The entanglement of purification

Transcription

1 JOURNAL OF MATHEMATICAL PHYSICS VOLUME 43, NUMBER 9 SEPTEMBER 2002 The entanglement of purfcaton Barbara M. Terhal a) Insttute for Quantum Informaton, Calforna Insttute of Technology, Pasadena, Calforna and IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heghts, New York Mchał Horodeck Insttute of Theoretcal Physcs and Astrophyscs, Unversty of Gdańsk, Gdańsk, Poland Debbe W. Leung IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heghts, New York Davd P. DVncenzo Insttute for Quantum Informaton, Calforna Insttute of Technology, Pasadena, Calforna and IBM T.J. Watson Research Center, P.O. Box 218, Yorktown Heghts, New York Receved 6 February 2002; accepted for publcaton 16 May 2002 We ntroduce a measure of both quantum as well as classcal correlatons n a quantum state, the entanglement of purfcaton. We show that the regularzed entanglement of purfcaton s equal to the entanglement cost of creatng a state asymptotcally from maxmally entangled states, wth neglgble communcaton. We prove that the classcal mutual nformaton and the quantum mutual nformaton dvded by two are lower bounds for the regularzed entanglement of purfcaton. We present numercal results of the entanglement of purfcaton for Werner states n H 2 H Amercan Insttute of Physcs. DOI: / I. INTRODUCTION The theory of quantum entanglement ams at quantfyng and characterzng unquely quantum correlatons. It does so by analyzng how entangled quantum states can be processed and transformed by quantum operatons. A crucal role n the theory s played by the class of local operatons and classcal communcaton LOCC, snce quantum entanglement s nonncreasng under these operatons. Indeed, by consderng ths class of operatons we are able to neatly dstngush between the quantum entanglement and the classcal correlatons that are present n the quantum state. Gven the success of ths theory, we may be darng enough to ask whether we can smlarly construct a theory of purely classcal correlatons n quantum states and ther behavor under local or nonlocal processng. At frst sght, such an effort seems doomed to fal snce merely local actons can convert quantum entanglement nto classcal correlatons. Namely, Alce and Bob who possess an entangled state a b wth Schmdt coeffcents can, by local measurements, obtan a jont probablty dstrbuton wth mutual nformaton equal to H( ). Thus t does not seem possble to separate the classcal correlatons from the entanglement f we try to do ths n an operatonal way. Note that t may be possble to separate quantum and classcal correlatons n a nonoperatonal way see, for example, Ref. 1 or 2. The drawback of such an approach s that no connecton s made to the dynamcal processng of quantum nformaton, whch s precsely what has made the theory of quantum entanglement so elegant and nnovatve. An a Electronc mal: terhal@watson.bm.com /2002/43(9)/4286/13/$ Amercan Insttute of Physcs

2 J. Math. Phys., Vol. 43, No. 9, September 2002 The entanglement of purfcaton 4287 operatonal approach to the quantfcaton of quantum and classcal correlatons was recently formulated n Ref. 3. In ths artcle we propose to treat quantum entanglement and classcal correlaton n a unfed framework, namely we express both correlatons n unts of entanglement. Such a theory of all correlatons may have potental applcatons outsde quantum nformaton theory as well. Researchers have started to look at entanglement propertes of many-partcle systems for example at quantum phase transtons see, for example, Ref. 4 and references theren. Instead of consderng the entanglement of formaton n these studes, one may choose to look at the behavor of a complete correlaton measure. In ths artcle we ntroduce such a measure, called the entanglement of purfcaton. We would lke to emphasze that our correlaton measure s not an entanglement measure, but a measure of correlatons expressed n terms of the entanglement of a pure state. It has been the experence n quantum nformaton theory that questons n the asymptotc approxmate regme are easer to answer than exact nonasymptotc queres. Thus we ask how to create a bpartte quantum state n the asymptotc regme, allowng approxmaton, from an ntal supply of EPR-pars by means of local operatons and asymptotcally vanshng communcaton. Ths latter class of operatons wll be denoted as LOq local operatons wth o(n) communcaton n the asymptotc regme versus the class LO for strctly local operatons. We can properly defne ths formaton cost E LOq as follows: E LOq lm 0 nf m n L LOq,D L LOq m, n. 1 Here s the snglet state n H 2 H 2 and L LOq s a local superoperator usng o(n) quantum communcaton. D s the Bures dstance D(, ) 2 1 F(, ) and the square-root-fdelty s defned as F(, ) Tr( 1/2 1/2 ). 