Generalized measurements to distinguish classical and quantum correlations

Transcription

1 Generalzed measurements to dstngush classcal and quantum correlatons. R. Usha Dev Department of physcs, angalore Unversty, angalore , Inda and. K. Rajagopal, Department of omputer Scence and enter for Quantum Studes, George Mason Unversty, V 22030, US Inspre Insttute Inc. McLean, V 22101, US Internatonal School and onference on Quantum Informaton (ISQI-08 March 4 12, 2008, hubaneswar, Orssa, Inda

2 Outlne Jont robabltes, classcal correlatons and Shannon mutual nformaton. Extenson to bpartte quantum systems - densty matrces of composte states and ther margnals; von Neumann nformaton entropy Measures of quantumness of correlatons : quantum dscord (OZ, quantum defct (RR Generalzed measures to dscern quantumness Summary

3 Two random varables are sad to be correlated f ther jont probablty dstrbutons cannot be expressed as a mere product of the margnal probabltes: ( a, b ( a ( b correlated Shannon Mutual nformaton entropy: ( : H ( + H ( H ( H, a ( a log( a ( b log( b + a, b b ( a, b log( a, b ( : 0 ff ( a, b ( a ( b H

4 Quantum descrpton: ( a, b ( a Tr ( b Tr Natural extenson of the dea of correlaton: von Neumann mutual nformaton: Relatve entropy Subsystem densty matrces correlated ( : S ( + S ( S ( S ( Tr S, partte densty matrx + Tr log Tr log log S ( : 0ff

5 Noton of correlaton per se does not set a borderlne between classcal and quantum descrptons. How do we dstngush between classcal and quantum correlatons n a bpartte quantum state? an we express ( S : ontrbuton from classcal correlatons + ontrbuton from quantum correlatons

6 R. F. Werner, hys. Rev. 40, 4277 (1989 bpartte densty operator s classcally correlated (separable f t admts a convex combnaton of product states: ( sep ( ( ; 0 1, 1 lce ob lasscal communcaton ( 1 ( 2. ( k. ( 1 ( 2. ( k. 1 2 k 1 2 k

7 Measurements on one part of the quantum system dstngushng classcal and quantum correlaton: H. Ollver and W. H. Zurek, hys. Rev. Lett. 88, (2001 Measurements on one end dsturbs the quantum correlated state n general: measurement on ' If an optmal measurement scheme (on one part exsts such ' that the state s classcally correlated re separable states classcal?

8 OZ approach rojectve measurements on { I } ompleteness I δ ' ' Orthogonalty The condtonal densty operator of subsystem when measurement I s known to have led to the value - s gven by, Tr I I Tr [( I ] [( ] I, 1

9 Gven the results of the complete measurements the condtonal nformaton entropy s gven by, { I } ( ( S S { } ( postve exhbts nherent dependence on measurements

10 structural generalzaton of Shannon condtonal entropy H ( H (, H ( a, b ( a, b log ( b a gves S ( S(, S( Ths s a consequence of the ayes rule ( b a ( a, b ( a uncrtcal extenson of Shannon form Not necessarly a postve defnte!!!!!

11 Quantum dscord (OZ: optmal dfference of two classcally dentcal expressons for condtonal entropes: δ ( mn { } S { } ( S(, { } Optmal measurement leaves the overall state wth least dsturbance and ths s quantfed by δ, ( partte states, whch are n conformty wth ayes Rule have δ, δ (, 0 ' 0 ff I I OZ: ( (.e, only when the state s left undsturbed as a result of optmal projectve measurement on one part of the system

12 Quantum Dscord DOES NOT VNISH FOR LL SERLE STTES!!!! Separablty s not synonymous wth classcal correlatons?! Quantum states wth vanshng quantum dscord: ( classcal

13 . K. Rajagopal and R. W. Rendell, hys. Rev. 66, (2002 ( d D Sd ( ( Quantum Defct: D S - a measure of quantumness of correlatons (d ( d ( ( : classcal decohered counterpart of, β, β ' β '; β β ; β ( ( ' β ( β β ; β β ' β ( β ; β β ; β classcal Subsystem egen bass ( ( β ( ( (. δ ' β ( I β etc... '

14 L. Henderson and V. Vedral: J. hys. : Math. Gen. 34, 6899 (2001 lasscal correlaton: ( { } ( ( max S S V Resdual nformaton entropy of after carryng out a OVM measurement on the subsystem { } V 1 Tr Tr [ ] V I V I [ ] V I V I lasscal and entangled correlatons do not add up to gve total correlatons! ( + E ( S( RE : ; re dfferent types of correlatons not addtve?

15 Measurements play a crucal role n dstngushng and quantfyng correlatons as classcal and quantum

16 Our approach: (. R. Usha Dev and. K. Rajagopal, To appear n hys. Rev. Lett. onsder all trpartte extensons of the state such that Tr [ ] erform generalzed projectve measurements on one part ( of the system. ' ( ( Tr I I Tr State under nvestgaton { ( I } ( ' state left after generalzed measurement

17 harle ob Optmal projectve measurement by Tr ( lce ' classcal ' quantum

18 Quantumness: Q { ( } I, ( ' mn S Mnmzaton s over the set of all trpartte extensons and the set of all projectve measurements at the end Q 0 ff '

19 Separablty and Quantumness Quantumness vanshes when. e., where Tr Tr ( [ ( ( [ ] ( ] ( ( Separable states are nsenstve to generalzed measurement (optmal quantumness vanshes

20 Generalzed measurements are NOT necessarly OVMs n example: ( 1 +, + +, 0, 0 0, ± Three qubt extended state ( 0 ± 1 ; ,0,0 1,0,0 +, + Tr n optmal measurement on : Ths leaves the overall state unperturbed: and 4 ( 1 0, +, + 0, + [ ] ( ( 1,0 1,0, 1 ( ( ' 0, + 0, +, 0 ( ( I I 1 ' Q 0 4 1,, 1 1, 0,,

21 Operatonal aspects of quantumness ' (, b ', b ( I ( b' ; ( b b ', b ( ( ( b' b ', b b' I b' b b b' b b b' b Tr ( [ ( I I ] b b b Tr ' ( ( s a separable state wth same margnal: Tr ' ( [ ] b' b b' b, b', b

22 Q { } ( ' mn S I, ( ( sep mn S ( sep { } ( sep separable state whch shares the same margnal Mnmum entropc dstance between and the closest Q 0 ff s separable Q 0 for all entangled states

23 lasscal correlatons: ( S( mns( ( sep { } ( sep 0 so that total correlatons (mutual nformaton s equal to a sum of classcal correlatons and quantumness Q (

24 Summary Importance of generalzed measurements n dscernng quantumness of correlatons. physcal approach to ths fundamental problem, based on the basc concept of a quantum measurement and the correspondng nformaton content Entangled states get projected to ther closest separable states (wth same margnal for one of the subsystems by an optmal generalzed projectve measurement on one part Our new measure Quantumness s the mnmum entropc dstance of the bpartte state wth ts closest separable state; t serves as an upper bound of relatve entropy of entanglement Flawless merger of quantumness of correlatons wth quantum entanglement tself based on a measurement based approach.