Nonadditive Conditional Entropy and Its Significance for Local Realism

Transcription

1 Nonaddtve Condtonal Entropy and Its Sgnfcance for Local Realsm arxv:uant-ph/ Jan 2000 Sumyosh Abe (1) and A. K. Rajagopal (2) (1)College of Scence and Technology, Nhon Unversty, Funabash, Chba , Japan (2)Naval Research Laboratory, Washngton, D. C , USA Based on the form nvarance of the structures gven by Khnchn s axomatc foundatons of nformaton theory and the pseudoaddtvty of the Tsalls entropy ndexed by, the concept of condtonal entropy s generalzed to the case of nonaddtve (nonextensve) composte systems. The proposed nonaddtve condtonal entropy s classcally nonnegatve but can be negatve n the uantum context, ndcatng ts utlty for characterzng uantum entanglement. A crteron deduced from t for separablty of densty matrces for valdty of local realsm s examned n detal by employng a bpartte spn-1/2 system. It s found that the strongest crteron s obtaned n the lmt. PACS numbers: Bz, a, y, d

2 In ths Letter, we present a generalzaton of the concept of condtonal entropy for nonaddtve (nonextensve) composte systems. Ths s done based on the form nvarance of the structures n Khnchn s axomatc foundatons of classcal nformaton theory and the pseudoaddtvty of the Tsalls entropy ndexed by n. nonextensve statstcal mechancs. Then, we dscuss the nonaddtve condtonal entropy n the uantum context. It s shown that the nonaddtve condtonal entropy s nonnegatve for all classcally correlated states (.e., separable states) but can be negatve for nonclasscal correlated states (.e., uantum entangled states). Snce classcally correlated states admt local hdden-varable models, the postvty of the nonaddtve condtonal entropy leads to a crteron for valdty of local realsm. Ths crteron depends on the Tsalls nonaddtvty parameter. We shall see how the strongest crteron can be obtaned by controllng the value of. The present attempt can also be thought of as a steppng stone towards nonaddtve (uantum) nformaton theory. In the development of the foundatons of classcal nformaton theory, Khnchn [1] presented a mathematcally rgorous proof of a unueness theorem for the Boltzmann- Shannon entropy based on the addtvty law for a composte system n terms of the concept of condtonal entropy. Suppose the total system be dvded nto two subsystems, A and B, and let p ( j A, B) be the normalzed jont probablty of fndng A and B n ther th and jth states, respectvely. Then the condtonal probablty of B ( )= ( ) ( ) wth A found n ts th state s gven by p B A p A, B p A, whch leads to the celebrated Bayes multplcaton law where p A p A, B p A p B A j j j j ( ) = ( ) ( ), (1),. It should be ( ) s the margnal probablty dstrbuton: p ( A) = p j( A B) noted that ths form of factorzaton can always be establshed n any physcal stuaton. From ths law, the Boltzmann-Shannon entropy s found to yeld S( A, B) = p ( A, B) ln p ( A, B) (2), j j j j

3 where S B A wth S( A, B) = S( A) + S( B A), (3) ( ) stands for the condtonal entropy defned by ( A) S( B A) = S( B A ) = p ( A) S( B A ) ( ) ( ) ( ) S B A p j B A ln p j B A. (5) j In the partcular case when A and B are statstcally ndependent, p p B j( ) and therefore S B A S B j( (4) B A) s eual to ( ) = ( ), mplyng the addtvty law:. S A, B S A ( ) = ( ) + S( B ). We emphasse here that there s a natural correspondence relaton between the multplcaton law and the addtvty law: ( ) = ( ) ( ) ( ) = ( ) + ( ). (6) p A, B p A p B A S A, B S A S B A j j When the above dscusson s generalzed to uantum theory of composte systems, a remarkable feature appears. The condtonal entropy may then take negatve values, suggestng ts mportance for characterzng uantum entanglement [2]. Ths feature may have ts root n a profound dfference between classcal and uantum probablty concepts: n the latter, one of Kolmogorov s axoms, namely the addtvty of the probablty measure, s volated n general [3]. Also, there are theoretcal observatons [4,5] that formal correspondences exst between thermodynamcs and uantum entanglement. These nvestgatons seem to suggest that the measure of uantum entanglement may not be addtve [5,6]. Nonaddtvty, or nonextensvty, s an mportant concept also n the feld of statstcal mechancs. A statstcal system s nonextensve f t contans long-range nteracton, long-range memory, or (mult)fractal structure. In such a system, a macroscopc thermodynamc uantty (e.g., the nternal energy) s not smply proportonal to the mcroscopc degrees of freedom. Boltzmann-Gbbs statstcal mechancs, whch can be constructed based on the addtve Boltzmann-Shannon entropy s known to expose dffcultes, f appled to such a system [7]. In ths respect, a nonextensve generalzaton of Boltzmann-Gbbs statstcal mechancs formulated by

