Strong monogamy of multi-party quantum entanglement

Transcript

1 Strog oogay of ulti-party quatu etagleet Jeog Sa Ki Departet of pplied Matheatics

2 Cotets Quatu Etagleet Bipartite quatu etagleet Etagleet Measures Multi-party quatu systes Moogay of ulti-party quatu etagleet Matheatical characterizatio: Moogay iequality Strog oogay of ulti-party etagleet Strog oogay iequality Saturatio of strog oogay iequality Suary

3 Etagleet No-local Nature of Quatu State Useful pplicatios Quatu Teleportatio Dese Codig Quatu Cryptography (QKD) Etc. Quatificatio ad Qualificatio 3

4 Etagleet of Foratio (EoF) For bipartite pure state ψ C C d d' Mixed state ( ψ ) = ( ) ( = ( )) E S S f B ( ) = tr ψ ψ, B S ( ) = trlog B ( C d C d' ) E ( ) i pe ( ψ ) = f i f i i i: over all possible pure state decopositios = p ψ ψ i i i i 4

5 Tagle (Liear etropy) Pure state ψ C d C d' ( ) = ( tr ) = S l ( ) τ ψ Mixed state B ( C d C d' ) ( ) ( ) i p τ = i τ ψi i i: over all possible pure state decopositios = p ψ ψ i i i i 5

6 Tagle alytic forula for two-qubit syste For a two-qubit state λi : the sigularvalues of C( ) = ax{0, λ λ λ λ }: cocurrece B ( C C ) * ( ) ( ) = σ σ σ σ y y y y 3 4 ( ) C ( ) τ = i decreasig order [ W. K. Wootters, PRL (998)] 6

7 Multi-party quatu etagleet 7

8 Moogay of etagleet (MoE) Restricted shareability of ulti-party etagleet Three-qubit systes: ψ = ( ) ϕ C B B C B Maxially etagled C No etagleet! Uique characteristic of quatu correlatio with o classical couterpart: classical correlatios ca be shared freely aog differet parties pplicatios i quatu iforatio processig Boud o the aout of iforatio to eveasdropper: security proof of quatu cryptography Characterizatio of ulti-party etagleet 8

9 Characterizatio of MoE Upper boud o a su of bipartite etagleet easures showig that bipartite sharig of etagleet is bouded. Three-qubit systes: Coffa-Kudu-Wootters iequality Tagle B C ( ψ BC ) B + C τ τ ( ) [V. Coffa, J. Kudu ad W. K. Wootters PR (000)] ( ) = 4det = C ( ) τ ψ ψ ( ) p ( ) B C τ ( C ) τ = i iτ ψi, = pi ψi ψ i i i 9

10 Characterizatio of MoE Upper boud o a su of bipartite etagleet easures showig that bipartite sharig of etagleet is bouded. Three-qubit systes: Coffa-Kudu-Wootters iequality 3-Tagle B C ( ψ BC ) B + C τ τ ( ) Geuie three-party etagleet [V. Coffa, J. Kudu ad W. K. Wootters PR (000)] ( ) = ( ) ( ) C C ( C ) τ ψ τ ψ τ τ B C τ ( C ) 0

11 Characterizatio of MoE Upper boud o a su of bipartite etagleet easures showig that bipartite sharig of etagleet is bouded. Three-qubit systes: Coffa-Kudu-Wootters iequality B C ( ψ BC ) B + C τ τ ( ) [V. Coffa, J. Kudu ad W. K. Wootters PR (000)] Geeralizatio of CKW iequality ito ulti-qubit systes ( ) ( ) + + ( ) τ ψ τ τ [T. J. Osbore ad F. Verstraete PRL (006)] B C τ ( C )

12 W-class state -qubit geeralized W-class state W = a a a with a = i= i Geeralizatio of W state Three-qubit W state: Saturatio of CKW iequality W = ( ) τ = τ + τ + + τ 3 [JSK ad B. C. Saders, J. Phys (008)]

13 Geeral Moogay Iequalities Squashed etagleet { } For, ext : = E tre E = E : = if I ; E : = S + S S S if: over all ext { } ( ) ( ) ( ) ( ) ( ) ( ) sq E BE E E E [M. Christadl ad. Witer, J. Math. Phys. 45, p (004)] I( ; ) I( ; E)

