One-Shot Quantum Information Theory I: Entropic Quantities. Nilanjana Datta University of Cambridge,U.K.

Transcription

1 One-Shot Quantu Inforaton Theory I: Entropc Quanttes Nlanjana Datta Unversty of Cabrdge,U.K.

2 In Quantu nforaton theory, ntally one evaluated: optal rates of nfo-processng tasks, e.g., data copresson, transsson of nforaton through a channel, etc. under the assupton of an asyptotc, eoryless settng Assue: nforaton sources & channels are eoryless They are avalable for asyptotcally any uses

3 E.g. Transsson of classcal nforaton classcal nfo N Nosy quantu channel Optal rate (of classcal nforaton transsson): classcal capacty C( N) axu nuber of bts transtted per use of N eoryless: there s no correlaton n the nose actng on successve nputs n N : n successve uses of the channel; ndependent

4 To evaluate C( N): asyptotc, eoryless settng classcal nfo n n N uses x E n encodng ( n ) x nput N n N n ( n) x ( ) channel output D n decodng POVM x ' p ( n ) e One requres : prob. of error 0 as n C( N): Optal rate of relable nforaton transsson

5 Entropc Quanttes Optal rates of nforaton-processng tasks n the asyptotc, eoryless settng Copresson of Inforaton: Meoryless quantu nfo. source Data copresson lt: S( ), H von Neuann entropy [Schuacher] Info Transsson thro' a eoryless quantu channel C( N) Classcal capacty --gven n ters of the Holevo capacty ; Quantu capacty Q( N) N [Holevo, Schuacher, Westoreland] [Lloyd, Shor, Devetak] --gven n ters of the coherent nforaton ;

6 These entropc quanttes are all obtanable fro a sngle parent quantty; Quantu relatve entropy: For, 0; Tr 1 D( ) : Tr log Tr lg o supp supp e.g. Data copresson lt: S( ) : Tr lo D( I) g ( I)

7 In real-world applcatons asyptotc eoryless settng not necessarly vald In practce: nforaton sources & channels are used a fnte nuber of tes; there are unavodable correlatons between successve uses (eory effects) Hence t s portant to evaluate optal rates for a fnte nuber of uses (or even a sngle use) of an arbtrary source or channel Evaluaton of correspondng optal rates: One-shot nforaton theory

8 One-shot nforaton theory classcal nfo N sngle use x E encodng x nput N N( x ) channel output D decodng POVM x ' One-shot classcal capacty error : C (1) ( N) Prob. of p e error: ax. nuber of bts that can be transtted on a sngle use of for soe N 0,

9 In the one-shot settng too Capactes, data copresson lt etc. are -- gven n ters of entropc quanttes Mn-/ax-/0- entropes (R.Renner) Obtanable fro certan (generalzed) relatve entropes Parent quanttes for optal rates n the one-shot settng D ( ) D ( ) D ( ) ax 0 n Max-relatve entropy 0-relatve Reny entropy Mn-relatve entropy super-parent : D ( ) Quantu Reny Dvergence (sandwched Reny relatve entropy) [Wlde et al; Muller-Lennert et al]

10 In the one-shot settng too Capactes, data copresson lt etc. are -- gven n ters of entropc quanttes Mn-/ax- entropes (R.Renner) Obtanable fro certan (generalzed) relatve entropes D ( ) D ( ) D ( ) ax 0 n 1 D( )? [ND, F.Ledtzky] D ( ) Quantu Reny Dvergence (sandwched Reny entropy) [Wlde et al; Muller-Lennert et al] 1 2 relatve Reny entropy D ( ) f [, ] 0

11 Outlne Matheatcal Tool: Sedefnte prograng Defntons of generalzed relatve entropes: D ( ), D ( ), D ( ) ax 0 n Propertes & operatonal sgnfcances of the Ther soothed versons Ther chldren: the n-, ax- and 0-entropes The super-parent : Quantu Reny Dvergence Relatonshp between D ( ) & D0 ( ) D ( )

12 Notatons & Defntons A H A quantu syste Hlbert space B ( H ): algebra of lnear operators actng on H P ( H ):set of postve operators D ( H ) P( H ): Lnear aps: If ts adjont ap: defned through set of densty atrces (states) : B ( H ) B( H ) A * : B A Tr ( ) Tr * ( ) B A B A Quantu operatons (quantu channels) : lnear CPTP ap B : A B

13 Matheatcal Tool Se-defnte prograng (SDP) A well-establshed for of convex optzaton The objectve functon s lnear n an nput constraned to a se-defnte cone Effcent algorths have been devsed for ts soluton

14 Matheatcal Tool (2) Se-defnte prograng (SDP) (, A, B); : P ( H ) P( H ) postvty-preservng ap Pral proble nze subject to AB, P ( H ), A B (forulaton:watrous) Dual proble AX axze Tr( BY ) Tr( ) ( X) B; subject to * ( Y) A; X 0; Y 0; Optal solutons: IF Slater s dualty condton holds.