4 Tsalls [8,9] has attracted much attenton n recent years. In ths formalsm, referred to as nonextensve statstcal mechancs, the Boltzmann-Shannon entropy n e. (2) s generalzed as follows: 1 S ( A, B) p j( A, B) 1, j = [ ] 1, (7) where s a postve parameter. Ths uantty converges to the Boltzmann-Shannon entropy n the lmt 1. Lke the Boltzmann-Shannon entropy, t s nonnegatve, possesses the defnte concavty for all > 0, and s known to satsfy the generalzed H- theorem. Nonextensve statstcal mechancs has found a lot of physcal applcatons. (A comprehensve lst of references s currently avalable at URL [10]. See also Ref. [11].) A standard dscusson about the nonaddtvty of the Tsalls entropy S[ p] assumes factorzaton of the jont probablty dstrbuton n e. (2). Then, the Tsalls entropy s found to yeld the so-called pseudoaddtvty relaton ( ) = ( ) + ( ) + ( ) ( ) ( ) S A, B S A S B 1 S A S B. (8) Clearly, the addtvty holds f and only f 1. However, there s a logcal dffculty n ths dscusson. As mentoned above, Tsalls nonextensve statstcal mechancs was devsed n order to treat a statstcal system wth, for example, a long-range nteracton. On the other hand, physcally, dvdng the total system nto the subsystems mples that the subsystems are made spatally separated n such a way that there s no resdual nteracton or correlaton. If the system s governed by a long-range nteracton, the statstcal ndependence can never be realzed by any spatal separaton snce the nfluence of the nteracton perssts at all dstances. In fact, the probablty dstrbuton n nonextensve statstcal mechancs does not have a factorzable form even f the systems A and B are dynamcally ndependent, and therefore correlaton s always nduced by nonaddtvty of statstcs [12]. Another mportant physcal mechansm whch prevents realzng the statstcal ndependence by spatal separaton s provded by the concept of entanglement n uantum theory. Thus, t s clear that the assumpton of the factorzed jont probablty dstrbuton s not physcally pertnent for

5 characterzng the nonaddtvty of the Tsalls entropy. These consderatons naturally lead us to the necessty of defnng the condtonal entropy assocated wth the Tsalls entropy. To overcome the above-mentoned logcal dffculty and to generalze the correspondence relaton n e. (6) smultaneously, frst let us recall the defnton of the normalzed -expectaton value n nonextensve statstcal mechancs [9]. In order to specfy the constrant on a physcal uantty to develop the Jaynes maxmum entropy prncple for nonextensve statstcal mechancs based on the Tsalls entropy, the authors of Ref. [9] ntroduced the dea of the normalzed -expectaton value. Consder a physcal uantty Q, whose value n the system s th state s denoted by Q. Its normalzed -expectaton value s defned by < Q > = P Q, (9) where P s the so-called escort dstrbuton [13] assocated wth the orgnal probablty dstrbuton p : P ( p ) = p ( ). (10) Next, let us consder the Tsalls entropy of the condtonal probablty dstrbuton 1 S ( B A ) = p j B A ( ) 1 j ( ) ( ) p j( A, B) 1 j = 1 p ( A) ( ) 1 1, (11) whch s a natural nonaddtve generalzaton of e. (5). Now, n conformty wth e. (4), we propose to defne the nonaddtve condtonal entropy as the normalzed - expectaton value of S ( B A ) wth respect to the margnal probablty dstrbuton p A ( ) as follows: A ( ) ( ) ( ) S B A S B A