14 Geeral Moogay Iequalities Squashed etagleet, ext : = { E tre E = } For E I E S S S S { } ( ) : = if ( ; ) : = ( ) + ( ) ( ) ( ) sq E BE E E Etagleet ootoe E ( ) E ( ) Lower boud of, upper boud of f D For ψ ψ, E = E ext B E = ψ ψ E sq ( ψ ) = S( ), = tr ( ψ ψ ) B

15 Geeral Moogay Iequalities Squashed etagleet, ext : = { E tre E = } For E I E S S S S { } ( ) : = if ( ; ) : = ( ) + ( ) ( ) ( ) sq E BE E E Moogay iequality ( ) ( ) ( ), CE I ; BC E = I ; B E + I ; C BE ( chai rule) ( ( )) ( ) + ( ) S S S sq BC sq sq C ( by iiizig E for I ( ; BC E) ) [M. Koashi ad. Witer, Phys. Rev. 69, 0309 (004)]

16 Geeral Moogay Iequalities Squashed etagleet, ext : = { E tre E = } For E I E S S S S ( ) : = if ( ; ) : = ( ) + ( ) ( ) ( ) Moogay iequality, CE I ; BC E = I ; B E + I ; C BE ( chai rule) ( ) = 0 iff : separable E sq { } sq E BE E E ( ) ( ) ( ) ( ( )) ( ) + ( ) S S S sq BC sq sq C [M. Koashi ad. Witer, Phys. Rev. 69, 0309 (004)] [F.G.S.L. Bradao, M. Christadl ad Jo Yard, Cou. Math. Phys. 306, 805 (0)]

17 Polygay Iequality Dual oogay iequality For three-qubit pure state C ( ) ( ) + ( ) τ ψ τ τ ( BC ) a a ψ [G. Gour, D. Meyer ad B. C. Saders PR (005)] τ a ( ) : tagle of assistace ( ) ax p ( ) τ = τ ψ a i i i ax: over all possible pure state decopositios = p ψ ψ i i i i

18 Geeral Polygay Iequality Etagleet of ssistace ( d d' ) For ay E ( ) ax pe ( ψ ) = a i f i i ax: over all possible pure state decopositios = p ψ ψ i i i i ( d ) d d Ea( ) ( ) Ea( ) + Ea( ) + + Ea( ) 3 [JSK, PR 85, 0630 (0)]

19 Moo-poly iequality For ay ψ d d d ( ψ ) = ( ) = ( ψ ) ( ) ( ) E S E sq a ( ) + ( ) + + ( ) ( ) E E E S sq sq sq 3 ( ) ( ) ( ) E + E ++ E a a a 3

20 Strog oogay of etagleet 0

21 Strog oogay of etagleet CKW-type oogay iequality ( ) E E( ) E( ) E( ) 3 3

22 Strog oogay of etagleet Stroger (or fier) oogay iequality? ( ) E E( ) E( ) E( ) E( ) E( 3 ) 3

23 -tagle 3-Tagle For three-qubit pure state -tagle For -qubit pure state ψ C ( ) = ( ) ( ) C C ( C ) τ ψ τ ψ τ τ ψ,, = : idex vector spas over all (-)-ordered subsets of τ ( ψ ) = τ ( ψ ) τ ( ) = i, = ph h { } ph, ψ h h τ τ ψ / {,3,, } = p ψ ψ h h h h 3

24 -tagle 3-Tagle For three-qubit pure state -tagle For three-qubit = pure state ψ ( ) = ( ) ( ) C C ( C ) τ ψ τ ψ τ τ ψ ( ) ( ) / τ τ ψ τ ψ τ = i = ph τ { } ψh ph, ψ h h = (,, = ) : idex vector spas over all (-)-ordered subsets of τ ( ) = i ph τ ( ψh ) : two-tagle { p h, ψ h } h τ i, p = h τ ψh { } = p ph, ψ h h h C {,3,, } ψ ψ h h h 4