5 We could have allowed classcal nstead of quantum communcaton n our defnton our results wll not depend on ths choce, so we may as well call all communcaton quantum communcaton. Before we consder ths entanglement cost for mxed states, we observe that by allowng asymptotcally vanshng communcaton, we have preserved the nterconvertblty result for pure states. 6 Ths s due to the fact that both the process of entanglement dluton as well as entanglement concentraton can be accomplshed wth no more than asymptotcally vanshng amount of communcaton, see Ref. 7. We see that the cost E LOq ( ) of creatng the state s defned analogously to the entanglement cost E c ( ), 8,9 wth the restrcton that Alce and Bob can only do a neglgble amount of communcaton. It s mmedate that E LOq ( ) wll n general be larger than E c ( ). In partcular, for a separable densty matrx, E c ( ) 0, whereas we wll show that for any correlated.e., not of the form AB A B densty matrx E LOq ( ) 0. The entanglement cost E c was found 9 to be equal to E c lm n E f n, 2 n where E f ( ) s the entanglement of formaton. 8 We wll smlarly fnd an expresson for E LOq, E LOq lm n E p n E n p, 3 where E p ( ) s a new quantty, the entanglement of purfcaton of. Our artcle s organzed n the followng manner. We start by defnng the entanglement of purfcaton and dervng some basc propertes of ths new functon, such as contnuty and monotoncty under local operatons. We wll relate the entanglement of purfcaton to the problem of mnmzng the entropy of a state under a local TCP trace-preservng completely postve map. Wth these tools n hand, we can prove our man result, Theorem 2. Then we spend some

3 4288 J. Math. Phys., Vol. 43, No. 9, September 2002 Terhal et al. tme provng the mutual nformaton lower bounds for E LOq ( ). We also compare our correlaton measure wth the nduced Holevo correlaton measures C A/B that were ntroduced n Ref. 1. We prove that for Bell-dagonal states the correlaton measure C A s equal to the classcal capacty of the related one-qubt Paul channel. At the end of the artcle we present our numercal results for E p ( ) where s a Werner state on H 2 H 2. The proofs of the lemmas and theorems n ths artcle are all farly straghtforward and use many basc propertes of entropy and mutual nformaton concavty, subaddtvty of entropy, nonncrease of mutual nformaton under local actons, etc.. II. ENTANGLEMENT OF PURIFICATION We defne the entanglement of purfcaton: Defnton 1: Let be a bpartte densty matrx on H A H B. Let H AA H BB. The entanglement of purfcaton E p ( ) s defned as E p mn E f, :Tr A B 4 where E f ( ) s the entanglement of whch s equal to the von Neumann entropy S( BB ) Tr BB log BB where BB Tr AA. Let, be the egenvalues and egenvectors of AB. The standard purfcaton of s defned as s AB 0 A B. 5 Every purfcaton of can be wrtten as (I AB U A B ) s for some untary operator U A B on A and B. Therefore, Eq. 4 can be rephrased as E p mn U A B E I AB U A B s s I AB U A B 6 mn U A B S Tr AA I AB U A B s s I AB U A B ) mn B S I B B BB, 7 where we have taken the trace over A and A to obtan Eq. 7, BB Tr AA s s, 8 and B ( ) Tr A U A B ( B 0 0 A )U. The mnmzaton n Eq. 7 s over all possble A B TCP maps B snce every TCP map can be mplemented by performng a untary transformaton on the system and some anclla and tracng over the anclla. Note that the mnmzatons over U A B and B are equvalent. Equatons 6 and 7 provde two dfferent formulatons of the same mnmzaton. Conceptually the frst formulaton s based on purfcatons of and varaton over U A B. The second formulaton s based on extensons of, ABB, such that Tr B ABB AB, and varaton over B ( ). Both formulatons wll be used throughout the artcle. The dea of bpartte purfcatons was consdered n Ref. 10 where the authors proved that every correlated state has, n our language, a nonzero entanglement of purfcaton. If we would have ncluded mxed states n the mnmzaton n Eq. 4 and used the entanglement of formaton as the entanglement measure, then the defned quantty would be equal to the entanglement of formaton of, snce the optmal extenson of s tself. We put some smple bounds on E p ( ). Intutvely, the amount of quantum correlaton n a state s smaller than or equal to the total amount of correlaton, or E f ( ) E p ( ). To prove ths lower bound, let, j A j B j be the purfcaton that acheves the mnmum n Eq.