15 Outlne Matheatcal Toolkt: Sedefnte prograng Defntons of generalzed relatve entropes: D ( ), D ( ), D ( ) ax 0 n

16 D ( H ), Defntons of generalzed relatve entropes P ( H ); supp supp ; Max-relatve entropy [ND] D ax ( ): nf : 2 ax 1/2 1/2 1/2 1/2 log ( ) 2 I Mn-relatve entropy [Dupus et al] D ( ): 2log n 1 2log F(, ) fdelty

17 D 0 Defntons of generalzed relatve entropes D ( H ), P ( H ); supp supp ; 0-relatve Reny entropy ( ): log Tr ( ) where denotes the projector onto supp contd. -relatve Reny entropy ( 1) 1 D 1 1 ( ): log Tr ( ) l D ( ) 0 = D ( ) 0

18 Propertes of generalzed relatve entropes Postvty: If, D ( H ), for * ax, 0, n D ( ) 0 * Data-processng nequalty: just as D( ) D ( ( ) ( )) D ( ) for any CPTP ap * * Invarance under jont untares: D ( UU UU ) D ( ) * * for any untary operator U Interestngly, D ( ) D ( ) D( ) D ( ) 0 n ax

19 Operatonal nterpretaton of the ax-relatve entropy Multple state dscrnaton proble: ts state Bob He does easureents to nfer the state: POVM Hs optal average success probablty: p a quantu syste & told 1 * 1 succ E1,.., E 1 : ax Tr 2 E 1,.., E : 0 E I; E I 1 E wth prob. 1 1

20 Theore 3 [M.Mosony & ND]: The optal average success probablty n ths ultple state dscrnaton proble s gven by: * p succ 1 n ax 1 D ax ( ) 2

21 Sketch of proof: Let 1 : bass n C Let 1 1 : 1,.., 1 * 1 : ax Tr E E succ E p 1 : 1 ax Tr ( )] [ POVM E E ( ); Tr ax Tr I Y Y Y C C H P H 1 ( ); E Y C P H Tr ; E I Y C H ( ) C P H

22 Sketch of proof: Let 1 : bass n C Let 1 1 : 1,.., 1 * 1 : ax Tr E E succ E p 1 : 1 ax Tr ( )] [ POVM E E ( ); Tr ax Tr I Y Y Y C C H P H 1 ( ); E Y C P H Tr ; E I Y C H ( ) C P H

23 Let 1 : p * 1 succ E1,.., E 1 Sketch of proof: bass n C Let : ax Tr E E 1 : POVM 1 : ax Tr [ ( E )] 1 1 P ( C H ) ax Tr YP ( C H ); Tr Y I C H Y Y 1 E Tr Y E I ; C P ( C H ); H SDP dualty condton holds [Koeng, Renner, Schaffner]

24 Pral proble Dual proble nze subject to AX axze Tr( BY ) Tr( ) ( X) B; subject to * ( Y) A; X 0; Y 0; nze subject to Tr( AX ) * =Tr : P ( C H ) P( H ) C Tr X I X C X 0; Tr( I X) Tr X H B : P ( H ) P( C H ); ax Tr YP ( C H ); Tr Y I C ; A H I H ; ( X) I X C Y * =TrC

25 * succ Sketch of proof contd: p n Tr X : X 0, I X ; C : n Tr X : X 0, X 1,2,.., 1 I X C 1 n Tr X : X 0, X 1,2,.., X X X; TrX TrX TrX 1 n X : 0, 1, 2,.., D ( H ) 1 n ax 1 M D TrX X =Tr X ; ; X TrX nf : 2 ax ( ) 2 Dax ( )

26 Operatonal nterpretaton of 0 Quantu bnary Bob receves a state hypothess testng: He does a easureent to nfer whch state t s POVM A Possble errors Type I Type II Error probabltes [ ] Tr(( I A) ) Tr( A ) ( I A) or & [ ] nference D (null hypothess) ( ): log Tr ( ) actual state Type I Type II (alternatve hypothess)

27 Suppose (POVM eleent) A Prob(Type I error) Prob(Type II error) Tr(( I A) ) Tr( A ) 0 Tr( ) Bob never nfers the state to be when t s BUT D 0 ( ) : log Tr 0 ( ) 2 D Hence 0 = Prob(Type II error Type I error = zero)

28 Suppose (POVM eleent) A Prob(Type I error) Prob(Type II error) Tr(( I A) ) Tr( A ) 0 Tr( ) Bob never nfers the state to be when t s BUT D 0 ( ) : log Tr In fact, n Prob(Type II error Type I error = zero) * D ( )