25 -tagle 4-tagle For four-qubit pure state ψ CD ( ) = ( ) ( C ) ( D ) CD CD ( CD ) τ ψ τ ψ τ τ τ tr = ψ ψ C D CD ( C ) ph ( h C ) { } 3/ 3/ 3/ ( ) ( C ) ( D ) τ τ τ C = ph ψh ψ C h h h τ = i τ ψ, ph, ψ h ( h ) = ( ) ( ) C C ( C ) τ ψ τ ψ τ τ 5

26 Strog oogay coecture 4-tagle ssuig o-egativity of 4-tagle ( CD) τ ψ τ B C 0 ( ψ ) τ ( C ) + τ ( BCD D) + τ ( C D ) 3/ 3/ 3/ ( ) + τ ( C ) + τ ( D ) + τ ( ) ( C ) ( D ) τ + τ + τ D [B. Regula, et. al., PRL (04)] 6

27 Strog oogay coecture -tagle τ ( ψ ) = τ ( ψ ) τ = ssuig o-egativity of -tagle / ( ) ( ) τ ψ τ + ( ) = = 3 = τ τ / Strog oogay iequality of ulti-qubit etagleet [B. Regula, et. al., PRL (04)] 7

28 Strog oogay coecture Provig strog oogay coecture? τ ( ψ ) = τ ( ψ ) τ = i, = ph { } h ph, ψ h h τ τ ψ Expoetially ay optiizatio processes w.r.t. / Nuerical test for 4-qubit systes rado 4-qubit pure states [B. Regula, et. al., PRL (04)] 8

29 Saturatio of ulti-qubit strog oogay iequality 9

30 Saturatio of CKW iequality -qubit geeralized W-class state W = a a a with a = i= i Saturatio of CKW iequality ( W ) = ( ) ( ) ( ) τ τ τ τ [JSK ad B. C. Saders, J. Phys (008)] Good cadidate of possible couterexaple for strog oogay iequality 30

31 W-class state ad strog oogay iequality Strog oogay coecture W-class state τ ψ τ τ ( ) ( ) + = = 3 / ( W ) = ( ) ( ) ( ) + + = τ τ τ τ... Strog oogay coecture for W-class states = W = a a a = 3 τ / = 0 for W-class states 3

32 W-class state ad strog oogay iequality Strog oogay coecture Lea ( ) ( ) + = = 3 = 0 Strog oogay coecture for W-class states τ ψ τ τ τ W-class state for all the idex vectors =,, with ( W ) = ( ) ( ) ( ) + + = τ τ τ τ ( ) / = for geeralized W-class states W = a a a [JSK, PR 90, (04)] = 3 τ / = 0 for W-class states 3

33 W-class state ad strog oogay iequality Saturatio of strog oogay iequality For ay geeralized W-class state W = a a a ( ) W ( ) τ = τ τ + = = 3 / Moreover, the saturatio strog oogay iequality is also true for ψ = a b b b [JSK, PR 90, (04)] 33

34 Negativity ad SM iequality i higher-diesioal systes 34

35 Couterexaples i higher diesio Multi-qubit SM iequality ( ) ( ) τ ψ τ τ + = = 3 / -qubit systes ( BC ) ( ) ( C ) τ ψ τ + τ 3-qubit systes Couterexaples ψ = + + C 6 ( ) [Y. Ou, PR (007)] ψ = ( ) + ( C ) 6 6 [JSK ad B. C. Saders, J. Phys (008)] 35

36 Couterexaples i higher diesio Multi-qubit SM iequality ( ) ( ) τ ψ τ τ + = = 3 / -qubit systes ( BC ) ( ) ( C ) τ ψ τ + τ 3-qubit systes Couterexaples ψ = + + C 6 ( ) ( ) < ( ) + ( C ) τ ψ τ τ BC [Y. Ou, PR (007)] ψ = ( ) + ( C violatio of SM iequality ) i ters of ta gle 6 6 [JSK ad B. C. Saders, J. Phys (008)] 36

