EE8;AB!5I ~A;B!22S~AB!.

Further applications of Fannes inequality,9 lemma 13, give I (A;B)>2 logs2f(d)logs and2S(AB)< f (d)logs, with a functionf of d vanishing asd approaches 0. Hence

12 I ~A;BuE!> logs2 f ~d!logs.

Since this is true for all extensions, we can put this together with Eq.~2!, and obtain



Esq~rAB!>1

n~12 f ~d!!logs,

which, with n→` andd→0, concludes the proof, because we considered an arbitrary distillationprotocol. h

Remark 11: In the proof of proposition 10 we made use of the continuity of Esq in the vicinityof maximally entangled states. Similarly, Esq can be shown to be continuous in the vicinity of anypure state. This, together with proposition 3, the additivity on tensor products (second part ofproposition 4), and the normalization on Bell states, suffices to prove corollary 6 and proposition10.15

Corollary 12:

12 ~ I ~A;B!2S~AB!!<Esq~rAB!.

Proof: The recently establishedhashing inequality7 provides a lower bound for theone-waydistillable entanglement ED(rAB),

S~B!2S~AB!<ED~rAB!.

Interchanging the roles ofA andB, we have

12 ~ I ~A;B!2S~AB!!<ED~rAB!,

where we use the fact that one-way distillable entanglement is smaller or equal to distillableentanglement. This, together with the bound from proposition 10, implies the assertion.h

III. ANALOGY TO INTRINSIC INFORMATION

Intrinsic information is a quantity that serves as a measure for the correlations betweenrandom variables in information-theoretical secret-key agreement.18 The intrinsic (conditionalmutual) informationbetween two discrete random variablesX and Y, given a third discreterandom variableZ, is defined as

I ~X;Y↓Z!5 inf$I ~X;YuZ!:Z with XY→Z→Z a Markov chain%.

The infimum extends over all discrete channelsZ to Z that are specified by a conditional prob-ability distributionPZuZ .

A first idea to utilize intrinsic information for measuring quantum correlations was mentionedin Ref. 11. This inspired the proposal of aquantum analog to intrinsic information4 in which theShannon conditional mutual information plays a role similar to the quantum conditional mutualinformation in squashed entanglement. This proposal possesses certain good properties demandedof an entanglement measure, and it opened the discussion that has resulted in the current work.

Before we state some similarities in the properties that theintrinsic informationandsquashedentanglementhave in common, we would like to stress their obvious relation in terms of thedefinitions. Let uC&ABC be a purification ofrAB and let rABE be an extension ofrAB withpurificationuF&ABEE8. Remark that all purifications ofrAB are equivalent in the sense that there isa suitable unitary transformation on the purifying system with

1AB^ U:uC&ABC°uF&ABEE8.

Applying a partial trace operation over systemE8 then results in the completely positive map

L:B~HC!→B~HE!,

id^ L:uC&^CuABC°rABE.



Conversely, every staterABE constructed in this manner is an extension ofrAB.This shows that the squashed entanglement equals

Esq~rAB!5 inf$ 12 I ~A;BuE!:rABE5~ id^ L!uC&^CuABC%, ~3!

where the infimum includes all quantum operationsL:B(HC)→B(HE).In Ref. 5 it is shown that the minimization inI (X;Y↓Z) can be restricted to random variables

Z with a domain equal to that ofZ. This shows that the infimum in the definition is in effect aminimum and that the intrinsic information is a continuous function of the distributionPXYZ. It isinteresting to note that the technique used there~and, for that matter, also in the proof thatentanglement of formation is achieved as a minimum over pure state ensemblesrAB

5(kpkuCk&^CkuAB of size (rankrAB)2), does not work for our problem, and so, we do not havean easy proof of the continuity of squashed entanglement. In the following section this issue willbe discussed in some more detail.