29 0.e., let 0A Ifor soe 0. Tr ( A ) 1 ( ) log * log Tr D 0 0 log *? 0AI Tr ( A ) 1 Tr(( I A) ); 0 Tr ( A) Tr ( ) 1 For choose A such that Tr ( A) 1 D ( ) 0 Soothed relatve entropes What f Bob has a sngle copy of the state but one allows non-zero but sall value of the Prob(Type I error)? * n Tr ( A ) * n Tr ( A ) log * ax log(tr ( A)) Hypothess testng relatve entropy [Wang & Renner] 0AI Tr ( A ) 1 D ( ) H

30 Copare operatonal sgnfcances of D & D( ) D( ) arses n asyptotc bnary hypothess testng Suppose Bob s gven any dentcal copes of the state He receves *( n) ( n) [0,1) : : 1 n ( n) n n H ( ) Bob s POVM A I A n,( ) Mnu type II error when type I error *( n) l log ( n) D( ) n [Quantu Sten s Lea] n

31 Operatonal nterpretatons n bnary hypothess testng D ( ) H One-shot settng; Sngle copy of the state: log * D( ) Asyptotc eoryless settng; Multple copes of the state: 1 n *( n) l log ( n) n [0,1) : n n (Bob receves dentcal copes of the state: or )

32 0. Sooth ax-relatve entropy D ( ) ax B ( ) : n D ( ) ax B ( ) : 0,Tr 1: F(, ) 1 fdelty 2 D ax ( ) & D ( ) 2 H can both be forulated as SDPs

33 Outlne Matheatcal Toolkt: Sedefnte prograng Defntons of generalzed relatve entropes: D ( ), D ( ), D ( ) ax 0 n Propertes & operatonal sgnfcances of the Ther chldren: the n-, ax- and 0-entropes

34 Just as: D ( ), D ( ) & D ( ) ax 0 n von Neuann entropy as parent quanttes for other entropes S D I ( ) ( ) ( I) H n ( ): D ax ( I) log ax H 0 ( ): ( ) D I 0 log rank( ) H ax ( ): D ( I) n [Renner] 2log Tr

35 Other n- & ax- entropes For a bpartte state AB : A B Condtonal entropy S( A B) S( ) S( ) Condtonal n-entropy H n ax D( I ) ( A B) : ax D ( I ) Max-condtonal entropy H H ax 0 AB B ax n B AB A B ( A B) : ax D ( I ) 0-condtonal entropy B 0 AB A B ( A B) : ax D ( I ) B B AB A B AB A B

36 They have nterestng atheatcal propertes: e.g. Dualty relaton: [Koeng, Renner, Schaffner]: For any purfcaton ax : ABC of a bpartte state H ( A B) H ( A C) n AB (just as for the von Neuann entropy): H ( A B) H( A C) -- and -- nterestng operatonal nterpretatons:

37 Condtonal n-entropy Operatonal nterpretatons axu achevable snglet fracton Condtonal ax-entropy [Koeng, Renner, Schaffner] decouplng accuracy Condtonal 0-entropy one-shot entangleent cost under LOCC [F.Busce, ND]

38 Operatonal nterpretatons contd. Condtonal n-entropy Max. achevable snglet fracton d 1 H H H AB A B A B d 1 AB AB AB 2 n ( AB ) 2 d ax F (d ), B : CPTP : ( ) A B AB AB fdelty Gven the bpartte state t s the axu overlap wth the snglet state quantu operatons AB, AB, that can be acheved by local on the subsyste B. B ax. entangled state [Koeng, Renner, Schaffner]

39 Operatonal nterpretatons contd. Condtonal ax-entropy Decouplng accuracy Dstance of AB, fro a product state H 2 ax ( AB ) 2 d ax F, A A AB A B B fdelty How rando appears fro the pont of vew of an adversary who has access to B. no correlatons; I A copletely xed state on HA d A Fro the cryptographc pont of vew: A B decoupled [Koeng, Renner, Schaffner]

40 Operatonal nterpretatons contd. Condtonal 0-entropy one-shot entangleent cost One-shot Entangleent Dluton Bell states Bell : LOCC Alce AB Bob One-shot entangleent cost E (1) ( ): C AB n = nu nuber of Bell states needed to prepare a sngle copy of AB va LOCC

41 Pure-state ensebles: and E E Operatonal nterpretatons contd. Theore [F.Busce & ND]: One-shot perfect entangleent cost of a bpartte state AB under LOCC: E (1) ( ) n H ( A R) C AB 0 p, ; AB p E p RAB R R AB AB E E Tr, RA B RAB AB AB AB classcal-quantu state E condtonal 0-entropy