4 J. Math. Phys., Vol. 43, No. 9, September 2002 The entanglement of purfcaton Alce and Bob locally measure the labels A and j B of the state such that they obtan j wth probablty p j j j. Snce entanglement s nonncreasng under local operatons, we have E f j p j E j j p j E p. 9 It s mmedate that we have equalty between the entanglement of formaton and the entanglement of purfcaton for pure states, where the optmal purfcaton of a pure state s the pure state tself. An easy upper bound s E p ( ) E( s s ) S( A ), where A Tr B ( ) s the reduced densty matrx n A. Ths corresponds to U A B I A B or equvalently B I B on the rhs of Eq. 6 or 7. Applyng the same argument wth AA and BB nterchanged, we obtan E p mn S A,S B, 10 where the purfcatons correspond to ether completely purfyng the state on A or on B. In general ths s not the optmal purfcaton, as we wll see n Sec. V. The entanglement of purfcaton s nether convex nor concave, unlke the entanglement of formaton. For nstance, a mxture of product states, each wth zero entanglement of purfcaton, need not have zero entanglement of purfcaton for example, consder an equal mxture of 00 and 11 ). On the other hand, the completely mxed state has zero entanglement of purfcaton equal to zero yet t s a mxture of four Bell states, each wth one ebt of entanglement of purfcaton. Before we present contnuty bounds for the entanglement of purfcaton, we analyze the optmzaton problem of Eq. 4 n more detal. We can omt doubly stochastc maps B n the optmzaton n Eq. 7 snce they never decrease the entropy. Furthermore, the von Neumann entropy s concave, so that the optmum n Eq. 7 can always be acheved when B s an extremal TCP map. An extremal TCP map s a TCP map that cannot be expressed as a convex combnaton of other TCP maps. Cho 11 has proved that an extremal TCP map wth nput dmenson d has at most d operaton elements n ts operator-sum representaton. Ths result wll allow us to upper bound the dmensons of the optmal purfyng Hlbert spaces, as stated n the followng lemma. Lemma 1: Let act on a Hlbert space of dmenson d AB d A d B. The mnmum of Eq. (4) can always be acheved by a state for whch the dmenson of A s d A d AB and the dmenson of 2 B s d B d AB or vce versa. Proof: We use the formulaton of the entanglement of purfcaton as an optmzaton of a TCP map n Eq. 7. Snce the densty matrx BB ( ) sonh db H dab, the optmal map B maps H dab nto a space of some unspecfed dmenson. The optmal map B can be assumed to be extremal. Theorem 5 of Ref. 11 shows that an extremal TCP map :B(H d1 ) B(H d2 ) Refs. 12 and 13 can be wrtten wth at most d 1 operatons elements, that s, has the form 1 d 1 V V. 11 In our case d 1 d AB. Consder mplementng the TCP map by applyng a untary operaton U to the nput state wth an anclla appended. In our case, ths anclla can be taken as Alce s purfyng system A, and U acts on A B. The dmenson of the anclla A can always be taken to be the number of operaton elements. Thus we have d A d AB. The B dmenson s equal to the output dmenson d 2 of the optmal map, whch s unconstraned by the extremalty condton. However, we note that the operator ( ) of Eq. 11 has a rank of at most d AB. Ths s obtaned by 2 observng that the range of ths operator s exactly that of the vectors gven by all the columns of the matrces V for all the V matrces have d 1 columns and d 2 rows. Thus, there exsts a untary operator U that permts the constructon of a new map U whose output s confned

5 4290 J. Math. Phys., Vol. 43, No. 9, September 2002 Terhal et al. to the frst d 2 1 dmensons of the output space. The operator U may be obtaned explctly va a Gram Schmdt procedure appled to the column vectors of the V matrces. s also optmal, snce the entropy of Eq. 7 s not changed by a untary operaton. Snce the output space of has dmenson d 2 2 1, we conclude that d B can be taken to be d B d AB. It s nterestng to note that a smlar mnmzaton problem was encountered n Ref. 14. There the goal was to use a set of nosy states for classcal nformaton transmsson and we wanted to mnmze the coherent nformaton dvded by the entropy of a quantum state under the acton of a local map. Theorem 1 contnuty of the entanglement of purfcaton : Let and be two densty matrces on H da H db wth Bures dstance D(, ). Then E p E p 20D, log d AB D, log D,, 12 for small enough. Proof: Let and be the purfcatons of and whch acheve the maxmum 5 n F, max,. 