6 = ( p ( A) ) S ( B A ) ( p ( A) ). (12) Usng e. (11) n e. (12), we obtan ( ) ( ) ( ) ( ) S A, B S A S ( B A) =. (13) S A Ths s our defnton of the nonaddtve condtonal entropy. From ths, t s mmedate to see ( ) = ( ) + ( ) + ( ) ( ) ( ) S A, B S A S B A 1 S A S B A, (14) whch s a natural nonaddtve generalzaton of e. (3) n vew of the pseudoaddtvty n e. (8). Therefore, the correspondence relaton n e. (6) s generalzed to ( ) = ( ) ( ) p A, B p A p B A j j ( ) = ( ) + ( ) + ( ) ( ) ( ) S A, B S A S B A 1 S A S B A. (15) In uantum theory, the probablty dstrbuton s replaced by the densty matrx ˆρ, whch s Hermtan, traceclass, and postve semdefnte. It ncorporates both pure and mxed states of a system. When the state s pure, the eualty ˆ ρ 2 = ˆ ρ holds, whereas ˆ ρ 2 < ˆ ρ for the mxed state. The uantum counterpart of e. (7) s gven by S [ ] 1 A, B Tr ˆ( ρ A, B) 1, (16) 1 ( ) = ( ) ( ) = [ ] whch converges to the von Neumann entropy S A, B Tr ˆ( ρ A, B) ln ˆ( ρ A, B) n the lmt 1. It s evdent that, lke the von Neumann entropy, ths uantty s nonnegatve and vanshes for a pure state. There s an orderng ambguty n defnng the condtonal densty matrx, snce the jont densty matrx and ts margnals do not commute wth each other, n general. Here, we avod to defne t explctly and drectly translate e. (13) to ts uantum counterpart. That s, S ( B A) = ( ) ( ) ( ) ( ) S A, B S A, (17) S A

7 wth [ ] 1 S ( A) = TrA A ( ˆ( ρ )) 1, (18) 1 and S ( A B) gven n e. (16). In e. (18), ˆ( ρ A ) s the margnal densty operator, defned by ˆ( ρ A) = Tr ˆ( ρ A, B), wheretr A (Tr B ) stands for the partal trace over the B states of the subsystem A ( B). By gong to the representatons n whch ˆ( ρ A, B ) and ˆ( ρ A ) are dagonal, S( B A) appears to take the form of e. (12). The dfference between the two s now that the expanson coeffcents are the egenvalues of the dagonalzed densty matrces n contrast to ther classcal counterparts presented earler. In the lmt 1, S( B A) converges to the condtonal von Neumann entropy S( B A) = S( A, B) S( A) dscussed n detal n Ref. [2]. Let us consder two partcular cases: a product state and a classcally correlated state. A product state ˆ( ρ A, B) = ˆ( ρ A) ˆ( ρ B), gves rse to the pseudoaddtvty relaton, whose form s the same as the classcal one n e. (8). Therefore, we have S( B A) = S( B) as expected. A classcally correlated state,.e. a separable state, s a convex combnaton of product states [14]: ˆ ρ( A, B) = w ˆ ρ ( ) ˆ A ρ ( B), (19) where w [ 0, 1 ] and w = 1. Ths state s known to admt local hdden-varable models and to satsfy the Bell neualty [14]. Usng the orthonormal bases of A and B, we wrte ˆ ( A) = p ( a) a a, ˆ ( B) = r ( b) b b, (20) ρ a where p ( a), r ( b) [ 0, 1 ] and p ( a) = r ( b) = 1. In ths case, the nonaddtve a a uantum condtonal entropy s gven by [ w p ( a) ] S ( B a) a S ( B A) =, (21) w p ( a) ρ [ ] 1 provded that we have used the notaton S B a ( 1 ) π b a 1 wth ( ) π b a w p ( a ) r ( b ) w p ( a ). Note that π b a b b { b [ ] } ( ) = ( ) ( ) [, ] 0 1 and π b a 1. b ( ) =

8 Therefore π( b a) s analogous to the classcal condtonal probablty dstrbuton. Ths fact makes S( B a) nonnegatve. Thus, we conclude that S B A ( ) 0 for any classcally correlated states. Ths establshes a -generalzaton of the result n Ref. [2]. [ ] = Here, t s worth mentonng that n Ref. [15] the Rény entropy S R ˆ α ρ ( 1 α) ln ( ˆ) Tr ρ α wth α > 1 s dscussed n connecton wth volaton of local realsm n uantum theory. The authors of Ref. [15] proposed the α -entropc neualty: R R R S ( A, B) max { S ( A), S ( B) }, whch can be volated by entangled mxed states. We α α α note that the Rény entropy s addtve but s not concave for α > 1. s A correspondng entropc neualty n the present nonnaddtve condtonal entropy S( B A) 0, S A B ( ) 0. (22) As mentoned above, e. (22) holds for all classcally correlated states that can be modelled by local hdden-varable theores. Nonclasscal correlated states, that s, uantum entangled states, yeld S( B A) < 0. Therefore, the neualty (22) can be used as a crteron for separablty of densty matrces. A pont s that the neualty depends on the parameter. To see how the separablty crteron s strengthened by controllng the value of, let us employ a bpartte spn-1/2 system. For ths purpose, we consder the followng parametrzed form of the Werner-Popescu state [14,15]: where 0 x 1, ÎA Î B 1 x ˆ ρ( A, B) = I ˆ I ˆ A B + x 4 ( ) the 2 2 Ψ Ψ, (23) dentty matrx n the space of the spn A ( B), and ( ) 1 Ψ = 2. (24) A B A B The separablty crtera known so far are: a) x < 1/ 2 (the Bell neualty), b) x < 1/ 3 (the α -entropc neualty wth α = 2) [16], and c) x < 1/ 3 (the nonnegatve egenvalues of the densty matrx wth partal transpose) [17]. The strongest crteron c) found by Peres [17] s known to actually be the necessary and suffcent condton for the separablty of the bpartte spn-1/2 system [18]. Our nterest here s to see whether 1