In the cryptographic context in which it appears, intrinsic information serves as an upperbound for the secret-key rateS5S(X;YuuZ).18 S is the rate at which two parties, having access torepeated realizations ofX andY, can distill secret correlations about which a third party, holdingrealizations ofZ, is almost ignorant. This distillation procedure includes all protocols in which thetwo parties communicate via a public authenticated classical channel to which the eavesdropperhas access but cannot alter the transmitted messages. Clearly, one can interpret distillable en-tanglement as the quantum analog to the secret-key rate. On the one hand,secret quantum corre-lations, i.e., maximally entangled states of qubits, are distilled from a number of copies ofrAB. Inthe classical cryptographic setting, on the other hand, one aims at distillingsecret classical cor-relations, i.e., secret classical bits, from a number of realizations of a triple of random variablesX,Y, andZ.

We proved in proposition 10 that squashed entanglement is an upper bound for distillableentanglement. Hence, it provides a bound in entanglement theory which is analogous to the one ininformation-theoretic secure key agreement, where intrinsic information bounds the secret-keyrate from above.

This analogy extends further to the bound on the formation of quantum states~proposition 5and corollary 6!, where we know of a recently proven classical counterpart, namely, that theintrinsic information is a lower bound on the formation cost of correlations of a triple of randomvariablesX, Y, andZ from secret correlations.22

IV. THE QUESTION OF CONTINUITY

Intrinsic information, discussed in the previous section, and entanglement of formation arecontinuous functions of the probability distribution and state, respectively. This is so, because inboth cases we are able to restrict the minimization to a compact domain; in the case of intrinsicinformation to bounded rangeZ and in entanglment of formation to bounded size decompositions,noting that the functions to be minimized are continuous.

Thus, by the same general principle, we could show continuity if we had a universal bounddon the dimension ofE in definition 1, in the sense that every value ofI (A;BuE) obtainable bygeneral extensions can be reproduced or beaten by an extension with ad-dimensional systemE.Note that if this were true, then~just as for intrinsic information and entanglement of formation!the infimum would actually be a minimum: in remark 7 we have explained that thenEsq(r

AB)50 would imply, using the result of,13 that rAB is separable.

As it is, we cannot yet decide on this question, but we would like to present a reasonableconjecture, an inequality of the Fannes-type9 for the conditional von Neumann entropy, which wecan show to imply continuity ofEsq. Let us first revisit Fannes’ inequality in a slightly nonstand-ard form:



Lemma 13: For density operatorsr, s on the same d-dimensional Hilbert space, withir2si1<e,

uS~r!2S~s!u<h~e!1e logd,

with the universal function

h~e!5H 2e loge e< 12 ,

12 otherwise.

Observe thath is a concave function. h

Now we can state the conjecture, recalling that for a density operatorrAB on a bipartite systemHA^ HB , the conditional von Neumann entropy3 is defined as

S~AuB!ªS~rAB!2S~rB!.

Conjecture 14: For density operatorsr, s on the bipartite systemHA^ HB , with ir2si1

<e,

uS~AuB!r2S~AuB!su<h~2e!13e logdA ,

with dA5dimHA , or some other universal function f(e,dA) vanishing ate50 on the right-handside.

Note that the essential feature of the conjectured inequality is that it only makes reference tothe dimension of systemA. If we were to use Fannes inequality directly with the definition of theconditional von Neumann entropy, we would pick up additional terms containing the logarithm ofdB5dimHB . In the Appendix we show that this conjecture is true in the classical case, or moreprecisely, in the more general case where the states are classical on systemB.

In order to show that the truth of this conjecture implies continuity ofEsq, consider two statesrAB andsAB with irAB2sABi1<e. By well-known relations between fidelity and trace distance10

this means thatF(rAB,sAB)>12e, hence16,27 we can find purificationsuC&ABC and uF&ABC ofrAB andsAB, respectively, such thatF(uC&ABC,uF&ABC)>12e. Using Ref. 10 once more, we get

iuC&^CuABC2uF&^FuABCi1<2Ae.