13 Let and correspond to the optmal purfcatons of and wth respect to E p. There exsts a untary transformaton U relatng to,.e., (U 1). We defne the nonoptmal purfcaton as (U 1). Now we have E p E p E E E E. 14 We use contnuty of entanglement, 15,16 Lemma 1 whch ndcates that the pure state has support on a space of dmenson at most d 4 AB, and the fact that F(, ) to bound E p E p 5D, log d 4 AB 2D, log D, 15 for small enough D(, ). We can obtan the full bound n Eq. 12 by alternatvely relatng to the optmal purfcaton by a untary transformaton U. It s farly straghtforward to prove monotoncty of the entanglement of purfcaton from monotoncty of entanglement: Lemma 2 (monotoncty of the entanglement of purfcaton): The entanglement of purfcaton of a densty matrx s nonncreasng under strctly local operatons. Let Alce carry out a local TCP map S A on the state. We have E p S A 1 E p. 16 Let Alce carry out a local measurement on through whch she obtans the state wth probablty p. We have p E p E p. 17 Let L LOq be a local operaton asssted by m qubts of communcaton. The entanglement of purfcaton obeys the equaton E p L LOq E p m. 18 Proof: Let be the optmal purfcaton of. Ths optmal purfcaton s related to some purfcaton of (S A 1)( ) by a untary transformaton on Alce s system only. Then Eq. 16 follows from the fact that entanglement s nonncreasng under local partal traces. The state

6 J. Math. Phys., Vol. 43, No. 9, September 2002 The entanglement of purfcaton 4291 A I B / A A I B, where A corresponds to a measurement outcome of Alce, s some purfcaton of. The entanglement s nonncreasng under local operatons and thus E p E p E p E p. 19 For the last nequalty, let Alce and Bob start wth the entangled state and carry out ther LOq protocol. By subaddtvty of entropy, the entanglement of ths state can ncrease by at most m bts when m qubts of communcaton are sent back and forth. Thus the entanglement of the fnal state whch s some purfcaton of L LOq ( ) s smaller than or equal to E p ( ) m. Now we are ready to prove our man theorem: Theorem 2: The entanglement cost of on H d H d wthout classcal communcaton equals E LOq ( ) E p ( ). Proof: The nequalty E LOq ( ) E p ( ) uses entanglement dluton. Let k be the number of copes of for whch the regularzed entanglement of purfcaton E p s acheved. One way of makng many (p) copes of k out of EPR pars and o(p) o(pk) classcal communcaton s to frst perform entanglement dluton on the EPR pars so as to create an approxmaton to the purfcaton p and then trace over the addtonal regsters to get kp. The other nequalty E p ( ) E LOq ( ) can be proved from monotoncty and contnuty of the entanglement of purfcaton. We start wth n EPR pars whch have E p equal to n. The LOq process for creatng an approxmaton k to k usng o(k) qubts of communcaton ncreases the entanglement of purfcaton by at most o(k) bts, see Lemma 2, or E p ( k) n o(k). Usng the contnuty of Theorem 1 and dvdng the last nequalty by k and takng the lmt k gves E p ( ) E LOq ( ). III. MUTUAL INFORMATION LOWER BOUNDS The entanglement cost E LOq s a measure of the quantum and classcal correlatons n a quantum state. The quantum and classcal mutual nformaton of a quantum state are smlar measures that capture correlatons n a quantum state. How do these measures relate to the new correlaton measure? The quantum mutual nformaton I q ( AB ) s defned as I q AB S A S B S AB. 20 We defne the classcal mutual nformaton of a quantum state I c ( AB )as I c AB max H p A H p B H p AB. M A :p A,M B :p B 21 Here local measurements M A and M B gve rse to local probablty dstrbutons p A and p B. The classcal mutual nformaton of a quantum state s the maxmum classcal mutual nformaton that can be obtaned by local measurements by Alce and Bob. Both quantum as well as classcal mutual nformaton share the mportant property that they are nonncreasng under local operatons LO by Alce and Bob. For the classcal mutual nformaton, ths bascally follows from the defnton Eq. 21. The defnton tself as a maxmum over local measurements makes sense snce the classcal mutual nformaton of a probablty dstrbuton s nonncreasng under local manpulatons of the dstrbuton. The proof of ths well known fact s analogous to the proof for the quantum mutual nformaton whch we wll gve here for completeness. We can wrte the quantum mutual nformaton as I q AB S AB A B, 22 where S(..) s the relatve entropy. The relatve entropy s nonncreasng under any map cf. Ref. 17,.e.,