37 Square of covex-roof exteded egativity (SCREN) Negativity Bipartite pure state with Schidt decopositio N ( ψ ) ( ) := ψ ψ Γ = i i< λλ For bipartite pure state with Schidt-rak ψ : Trace or, = Γ : Partial traspositio i λ i ii Negativity: two-tagle: ψ = λ ef + λ ef ( ψ ) = 4λλ 0 ( ) = ( tr ) = 4 0 ( N ) τ ψ λλ 37

38 Square of covex-roof exteded egativity (SCREN) Negativity vs. Tagle For bipartite pure state with Schidt-rak ψ λ ef λ ef ( ) = ( ψ ) = 4 0 N λλ=τ ( ψ ) For two-qubit state = p ψ ψ i i i i ( ) ( ) i p τ = i τ ψi i ( ) p N ψ SC ( B ) = i i i = N -SCREN 38

39 Square of covex-roof exteded egativity (SCREN) For -qudit pure state -SCREN ψ d d d N ( ) ( ) SC ψ = N ψ N SC SC = =,, : idex vector spas over all ( -)-subsets of {,,..., } ( ) Mixed state N SC i, = ph N SC h { } ψ ph, ψ h h = p ψ ψ h h h h / [JSK PR (05)] 39

40 -SCREN vs. -tagle For -qubit states ψ N ( ψ ) = τ ( ψ ) SC -qubit SM iequality ( ) ( ) τ ψ τ τ + = = 3 / ( ) ( ) N ψ N + N SC SC SC = = 3 / 40

41 -SCREN vs. -tagle Saturatio of SCREN SM iequality For ulti-qubit geeralized W-class state W = a a a ( ) = ( ) N W N + N SC SC SC = = 3 / Moreover, the saturatio SCREN SM iequality is also true for ψ = a b b b [JSK PR (05)] 4

42 -SCREN vs. -tagle Couterexaples of tagle SM iequality ψ = + + C 6 ( ) ψ = C 6 6 ( 00 0 ) ( 00 ) 3 ( ψ ) ( ) + ( ) N N N SC BC SC B SC C Validity of SCREN SM iequality 4

43 Beyod ulti-qubit systes Multi-qudit geeralized W-class states d d W = a i + a i + + a i... i= ( i i i ) d with a = s= i= si For d= [JSK ad B. C. Saders, J. Phys (008)] W = a a a qubit geeralized W-class state Saturatio of SM iequality ( ) = ( ) N W N + N SC SC SC = = 3 / [JSK PR (05)] 43

44 Beyod ulti-qubit systes d Partially coheret superpositio of with vacuu ( ) = pw W + p ( p, λ ) d d W... ( ) d d λ p( p) W W for 0 p, λ d λ = : pw = + p0 ( coheret superpositio) ( ) d d λ = 0: = pw W + p 0 0 ( icoheret superpositio) 44

45 Beyod ulti-qubit systes d Partially coheret superpositio of with vacuu ( ) = pw W + p ( p, λ ) d d I ters of decoherece W... ( ) d d λ p( p) W W for 0 p, λ d For ψ = pw + p0 ( p, λ ) ( ) = Λ ψ ψ where E λ I, ψ ψ 0 ψ ψ ψ ψ = E E + E E + E E = E λ ( I 0 0 ) = ad E = λ 0 0 ( p, λ ): resultig state fro a coheret state ψ by the decoherece pr oe c ss Λ. 45

46 Beyod ulti-qubit systes d Partially coheret superpositio of with vacuu ( ) = pw W + p ( p, λ ) d d Saturatio of SCREN iequalities W... ( ) d d λ p( p) W W for 0 p, λ N ( p, λ ) ( ) N ( ) = SC SC i i= N ( p, λ ) ( ) p N ( ψ ) { } = i = 0 h = p SC h SC h p h, ψ h ψ ψ h h h h [JSK i preparatio] 46

47 Suary Moogay of ulti-party quatu etagleet Matheatical characterizatio: CKW-type iequality Squashed etagleet Geeral polygay iequality Strog oogay coecture i ulti-qudit systes No-egativity of -tagle : strog oogay iequality SCREN SM iequality for qudits Saturatio of SCREN SM oogay iequality Future works alytic proof of strog oogay iequality? SM iequality of etagleet ad other correlatios 47