Now, let L be any quantum operation as in Eq.~3!: it creates extensions ofrAB andsAB,

rABE5~ id^ L!uC&^CuABC,

sABE5~ id^ L!uF&^FuABC,

with

irABE2sABEi1<2Ae.

Hence, usingI (A;BuE)5S(AuE)1S(BuE)2S(ABuE), we can estimate

uI ~A;BuE!r2I ~A;BuE!su<uS~AuE!r2S~AuE!su1uS~BuE!r2S~BuE!su1uS~ABuE!r

2S~ABuE!su<3h~2Ae!16Ae log~dAdB!5..e8.

Since this applies to any quantum operationL and thus to every state extension ofrAB andsAB,respectively, we obtain

uEsq~rAB!2Esq~sAB!u<e8,



with e8 universally dependent one and vanishing withe→0. h

Remark 15: Since Esq is convex it is trivially upper semicontinuous. This also follows from thefact that squashed entanglement is an infimum of continuous functions obtained by bounding thesize of the dimension of system E.

This observation, together with results from the general theory of convex functions, impliesthat squashed entanglement is continuous ‘‘almost everywhere.’’ Specifically, with theorem 10.1 inRef. 23, we have:

Proposition 16: Esq is continuous on the interior of the set of states (i.e., on the faithful states),and more generally, it is continuous when restricted to the relative interior of all faces of the stateset.

Continuity near pure states (see remark 11) thus implies continuity of Esq on the set of allrank-2 density operators. h

V. CONCLUSION

In this paper we have presented a new measure of entanglement, which by its very definitionallows for rather simple proofs of monotonicity under LOCC, convexity, additivity for tensorproducts and superadditivity in general, all by application of the strong subadditivity property ofquantum entropy. We showed the functional, which we call ‘‘squashed entanglement,’’ to be lowerbounded by the distillable entanglement and upper bounded by the entanglement cost. Thus, it hasmost of the ‘‘good’’ properties demanded by the axiomatic approaches14,20,29 without sufferingfrom the disadvantages of other superadditive entanglement monotones. The one proposed in Ref.8, for example, diverges on the set of pure states.

The one desirable property from the wish list of axiomatic entanglement theory that we couldnot yet prove is continuity. We have shown, however, that squashed entanglement is continuousnear pure states and in the relative interior of the faces of state space. Continuity in general wouldfollow from a conjectured Fannes-type inequality for the conditional von Neumann entropy. Theproof of this conjecture thus remains the great challenge of the present work. It might well be ofwider applicability in quantum information theory and certainly deserves further study.

Another question to be asked is whether or not there exist states that are nonseparable but,nonetheless, have zero squashed entanglement. We expect this not to be the case: if not by meansof proving that the infimum in squashed entanglement is achieved, then by means of an approxi-mate version of the result of Ref. 13. The relation to entanglement measures other than entangle-ment of formation, entanglement cost and distillable entanglment remains open in general. IfEsq50 would imply separability, however, it would follow that for the class of PPT states,squashed entanglement is larger than entanglement measures based on the partial transpose op-eration, like relative entropy of entanglement, the logarithmic negativity and the Rains bound.

Note added in proof.Alicki and Fannes~quant-ph/0312081! have recently proven our Con-jecture 14, with the upper bound 4e log dA12h(e)12h(12e). Thus, squashed entanglement isnow known to be continuous.

ACKNOWLEDGMENTS

We thank C. H. Bennett for comments on an earlier version of this paper. The work of M.C.was partially supported by a DAAD Doktorandenstipendium. Both authors acknowledge supportfrom the U.K. Engineering and Physical Sciences Research Council and the EU under projectRESQ~IST-2001-37559!.