7 4292 J. Math. Phys., Vol. 43, No. 9, September 2002 Terhal et al. S AB A B S AB A B. 23 When s of a local form,.e., A B, the lhs of ths equaton equals the quantum mutual nformaton of the state ( A B )( AB ) and thus the nequalty I q (( A B )( AB )) I q ( AB )s proved. Proof of lower bounds We show that the quanttes I q ( )/2 and the regularzed classcal nformaton I c ( ) lm n (I c ( n )/n) are both lower bounds for the entanglement cost E LOq. The argument s smlar to the proof of the E p lower bound on E LOq n Theorem 2. The reasonng s n fact a specal case of Theorem 4 n Ref. 18 cf. Ref. 19 appled to the class LOq nstead of the orgnal LOCC. We start wth a number, say k, of EPR pars whch have I q 2k and I c equal to k. 20 In the lmt of large n, the rato k/n s the entanglement cost E LOq ( ). We apply the LOq map L whch uses o(n) communcaton to obtan an approxmaton n to n. Snce the quantum mutual nformaton and the classcal mutual nformaton can only ncrease by o(n) by the process L appled to the ntal EPR pars, see Lemma 3, t follows that I q n o n 2k, 24 and smlarly I c n o n k. 25 The last step s to relate the mutual nformatons of n to the mutual nformaton of n. For ths, we need a contnuty result of the form I q/c I q/c C log d 1 O 1 26 for, on H d, 1 suffcently small and C s some constant. 21 Below we wll prove these desred contnuty results. We can dvde Eqs. 24 and 25 by n and take the lmt of large n. We use the contnuty relaton of Eq. 26 and the fact that n the large n lmt n tends to n. Thus we have lm n I q n I n q 2E LOq, 27 where we used that the quantum mutual nformaton s addtve, and smlarly I c E LOq. 28 What remans s to prove the contnuty relatons and the nonncrease modulo o(n) under LOq operatons. Contnuty of mutual nformaton The contnuty of the quantum mutual nformaton I q ( ) can be proved by nvokng Fannes nequalty 22 and Ruska s proof of nonncrease of the trace-dstance under TCP maps. 23 Let and be two densty matrces whch are close,.e., 1 Tr for suffcently small.we have I q AB I q AB S A S A S B S B S AB S AB, 29 whch can be bounded as I q AB I q AB 3 log d AB AB AB 1 3 AB AB 1, 30

8 J. Math. Phys., Vol. 43, No. 9, September 2002 The entanglement of purfcaton 4293 where (x) x log x and It s not hard to prove the contnuty of the classcal nformaton of a quantum state, agan usng the nonncrease of. 1 under TCP maps. Let M A and M B be the optmal measurement achevng the classcal mutual nformaton I c ( ). Under ths measurement the states and, whch s, say, close to, go to probablty dstrbutons p (, j) and p (, j) whch are close agan,.e., p p 1 1. We have that I c I c I p I p log k p p 1 O 1, 31 where k s the number of jont outcomes n the optmal measurement (M A,M B ) and I s the classcal mutual nformaton of a jont probablty dstrbuton. The last nequalty n Eq. 31 could n prncple be derved from Fannes nequalty, usng dagonal matrces, but t s a standard contnuty result n nformaton theory 24 as well. To fnsh the argument, we should argue that k, the number of jont measurement outcomes, s bounded. The classcal mutual nformaton I s a concave functon of the jont probablty p(, j). 24 Therefore only extremal measurements M A and M B need to be consdered n the optmzaton over measurements. An extremal measurement has at most d 2 2 outcomes when actng on a space of dmenson d Ref. 25 and thus k d AB. The same argument, nterchangng and, can be used to upperbound I c ( ) I c ( ). Lemma 3 (monotoncty propertes of mutual nformaton): Let L consst of a seres of local operatons asssted by m qubts of two-way communcaton. The quantum mutual nformaton obeys the nequalty I q L I q 2m, 32 for all states. For the classcal mutual nformaton we have I c L I c m, 33 for all pure states. Proof: Let us frst consder the quantum mutual nformaton. We can decompose the two-way scheme L nto a sequence of one-way schemes. It s suffcent to prove for such a one-way scheme usng m qubts of communcaton, say from Alce to Bob, that I q L I q 2m. 34 Alce s local acton can consst of addng an anclla A n some state and apply a TCP map to the systems AA thus obtanng the state AA :B. Such an acton does not ncrease the quantum nor classcal mutual nformaton as we showed before. Now Alce sends system A to Bob. We have I q AB I q AA :B S AA S B S AA B S AA S A S BA S AA B S A 2S A S BA S AA B I q A:BA 2S A, 35 where we used S(A) S(B) S(AB) S(A) S(B). The quantum mutual nformaton of the fnal state s I q ( A:BA ). Snce S(A ) m, we obtan the needed nequalty. Alce could send only a part of anclla A, but ths does not change the bound. Let us now consder the classcal mutual nformaton. We may convert the entre process L nto a coherent process L where all the measurements are deferred to the end; ths does not change the amount of communcaton that Alce and Bob carry out. Thus, pror to the measurements Alce and Bob have converted the pure state nto some pure state whose local entropy s at most E m where E s the entanglement of the state, whch s equal to I c ( ) see Ref. 20. Now Alce and Bob locally measure and/or trace out some regsters whch are operatons that do

