How many singlets are needed to create a bipartite state via LOCC?

Transcription

1 How many snglets are needed to create a bpartte state va LOCC? Nlanjana Datta Unversty of Cambrdge,U.K. jontly wth: Francesco Buscem Unversty of Nagoya, Japan [PRL 106, (2011)]

2 ntanglement cannot be created or ncreased by local operatons (LO) & classcal communcaton (CC) ntanglement manpulaton : the converson of entanglement from one form to another va LOCC by 2 dstant partes Asymptotc entanglement manpulaton protocols: ntanglement dstllaton ntanglement dluton

3 Asymptotc ntanglement Dstllaton n Alce Bob

4 Asymptotc ntanglement Dstllaton n : LOCC snglet Alce C C m n Bob

5 ntanglement Dstllaton n : LOCC Alce Heurstcally: snglets m n Bob n m, n 1 F n lmsup n m n n the maxmum number of snglets that can be extracted from each copy of the state = dstllable entanglement D ( )

6 Asymptotc ntanglement Dluton snglets m n

7 ntanglement Dluton snglets m n : LOCC Alce n Bob Heurstcally: m n n F, 1 n lmnf m n n n the mnmum number of Bell states needed to create a copy of the state = entanglement cost C ( )

8 Asymptotc ntanglement Manpulaton of Pure States ( ) ( ) C D : S( ) S( ) A B ntropy of entanglement S( ) Tr( log ) A A 2 A = von Neumann entropy of ts reduced state A entanglement of pure states s asymptotcally reversble Ths s not true for mxed states.

9 Asymptotc ntanglement Dluton ntanglement Cost of a mxed state [Hayden, Horodeck & Terhal] 1 n C( ) F( ): lm F( ) n n ntanglement of Formaton of a bpartte state ( ) mn F, A p A convex roof extenson p ps Tr A B

10 ( ), ( ): C D evaluated n the asymptotc..d. framework multple, dentcal copes of the ntal state LOCC multple, dentcal copes of the desred target state e.g. n ent. dstllaton n : no correlatons wth asymptotcally vanshng error n n practce: dffcult to acheve due to nose typcally one doesn t know how such a lmt s approached

11 A more relevant scenaro Asymptotc..d. framework One-shot scenaro.g. ntanglement Dluton nstead of snglets m n : LOCC Alce n Bob

12 A more relevant scenaro Asymptotc..d. framework One-shot scenaro.g. One-shot ntanglement Dluton snglets m : LOCC Alce Bob m? How many snglets are needed to create a sngle copy of a bpartte state va LOCC?

13 Outlne of the talk Queston How many snglets are needed to create a sngle copy of va LOCC? One-shot entanglement dluton Queston Is there an operatonal nterpretaton of the Schmdt number of? Defnton & propertes of the Schmdt number of Some relevant entropc quanttes Answerng the questons One-shot entanglement dstllaton & entanglement spread Open questons

14 Schmdt number A bpartte pure state H A H B Schmdt decomposton Schmdt coeffs. Schmdt rank : A pure state d 1 0 ; 1 ( ) r A necessary condton for d A B 1 = no.of non-zero Schmdt coeffs. = rank of A = rank of B Tr A B s entangled f and only f r( ) 1. LOCC d mn{ d, d } : r( ) r( ). A B

15 = Schmdt number : mxed states p, [Terhal & Horodeck] an ensemble of pure states p = r rmax max ( ) = max. Schmdt rank of the pure states n Schmdt number of : N( ):= mnr max In any p, = of, at least one such that r( ) N( ) = p, of, at least one r( ) N( ) such that,

16 Schmdt number contd. For a pure state : N( ) r( ) For a separable state N( ) 1. Schmdt number s an entanglement monotone LOCC N( ( )) N( ) Classfcaton of densty matrces: H : fnte-dl Hlbert space; DH H n n n S : D H H ; N( ) k Sk 1 k n n set of densty matrces S k S1 set of separable densty matrces Classfcaton of lnear maps

17 Classfcaton of lnear maps: Schmdt number contd. A lnear Hermtcty-preservng map s -postve f & only f k k I 0 k Sk : 1 postve map separable states postve maps [Horodeck] A state s entangled f & only f a 1 I 1 0 such that Schmdt number k -postve maps A state has Schmdt number a k I 0 k k 1, f such that

18 Schmdt number -- Summary Schmdt number of s an entanglement monotone S can be characterzed by k -postve maps k can be detected by k-schmdt number wtnesses There s a zoo of entanglement monotones! (Q) Does the Schmdt number have an operatonal sgnfcance?

19 Outlne of the talk Queston How many snglets are needed to create a sngle copy of va LOCC? Queston Is there an operatonal nterpretaton of the Schmdt number of? One-shot entanglement dluton

20 One-shot ntanglement Dluton snglets m : LOCC Alce Bob m?

21 quvalently, One-shot ntanglement Dluton M : LOCC Alce Bob 1 M 1 M a maxmally entangled M state of Schmdt rank M, log M?