APPENDIX: THE CLASSICAL CASE OF THE CONDITIONAL FANNES INEQUALITY

In this appendix we prove the conjecture 14 for states

rAB5(k

pkrkA

^ uk&^kuB, ~A1!



sAB5(k

qkskA

^ uk&^kuB, ~A2!

with an orthogonal basis (uk&)k and ofHB , probability distributions (pk) and (qk), and statesrkA

andskA on A. Note that this includes the case of a pair of classical random variables. In this case,

the statesrkA andsk

A are all diagonal in the same basis (u j &) j of HA and thusrAB andsAB describejoint probability distributions on a Cartesian product.

The key to the proof is that for states of the form~A1!,

S~AuB!r5(k

pkS~rkA!,

and similarly for the states given in Eq.~A2!.First of all, the assumption implies that

e>irB2sBi15(k

upk2qku.

Hence, we can successively estimate,

uS~AuB!r2S~AuB!su<(k

upkS~rkA!2qkS~sk

A!u<(k

pkuS~rkA!2S~sk

A!u1(k

upk2qkuS~skA!

<(k

pk~h~ek!1ek logdA!1e logdA<h~2e!13e logdA ,

using the triangle inequality twice in the first line, then usingS(skA)< logdA , applying the Fannes

inequality, lemma 13~with ekªirkA2sk

Ai1), and finally making use of the concavity of its upperbound. To complete this step, we have to show(kpkek<2e, which is done as follows:

e>irAB2sABi15(k

ipkrkA2qksk

Ai1>(k

~ ipkrkA2pksk

Ai12ipkskA2qksk

Ai1!>(k

pkek2e,

where we have used the triangle inequality. h

Note that in the case of pure states the conjecture is directly implied by Fannes inequality,lemma 13, sinceS(AB)50 andS(A)5S(B). Clearly, a proof of the general case cannot proceedalong these lines as they do not have the possibility to present the conditional von Neumannentropy as an average of entropies onA.

1Bennett, C. H., Brassard, G., Popescu, S., Schumacher, B., Smolin, J. A., and Wootters, W. K., ‘‘Purification of noisyentanglement and faithful teleportation via noisy channels,’’ Phys. Rev. Lett.76, 722–725~1996!; 78, 2031~E! ~1997!.

2Bennett, C. H., DiVincenzo, D. P., Smolin, J. A., and Wootters, W. K., ‘‘Mixed-state entanglement and quantum errorcorrection,’’ Phys. Rev. A54, 3824–3851~1996!.

3Cerf, N. J., and Adami, C., ‘‘Negative entropy and information in quantum mechanics,’’ Phys. Rev. Lett.79, 5194–5197~1997!; ‘‘Quantum information theory of entanglement and measurement,’’ Physica D120, 68–91~1998!.

4Christandl, M., ‘‘The quantum analog to intrinsic information,’’ Diploma thesis, ETH Zu¨rich, 2002~unpublished!.5Christandl, M., Renner, R., and Wolf, S., ‘‘A property of the intrinsic mutual information,’’ in Proc. ISIT 2003, Yoko-hama, Japan, p. 258.

6Davies, E. B., and Lewis, J. T., ‘‘An operational approach to quantum probability,’’ Commun. Math. Phys.17, 239–260~1970!.

7Devetak, I., and Winter, A., ‘‘Distillation of secret key and entanglement from quantum states,’’ quant-ph/0306078, 2003;Horodecki, M. and Horodecki, P., ‘‘Hashing inequality’’~in preparation!.

8Eisert, J., Audenaert, K., and Plenio, M. B., ‘‘Remarks on entanglement measures and non-local state distinguishability,’’J. Phys. A36, 5605–5615~2003!.

9Fannes, M., ‘‘A continuity property of the entropy density for spin lattice systems,’’ Commun. Math. Phys.31, 291–294~1973!.



10Fuchs, C. A., and van de Graaf, J., ‘‘Cryptographic distinguishability measures for quantum-mechanical states,’’ IEEETrans. Inf. Theory45, 1216–1227~1999!.