9 4294 J. Math. Phys., Vol. 43, No. 9, September 2002 Terhal et al. not ncrease the classcal mutual nformaton. Therefore the fnal state L( ) has a classcal mutual nformaton that s bounded by the ntal classcal mutual nformaton plus m. Remark: Note that Eq. 32 for the quantum mutual nformaton apples to both pure and mxed states whle we have found mxed states that volate Eq. 33 for the classcal mutual nformaton. Let us state the fnal result once more: Corollary 1: E LOq ( ) I q ( )/2 and E LOq I c ( ). Wth ths corollary we can show that the LOq-entanglement cost of any correlated densty matrx s nonzero. 26 Indeed, the quantum mutual nformaton I q ( ) of a correlated densty matrx s strctly larger than zero, snce S( AB ) s strctly less than S( A ) S( B ) equalty s only obtaned when AB A B and therefore E LOq ( ) 0. We present a smple example for whch E LOq ( ) E p ( ) I q ( )/2. Example 1 (All correlaton s classcal correlaton): Consder the separable state p a a b b where a a j j and b b j j. In ths case I q ( )/2 H(p)/2. However, we can show that E p ( ) H(p). We have cf. Eq. (8) ( ) p b b. Under some local TCP map we obtan a state p b b where are densty matrces. The entropy of equals S( ) p S( ) H(p) H(p). The entanglement of purfcaton E p ( ) may be nonaddtve, so we have to consder E p ( n ). We have ( n ) n and now 1,..., n p 1 p n 1,..., n 1,..., n 1,..., n. Agan the von Neumann entropy of s larger than or equal to nh(p). Note that n ths example we do acheve the classcal mutual nformaton lower bound. Here s an example where the upper and lower bounds fx the regularzed entanglement of purfcaton: Example 2: Let be an equal mxture of the state 0 (1/&)( ) and 1 (1/&)( ). Alce and Bob can get one bt of classcal mutual nformaton by both measurng n the 0,1 bass. Thus E LOq ( ) I c ( ) 1, but E LOq ( ) S( A ) 1, Eq. (10). Therefore E LOq 1. IV. OTHER CORRELATION MEASURES: THE LOCALLY INDUCED HOLEVO INFORMATION In Ref. 1 the authors consdered the locally nduced Holevo nformaton as a measure of classcal correlatons n the state. It s defned ether wth respect to Alce s measurement (C A )or Bob s measurement (C B ) C A/B max M A /M B S p B/A B/A p B/A S B/A, 36 where M A (M B )on gves reduced densty matrces B ( A ) wth probablty p B (p A ). The classcal mutual nformaton I c ( ) wll n general be less than these quanttes, snce to acheve the Holevo nformaton one may have to do codng. In Ref. 1 t was shown that C A/B are nonncreasng under local operatons. We leave t as an exercse for the reader to prove contnuty and nonncrease modulo o(n) under LOq operatons appled to some pure state, thus showng that the regularzed versons of these two quanttes are also lower bounds for E LOq. Bell-dagonal states We show that for Bell-dagonal states Bell the quantty C A equal to C B by symmetry of the Bell-dagonal states s equal to the classcal capacty of the correspondng qubt channels. By the prevous arguments ths gve us some lower bounds on the regularzed entanglement of purfcaton of these states. The Bell-dagonal states are of the followng form, Bell p, 37

10 J. Math. Phys., Vol. 43, No. 9, September 2002 The entanglement of purfcaton 4295 where are the four Bell states where 0 s (1/&)( ). The correspondng channel, the so called generalzed depolarzng channel, or Paul channel, s of the form p, 38 where 0 1, and 1,2,3 are the three Paul matrces. It s known 27 that all two qubt states wth maxmally mxed subsystems are Bell-dagonal, up to a untary transformaton U A U B. From the somorphsm between states and channels, 8,11,28 t follows that all untal channels are of the form 38 cf. Ref. 29, up to untary transformatons appled before and after the acton of the channel. The classcal one-shot capacty of the quantum channel s gven by 30,31 C 1 sup q,, q, 39 where s the Holevo functon of the ensemble q, S q q S. 40 The optmal states that acheve the capacty C 1 are always pure states, moreover t can be shown 29 that the ensemble q, that acheves C 1 for untal one-qubt channels satsfes q Let us argue that C A ( ) C 1 ( ) for a Bell-dagonal state Bell (1 A )( 0 0 ). Alce s POVM measurement on ths state commutes wth the channel. By dong a measurement on 0 she can create any pure-state-ensemble on system B, obeyng the relaton Eq. 41. Ths ensemble s then sent through the channel. If the ensemble s optmal for C 1, then ts Holevo nformaton equals C 1 and thus C A C 1. For untal one-qubt channels C 1 s gven by 29,32 C 1 1 mn S. 42 We can perform the mnzaton n the last nequalty and we obtan the followng formula for the capacty of a Paul channel or the nduced Holevo nformaton of the Bell-dagonal states, C A Bell C 1 1 H 1, 43 where s the sum of the two largest probabltes p and H(.) s the bnary entropy functon H(x) x log x (1 x)log(1 x). For two-qubt Werner states of the form W e e /3, we obtan C A 1 H 1 2e 3 for e 1 4,1, C A 1 H 2 2e 3 for e 0,