42 Outlne Matheatcal Toolkt: Sedefnte prograng Defntons of generalzed relatve entropes: D ( ), D ( ), D ( ) ax 0 n Propertes & operatonal sgnfcances of the and ther chldren: the n-, ax- and 0-entropes Ther soothed versons The super-parent : Quantu Reny Dvergence D ( )

43 D ( ) D ( ) D ( ) ax 0 n Max-relatve entropy O-relatve Reny entropy Mn-relatve entropy super-parent : D ( ) Quantu Reny Dvergence (sandwched Reny entropy) [Wlde et al; Muller-Lennert et al]

44 super-parent : Quantu Reny Dvergence (sandwched Reny entropy) [Wlde et al; Muller-Lennert et al] D ( ) D ( ) D ( ) ax 0 n 1 2 D( ) 1 D ( ) D ( ) f [, ] 0

45 D ( H ), For (0,1) (1, ) : D Quantu Reny Dvergence 1 ( : Tr ) log ( ) ; 1 where Note: If [, ] 0 Tr ( Tr ( ) P ( H ); supp supp ; ) Tr 1 Tr D ( ) 1 log Tr ( 1 ) 1 D ( ) -relatve Reny entropy

46 D ( H ), For D Quantu Reny Dvergence 1 ( : Tr ) log ( ) ; 1 where Note: If [, ] 0 (0,1) (1, ) : Tr ( ) P ( H ); supp supp ; Non-coutatve generalzaton of D ( ) Tr ( 2 ) 2 Tr 1 Tr D ( ) 1 log Tr ( 1 ) 1 D ( ) -relatve Reny entropy

47 Two propertes of Quantu Reny Dvergence (1) Data-processng nequalty; For [Frank & Leb; Beg 2013] holds also for D ( ( ) ( ) ) D ( ) : D ( ), D( ), D ( ) n , 2 CPTP ap ax

48 Two propertes of Quantu Reny Dvergence 1 (1) Monotoncty under CPTP aps : For, 2 [Frank & Leb; Beg 2013] holds also for D ( ( ) ( ) ) D ( ) D ( ), D( ), D ( ) n (2) Jont convexty: D For 1 1, 2 ( p p ) p D ( ) holds also for D n ( ), D ( ) Note: D ( ) ax s quas-convex: [ND] ax ax [Frank & Leb] D ( p p ) ax D ( ) ax 1 n

49 What about D ( )? 0

50 Outlne Matheatcal Toolkt: Sedefnte prograng Defntons of generalzed relatve entropes: D ( ), D ( ), D ( ) ax 0 n Propertes & operatonal sgnfcances of the Ther chldren: the n-, ax- and 0-entropes Ther soothed versons The super-parent : Quantu Reny Dvergence D ( ) Relatonshp between D ( ) & D0 ( ) & ts plcaton

51 Theore: [ND, F.Ledtzky] If For, 0,Tr 1; f then supp supp, l D ( ) = D ( ) 0...(1) 0 0 supp supp ; then (1) does not necessarly hold D ( ): Quantu Reny Dvergence D ( ): 0 0-relatve Reny entropy D ( ) D ( ) D ( ) ax 0 n D( ) D ( ) D ( ) f [, ] 0

52 Theore: [ND, F.Ledtzky] If For, 0,Tr 1; f then supp supp, Proof (key steps): (1) (2) If l D ( ) = D ( ) 0...(1) 0 supp supp : 0 supp supp ; then (1) does not necessarly hold D l D ( ) l D ( ) 0 0 (Arak-Leb-Thrrng nequalty) supp supp : ( ) D ( ) If l D ( ) D ( )...( b) ( a)&( b) (1) supp supp ; (a varant of the Pnchng lea) f D ( )...( a ) 0

53 Proof of the fact: If l D ( ) = D ( ) 0 A sple counterexaple: 0 supp supp, then does not necessarly hold 1 0 ; c c 1 c (0,1)., 0,, 0. D ( ) 0 0 l D ( ) = - log(1 c) 0 0;

54 Suary Generalzed relatve entropes: D ( ), D ( ), D ( ) ax 0 n Propertes & soe operatonal sgnfcances D ( ): ax n ultple state dscrnaton, D 0 ( ): n bnary hypothess testng D0 D H Ther soothed versons; Ther chldren: the n-, ax- and 0-entropes The super-parent : Quantu Reny Dvergence D ( ) & D ( ) Relatonshp between 0 ( ) ( ) Operatonal sgnfcances of condtonal entropes D ( )

55 Thank you! Thanks also to: F.Busce, F.Brandao, M-H.Hseh, F.Ledtzky, M.Mosony, R.Renner, T.Rudolph,