11Gisin, N. and Wolf, S., ‘‘Linking classical and quantum key agreement: is there ‘‘bound information’’?,’’Advances inCryptology-CRYPTO’00, Lecture Notes in Computer Science~Springer-Verlag, Berlin, 2000!, pp. 482–500.

12Hayden, P. M., Horodecki, M., and Terhal, B. M., ‘‘The asymptotic entanglement cost of preparing a quantum state,’’ J.Phys. A34, 6891–6898~2001!.

13Hayden, P., Jozsa, R., Petz, D., and Winter, A., ‘‘Structure of states which satisfy strong subadditivity of quantum entropywith equality,’’ Commun. Math. Phys.~to be published!, quant-ph/0304007.

14Horodecki, M., ‘‘Entanglement measures,’’ Quantum Inf. Comput.1, 3–26~2001!.15Horodecki, M., Horodecki, P., and Horodecki, R., ‘‘Limits for entanglement measures,’’ Phys. Rev. Lett.84, 2014–2017

~2000!.16Jozsa, R., ‘‘Fidelity for mixed quantum states,’’ J. Mod. Opt.41, 2315–2323~1994!.17Lieb, E. H., and Ruskai, M. B., ‘‘Proof of the strong subadditivity of quantum-mechanical entropy. With an appendix by

B. Simon,’’ J. Math. Phys.14, 1938–1941~1973!.18Maurer, U., and Wolf, S., ‘‘Unconditionally secure key agreement and the intrinsic conditional information,’’ IEEE Trans.

Inf. Theory45, 499–514~1999!.19Nielsen, M. A., and Chuang, I. L.,Quantum Computation and Quantum Information~Cambridge University Press,

Cambridge, 2000!.20Popescu, S., and Rohrlich, D., ‘‘Thermodynamics and the measure of entanglement,’’ Phys. Rev. A56, R3319–R3321

~1997!.21Rains, E. M., ‘‘A semidefinite program for distillable entanglement,’’ IEEE Trans. Inf. Theory47, 2921–2933~2001!.22Renner, R., and Wolf, S., ‘‘New bounds in secret-key agreement: The gap between formation and secrecy extraction,’’

Advances in Cryptology-EUROCRYPT’03, Lecture Notes in Computer Science~Springer-Verlag, Berlin, 2003!, pp.562–577.

23Rockafeller, R. T.,Convex Analysis~Princeton University Press, Princeton, 1970!.24Shor, P. W., ‘‘Equivalcne of additivity questions in quantum information theory,’’ Commun. Math. Phys.~to be pub-

lished!, quant-ph/0305035.25Shor, P. W., Smolin, J. A., and Terhal, B. M., ‘‘Nonadditivity of bipartite distillable entanglement follows from a

conjecture on bound entangled Werner states,’’ Phys. Rev. Lett.86, 2681–2684~2001!.26Tucci, R. R., ‘‘Quantum entanglement and conditional information transmission,’’ quant-ph/9909041; ‘‘Enganglement of

distillation and conditional mutual information,’’ quant-ph/0202144.27Uhlmann, A., ‘‘The ‘‘transition probability’’ in the state space of asp-algebra,’’ Rep. Math. Phys.9, 273–279~1976!.28Vedral, V., Plenio, M. B., Rippin, M. A., and Knight, P. L., ‘‘Quantifying entanglement,’’ Phys. Rev. Lett.78, 2275–2279

~1997!.29Vidal, G., ‘‘Entanglement monotones,’’ J. Mod. Opt.47, 355–376~2000!.30Vollbrecht, K. G. H., and Werner, R. F., ‘‘Entanglement measures under symmetry,’’ Phys. Rev. A64, 062307~2001!.31Yura, F., ‘‘Entanglement cost of three-level antisymmetric states,’’ J. Phys. A36, L237–L242~2003!.



Documents

“Squashed entanglement”: An additive entanglement measure