11 4296 J. Math. Phys., Vol. 43, No. 9, September 2002 Terhal et al. FIG. 1. Numercal bounds on E p for Werner states. In the upper curve we restrct to dm(a ) dm(b ) 2; for the next curve, we permt dm(a ) dm(b ) 4. The nset shows the curous behavor of E p around the pont where the egenvalue of 0 approaches zero. The dotted curve s the C A lower bound of Sec. IV A. The dashed curve s the entanglement of formaton lower bound whch vanshes when the egenvalue s smaller than or equal to 1 2. It was shown by Kng 32 that the classcal capacty of untal one-qubt channels s equal to the one shot capacty, or C 1 C 1 lm n (1/n) C 1 ( n ). Therefore C A C A C 1, whch s a lower bound on E LOq. V. WERNER STATES A numercal mnmzaton based on Eq. 6 was performed for the Werner states Eq. 44 for E p. We plot the results as a functon of the 0 egenvalue e n Fg. 1. We permtted varous output dmensons; The two curves shown have dm(a ) dm(b ) 2 and dm(a ) dm(b ) 4. In the frst case, the ntal varable of the mnmzaton was determned by a random 4 4 untary U A B pcked accordng to the Haar measure. In the second case, the ntal pont was determned by a random 16 4 sometry pcked accordng to a parameterzaton derved from Ref. 33. We dd not explore the largest dmensons permtted by Lemma 1, whch would have requred an optmzaton over a 64 4 sometry. It s evdent from the numercs presented n the fgure that the C A bound of Eq. 45 s not acheved for the Werner states: the C A lower bound s only tght at the trval ponts e 1 4 and e 1. Our results ndcate that E p s a very complex functon, nether concave nor convex, wth several dstnct regmes. In fact, we fnd four dfferent regmes n our numercs: I In ths regme the standard purfcaton of Eq. 5 appears to be optmal, so the U of Eq. 6 s the dentty, and the purfyng dmensons are dm(a ) 1 and dm(b ) 4. Ths regme only extends over a tny range, approxmately 0 e II In the range e 0.25 we fnd an optmal purfcaton of the form e 0 AB 0 A B 1 e 3 1 AB 1 A B 2 AB 2 A B 3 AB 3 A B ). 46 In ths regon the E p curve s gven by E p x log x (1 x)log((1 x)/3), wth x (1 2e 2) e(1 e))/12. Here the purfyng dmensons are dm(a ) 2 and dm(b ) 2. Of course E p drops to zero for the completely mxed state at e 4. 1 III In the range 0.25 e 0.69 we also fnd purfyng dmensons dm(a ) 2 and dm(b ) 2, but we were unable to determne the

12 J. Math. Phys., Vol. 43, No. 9, September 2002 The entanglement of purfcaton 4297 analytcal form of the purfyng state or of E p. IV In the range 0.69 e 1 the purfyng dmensons were dm(a ) 2 and dm(b ) 3. Agan, we were unable to come to any analytcal understandng of the result. Of course, E p 1 for e 1, correspondng to the pure maxmally entangled state. VI. CONCLUSION We have shown that the entanglement cost E LOq ( ) s equal to the regularzed entanglement of purfcaton. It s an open queston whether the entanglement of purfcaton s addtve:? E p E p E p. 47 In the alternatve formulaton usng the state ( ) the addtvty queston s the followng. Is the mnmum n mn CD S I AB CD AC BD, 48 acheved by a TCP map CD S S? Ths problem s smlar agan to the addtvty queston encountered n Ref. 14 where a local map could possbly lower the rato of the coherent nformaton and the entropy of many copes of a state together. It s nterestng not only to ask the formaton queston wth respect to ths class LOq, but also consder the dstllaton queston. One can consder dfferent versons. For example, how much entanglement can we dstll from usng o(n) communcaton? One would expect that ths quantty D LOq ( ) s always zero for states for whch the entanglement cost E c usng LOCC s lower than the dstllable entanglement D. We do not have a proof of ths statement, relatng rreversblty to a need for classcal communcaton. Instead of tryng to convert the correlatons n back to entanglement, we may ask what classcal correlatons Alce and Bob can establsh usng. We could allow Alce and Bob to perform an asymptotcally vanshng amount of communcaton n ths extracton process. A lttle bt of communcaton could potentally ncrease the classcal mutual nformaton n a quantum state by a large amount when the classcal correlaton s ntally hdden, thus ths may not be the best problem to pose. Researchers 34,35 have nvestgated the possbly more nterestng problem of the secret key K that Alce and Bob can establsh gven where one allows arbtrary publc classcal communcaton between the partes. There s agan more than one verson of ths problem, one n whch Eve possesses the purfcaton of the densty matrx 34 and a stuaton n whch Eve s ntally uncorrelated wth the densty matrx. In Ref. 36 a general framework s developed to address these ssues also n the multpartte settng. Qute recently, entanglement propertes of bpartte densty matrces were studed by lookng at mxed extensons of the densty matrx. 