9 the Peres crteron obtaned by an algebrac method can be derved wthn the present framework of nonaddtve uantum nformaton theory. From the densty matrx n e. (23), the nonaddtve condtonal entropy s calculated to be ( ) = ( ) = S B A S A B x x (25) For a fxed value of, ths s a monotoncally decreasng functon of x. In Fg. 1, we present an mplct plot of S( B A) = 0 wth respect to [ 0, ) and x [ 0, 1 ]. One clearly sees how the crteron obtaned from the condtonal von Neumann entropy ( 1) can be strengthened by ncreasng the value of. From e. (25), t s also evdent that S( B A) 0 n the lmt f and only f x < 1/ 3. Thus, the Peres strongest crteron s obtaned from the present nformaton-theoretc approach. In concluson, we have constructed the nonaddtve condtonal entropy based on the form nvarance of the structures of the axomatc foundatons of classcal nformaton theory and the pseudoaddtvty of the Tsalls entropy ndexed by. Then, we have dscussed t n the uantum context and appled t to separablty of the densty matrx for valdty of local realsm. We have found that for a bpartte spn-1/2 system,.e., a 2 2 system, the postvty of the nonaddtve condtonal entropy leads to the Peres strongest crteron for separablty n the lmt. In Ref. [18], t has been dscussed that Peres s method of partal transposton of the densty matrx yelds the necessary and suffcent condton for separablty of 2 2 and 2 3 systems but not for other general systems. Therefore, for a further development, t s mportant to clarfy the general propertes of the condton n e. (22). S.A. was supported by the GAKUJUTSU-SHO Program of College of Scence and Technology, Nhon Unversty. A.K.R. acknowledges the partal support of the U.S. Offce of Naval Research.

10 References [1] A. I. Khnchn, Mathematcal Foundatons of Informaton Theory (Dover, New York, 1957). [2] N. J. Cerf and C. Adam, Phys. Rev. Lett. 79, 5194 (1997); Phys. Rev. A 60, 893 (1999). [3] W. M. Dckson, Quantum Chance and Non-localty (Cambrdge Unversty Press, Cambrdge, 1998). [4] S. Popescu and D. Rohrlch, Phys. Rev. A 56, R3319 (1997). [5] M. Horodeck, P. Horodeck, and R. Horodeck, Phys. Rev. Lett. 80, 5239 (1998). [6] S. Abe and A. K. Rajagopal, Phys. Rev. A 60, 3461 (1999). [7] R. Balan, From Mcrophyscs to Macrophyscs (Sprnger-Verlag, Berln, 1991), Vol. I. [8] C. Tsalls, J. Stat. Phys. 52, 479 (1988). [9] C. Tsalls, R. S. Mendes, and A. R. Plastno, Physca A 261, 534 (1998). [10] [11] Brazlan J. Phys. 29, Specal Issue (1999). Ths volume can be obtaned from [12] S. Abe, Physca A 269, 403 (1999). [13] C. Beck and F. Schlögl, Thermodynamcs of Chaotc Systems: An Introducton (Cambrdge Unversty Press, Cambrdge, 1993). [14] R. F. Werner, Phys. Rev. A 40, 4277 (1989). [15] S. Popescu, Phys. Rev. Lett. 72, 797 (1994). [16] R. Horodeck, P. Horodeck, and M. Horodeck, Phys. Lett. A 210, 377 (1996). [17] A. Peres, Phys. Rev. Lett. 77, 1413 (1996). [18] M. Horodeck, P. Horodeck, and R. Horodeck, Phys. Lett. A 223, 1 (1996).

11

12 Fgure Capton An mplct plot of S( B A) = 0 wth respect to [ 0, ) and x [ 0, 1 ]. In the lmt, x converges to 1/ 3.

13 Fg. 1