22 One-shot ntanglement Dluton M Alce Dluton fdelty One-shot dl : LOCC -error entanglement cost 0; Bob F( ( ), ) 1 F ( M, ): max F( ( ), ) M M LOCC (1) C, ( ): mn log M : F ( M, ) 1 dl

23 One-shot, ( )? (1) C -error entanglement cost Theorem: For any 0, ( ) mn H ( R) (1) C, 0 RA = p,

24 Defnton of H ( ): 0 RA R = p, p Trpartte c-q state: p R R R R : an auxlary classcal system R H R p A RA R R Tr A B

25 0 Defnton of Quantum relatve Reny entropy of order 1 S( ): lm S ( ) 0 Defnton of 0 H ( ): 0 RA R 1 S 1 1 ( ): log Tr ( ) log Tr ( ) projecton onto support of H ( ): 0 RA R H ( R) max H ( ) 0 RA 0 RA R H ( ) S ( I ) R 0 RA R 0 RA R A H0 ( RA R) max log Tr ( R I A) R RA

26 Defnton of H ( ): 0 RA R smoothed entropy H ( R) maxlog Tr ( I ) 0 R RA RA R A H0 ( RA R ): RA mn H ( R) B ( ) RA 0 RA RA B ( ) : : RA RA RA RA 1 Theorem: For any 0, ( ) mn H ( R) (1) C, 0 RA

27 One-shot -error entanglement cost Theorem: For any 0, mn H ( R) ( ) mn H ( R) ( ) mn H ( R) (1) C, 0 RA 2 (1) 0 RA C, 0 RA

28 0: Corollary: One-shot 0-error entanglement cost ( ) mn H ( R) (1) C 0 RA log N( ) mn max log Tr ( I ) R Schmdt number RA R (*) A Answer: The mnmum number of snglets needed to create a sngle copy of s log N( )

29 Key ngredent of the proof (1) C, ( ): mn log M : F ( M, ) 1 dl Lemma: [G.Bowen & ND ; M. Hayash] F ( M, ) max dl F ( M, ) dl (), k = p k1 p RHS of (1) k M p -th Schmdt coeff. of quantum scssors effect.(1) F ( M, ) dl LHS of (1) Lo-Popescu theorem

30 (Q) Is log N( ) always addtve?? n log N( ) nlog N( ) Open Questons * (A) No! Isotropc state: [Terhal & Horodeck] Invarant under 1 f I f M 1 0 f 1 2 M M M M Proved: f 1 ; 2 2 N( ) N( ) [Terhal & Horodeck] 2 log N( ) log N( ) (Q) Can one understand ths operatonally? U U *

31 Isotropc state 1 f I f M 1 0 f 1 2 M M M M Numercal evdence: f 3 ; 2 N 2 log N( ) log N( ) 2 ( ) N( ) (Q) Can one prove ths?

32 One-shot ntanglement Dstllaton : LOCC Alce M Bob (1) D maxlog M : : LOCC, F( ( ), ) 1 M, ( ): One-shot -error dstllable entanglement

33 For a pure state ; Theorem: One-shot -error dstllable entanglement (1) S D, mn A ( ) ( ) S ( ) ( ) S ( ) log(1 2 ) (1) ' mn A D, mn A S mn smoothed mn-entropy mn mn-entropy ( ) max S ( ) A B A ( ) A max mn S ( ): log ( ) A A A

34 For a pure state ; (1) D, S mn A 0 ( ) ( ) ( ) S ( ) : log max ( A ) (1) D mn A mn-entropy (1) C, S max A ( ) ( ) max-entropy ( ) S ( ) : log A (1) C max A rank( ) The dfferences : (1) (1) ( ) C ( ) D ( ) 0 Operatonal sgnfcance? ( ) ( ) ( ) (1) (1) C, D, 0

35 The dfference : For a pure state ; = entanglement spread (1) (1) ( ) C ( ) D ( ) [Hayden, Wnter; Harrow] ( ) ( ) ( ) = (1) (1) C, D, - perturbed entanglement spread ( ): lower bound to the classcal communcaton cost of creatng from snglets va LOCC ( ) :... to a fdelty (1 ). (Q) Do these dfferences for a mxed state have smlar operatonal nterpretatons? have

36 One-shot entanglement dluton For an accuracy 0: SUMMARY Mnmum number of snglets needed to create a sngle copy of (1) ( ) log N ( ) C For a pure state ; One-shot entanglement dstllaton For a pure state ; For an accuracy ( ) mn H ( R) (1) C, 0 RA (1) C ( ) S ( ) max A : log A rank( ) (1) D ( ) S ( ) mn : ( ) A log max A (1) 0: D, ( ) S mn( A) Open questons: Schmdt number, entanglement spread [logarthm of the Schmdt number]

37 Retrevng the asymptotc entanglement cost Our Theorem S ( ) S( ) 0 Fannes nequalty 1 ( ) lmlm ( ) n (1) n C C, 0 n (1) C,... ( )... ( ) C n 1 lm lm 0 n n F ( ) Generalzed Sten s lemma [Hayden, Horodeck, Terhal]