37 It would be nterestng to explore the connecton between our results here on the entanglement of purfcaton and ths other approach. ACKNOWLEDGMENTS B.M.T., D.W.L., and D.P.D. are grateful for the support of the Natonal Securty Agency and the Advanced Research and Development Actvty through Army Research Offce Contract Nos DAAG55-98-C-0041 and DAAD19-01-C-0056 and partal support from the Natonal Reconnassance Offce. Ths work was also supported n part by the Natonal Scence Foundaton under Grant. No. EIA M.H. acknowledges the support of EU grant EQUIP, Contract No. IST We thank Charles Bennett, Paweł Horodeck, Ryszard Horodeck and John Smoln for a pleasant IBM lunch dscusson on ths topc. The concept of entanglement of purfcaton was rased n a dscusson of M.H. wth Ryszard Horodeck. M.H. would also lke to thank Robert Alck and Ryszard Horodeck for stmulatng dscussons. B.M.T. would lke to thank Andreas

13 4298 J. Math. Phys., Vol. 43, No. 9, September 2002 Terhal et al. Wnter for nterestng dscussons about the secret key rate K and ts relaton to other correlaton measures. D.W.L. would lke to thank Charles Bennett and John Smoln for dscussons on mxed state nputs that volate Eq L. Henderson and V. Vedral, J. Phys. A 34, ; quant-ph/ N. J. Cerf and C. Adam, Phys. Rev. A 56, ; quant-ph/ J. Oppenhem, M. Horodeck, P. Horodeck, and R. Horodeck, quant-ph/ T. Osborne and M. Nelsen, quant-ph/ A. Uhlmann, Rep. Math. Phys. 9, C. H. Bennett, H. J. Bernsten, S. Popescu, and B. Schumacher, Phys. Rev. A 53, H.-K. Lo and S. Popescu, Phys. Rev. Lett. 83, ; quant-ph/ C. H. Bennett, D. P. DVncenzo, J. A. Smoln, and W. K. Wootters, Phys. Rev. A 54, ; quant-ph/ P. Hayden, M. Horodeck, and B. M. Terhal, J. Phys. A 34, ; quant-ph/ J. Bouda and V. Buzek, to appear n Phys. Rev. A; quant-ph/ M.-D. Cho, Lnear Algebr. Appl. 10, We have a specal case when d 2. The Stnesprng theorem Ref. 13 mples that we have an operator-sum representaton of such a map. Then Cho s results on extremalty apply, boundng the number of operaton elements, from whch the fnal result can be proved. 13 W. F. Stnesprng, Proc. Am. Math. Soc. 6, M. Horodeck, P. Horodeck, R. Horodeck, D. W. Leung, and B. M. Terhal, Quantum Inf. Comput. 1, ; quant-ph/ H. Barnum, J. A. Smoln, and B. M. Terhal, Phys. Rev. A 58, ; quant-ph/ M. A. Nelsen, Phys. Rev. A 61, ; quant-ph/ V. Vedral and M. Pleno, Phys. Rev. A 57, ; quant-ph/ M. Horodeck, Quantum Inf. Comput. 1, M. J. Donald, M. Horodeck, and O. Rudolph, quant-ph/ One can prove that I c k by observng that any local measurement that s not projectng n the Schmdt bass s a nosy verson of the measurement that does project n the Schmdt bass. In other words, the probablty dstrbuton of any set of local measurements can be obtaned from the probablty dstrbuton of the Schmdt bass measurement by local processng, whch does not ncrease the classcal mutual nformaton. 21 We can alternatvely wrte down a contnuty relaton usng the Bures dstance. Snce the trace dstance. 1 and the Bures dstance are equvalent dstances, one contnuty relaton follows from the other and vce versa. 22 M. Fannes, Commun. Math. Phys. 31, M.-B. Ruska, Rep. Math. Phys. 6, T. M. Cover and J. A. Thomas, Elements of Informaton Theory Wley, New York, A. Peres, Quantum Theory: Concepts and Methods Kluwer Academc, Dordrecht, Note that ths does not drectly follow from the result n Ref. 10, snce the entanglement of purfcaton may be nonaddtve. 27 M. Horodeck and R. Horodeck, Phys. Rev. A 54, ; quant-ph/ A. Jamołkowsk, Rev. Mod. Phys. 3, C. Kng and M.-B. Ruska, IEEE Trans. Inf. Theory 47, ; quant-ph/ B. Schumacher and M. Westmoreland, Phys. Rev. A 56, A. S. Holevo, IEEE Trans. Inf. Theory 44, ; quant-ph/ C. Kng, quant-ph/ M. Reck, A. Zelnger, H. J. Bernsten, and P. Bertan, Phys. Rev. Lett. 73, N. Gsn and S. Wolf, quant-ph/ A. Wnter and R. Wlmnk, unpublshed manuscrpt, prvate communcaton. 36 N. J. Cerf, S. Massar, and S. Schneder, quant-ph/ R. Tucc, quant-ph/