Irreversibility for all bound entangled states

Transcription

1 Irreversblty for all bound entangled states Barbara Synak-Radtke Insttute of Theoretcal Physcs and Astrophyscs Unversty of Gdańsk Cooperaton: D. Yang, M. Horodeck, R.Horodeck [PRL 95, (2005, quant/ph ] RSQ ( , QUPRODIS ( , PBZ-MIN (2003

2 ntangled states Mathematcal defnton: s entangled t cannot be decomposed as the A B separable form p Operatonal asymptotc defnton: s entangled t cannot be prepared by Local Operaton and Classcal Communcaton (LOCC n asymptotc regme Maxmally entangled state (snglet 1 ψ _ = (

3 Dstllaton of nosy entanglement [Bennett,Brassard,Popescu, Schumacher, Smoln,Wooters PRL (1996 ] Alce Bob n pars Arbtrary local operatons : Classcal communcaton n Optmal transton from to snglet (asymptotc regme Alce Bob m pars (snglets Dstllable entanglement: optmal rate D ( =m/n (of snglets Ψ - m

4 Formaton of entanglement [Bennett, DVncenzo, Smoln,Wootters, PRA 54, (1996] Alce Bob n pars Arbtrary local operatons : Classcal communcaton n Optmal transton from snglet to (asymptotc regme Alce Bob m pars (snglets ntanglement cost : optmal rate C ( =m/n (of pars Ψ - m

5 Reversblty for pure and separable states Separable (dsentangled states 1 No snglets needed to create state: C = 0 3 No snglets can be drawn from state: D = 0 Pure entangled states n ψ + m n D ( = ( = S( A C

6 Mxed states: rreversblty n entanglement theory Bound entangled states: [M.,P.,R. Horodeccy PRL (1998 ] s entangled but n No pure entangled states Generc mxed state: [Rgorously: Vdal, Crac PRL (2001 ] ψ + k n ψ + k ' k < k D < C

7 Problem of rreversblty for bound entangled states Are the processes of formaton and dstllaton reversble or rreversble? c( d = ( reversblty ( c > d ( rreversblty

8 ntanglement needed to create one copy of state From bound entangled states, we cannot dstll entanglement D = 0 but we need entanglement to create one copy of them, what refers nonzero value of F > ntanglement of formaton 0 F ( = nf { p, ϕ } p S( ϕ ϕ A where p ϕ ϕ =

9 Problem of addtvty for F Relatonshp between F and C ( = lm ( / n c n F n [Hayden, M.Horodeck, Terhal, J.Phys.A, Math.Gen.2001] If F s addtve then C = F and for all entangled states C = F > 0 Problem We do not know f F s addtve

10 Other approaches to solve problem of rreversblty: To fnd a new entanglement measure: >? > C D To fnd lower bound of C : C G G > 0 for bound entangled states

11 Towards constructon of G Lets recall a measure of classcal correlaton of bpartte state: [Henderson, Vedral JPA 34, 6899 (2001] C( = max { BB + } S( A p S( A where p A 1 + = TrB ( I A B I A B p + = Tr( I A B ( I A B

12 Propertes of C C( = 0 ff = A B C s nvarant under local untary operatons C s non-ncreasng under local operatons, n partcular C( AA' : B C( A: B

13 For a trpartte pure state ϕ C, the followng dualty relaton s satsfed: Dualty relaton [M. Koash and A.Wnter, PRA 69, (2004] where = Tr C ϕ C, s dual to AC = Tr B ϕ C and vce versa. ( ( ( AC F A C S + = ( mn ( ( ( mn ( }, { }, { + = A p A A p A S p S S p S Note: F C

14 Defnton a canddate for bound on C Lets defne some new quanttes for G( = nf { p, } p C( where nfmum s taken over all n general mxed ensemble realzng state = A: B p A: B

15 G s postve for all entangled state Theorem G( = 0 A: B A: B s separable state Proof. " " If G s equal to zero then C must be zero for every element of ensemble { p, } realzng, so all states A B n ensemble must be product { = }

16 For all separable state G=0 C( :B s contnuous A ths comes from dualty relaton but t s also a consequence of proposton proved n our paper about asymptotc contnuty: [see: B.Synak-Radtke, M.Horodeck, quant-ph/ ] Basng on Caratheodory theorem we can show that for G, beng convex roof of other contnuous functon, there exsts optmal decomposton { p, If the state s entangled, there must be a non-product state n decomposton and C n nonzero on ths state. } G > 0 for entangled states and G = 0 for separable states

17 Inequalty relatng C and F Lemma For any four-partte pure state ψ AA BB the followng nequalty of entanglement s satsfed: where F = ( ψ ( + C( AA BB' F A' ' TrA' B' ψ ': A B : B AA' BB' and : = A' B' Tr ψ AA' BB' Proof. We apply a dualty relaton to a 4-partte state S( = ( + C( AA' F AA': B AA': B' F ( + C( A : B A': B' ψ AA' B B'

18 Inequalty relatng G and F Proposton For a mxed four-partte state A B ( ( + G( F AA' : BB' F A : B A' : B' Proof. Let { p, ϕ AA'BB' } be the optmal realzaton of F of state then AA' : BB' F ( = p S( AA': BB' AA' F ( + p C( A : B A': B' p ( + p C( F A : B A': B' F ( + G( A : B A': B'

19 G s lower bound of C Theorem For any entangled state C ( B G( > A : B A : 0 Proof. So n c F / n Notce that = lm ( n, then F ( =... ( + ( n 1 G( F n F ( n n n Let n then n F ( n 1 1 G( F ( n 1 C ( A B : + G( : G( A B > 0...

20 Irreversblty for all bound entangled state For any entangled state C ( > A: B 0 For all bound entangled states D = 0, but C > 0 Irreversblty between process of formaton and dstllaton for all bound entangles states!!!! D. Yang, M. Horodeck, R.Horodeck and B. Synak-Radtke Irreversblty for all bound entangled states, PRL 95, (2005, quant/ph ,

21 Applcaton 1 C ( > entangled 0 Mathematcal defnton of entangled states s equvalent to operatonal asymptotc one!!!

22 Applcaton 2 For a four-partte state AA'BB' F ( AA' : BB' F ( A: B + G( A': B' f the reduced state s A'B' entangled then F ( > ( AA' : BB' F A: B It s mpossble to clone a known entangled state by LOCC. [See: M. Horodeck, A.Sen (De, U. Sen, Phys.Rev.A (2004]

23 Thank you for your attenton!