Basic Entropies for Positive-Definite Matrices

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1 Journal of Mahemaics and Sysem Science 5 ( doi: 0.65/59-59/ D DAVID PUBLISHING Basic nropies for Posiive-Definie Marices Jun Ichi Fuii Deparmen of Ars and Sciences (Informaion Science, Osaa Kyoiu Universiy, Asahigaoa, Kashiwara, Osaa , Japan. Received: January, 05 / Acceped: February 5, 05 / Published: April 5, 05. Absrac: We exend basic enropies in he classical informaion heory o marix ones in he quanum informaion heory. Then we show ha relaions beween marix enropies similar o he classical ones hold. Keywords: Marix enropy, Informaion heory, Quanum enropy, Relaive operaor enropy. Inroducion Le a and b i be elemenary sochasic evens for probabiliy vecors X= ( pa ( and Y= ( pb ( i for =,..., n and i =,..., m. Then a (classical channel is a probabiliy marix Pc = ( pi which ransforms X ino Y and he enropies in he classical informaion heory are defined by: compound enropy H( XY, pa (, b pa (, b, = i i condiional enropies H( X Y = pa (, b pa ( b, HY ( X = pa (, b pb ( a, muual enropy i i i i pa (, b I( XY ; = pa (, b. ( ( i i pa pbi The relaions beween hem are: H( X, Y = H( X H( Y X = HY ( H( X Y = IXY ( ; H( X Y HY ( X, Corresponding auhor: Jun Ichi Fui Deparmen of Ars and Sciences (Informaion Science, Osaa Kyoiu Universiy, Asahigaoa, Kashiwara, Osaa , Japan. -mail: fuii@cc.osaa-yoiu.ac.p. H( X = I( X; Y H( X Y, HY ( = IXY ( ; HY ( X. Though all enropies are non-negaive, we noe ha he some individual muual informaions pa (, bi Ia ( ; bi = pa ( pb ( may ae negaive values. In his paper, we ry o exend hese classical informaion enropies o marix ones. The ey quaniy is he relaive enropy called he Kullbac-Leibler divergence: pa ( sx ( Y = pa (. pb ( i The muual enropy can be expressed by he relaive enropy: I( X; Y = I( X PX = s (( pa (, bi ( pa (, pb ( i In quanum informaion heory, X is considered as he densiy marix A (i.e., A 0 and TrA =, he channel P c as a race-preserving compleely posiive map Φ and he elemenary evens { b i } as a POVM (posiive operaor-valued measure F = { F i } (i.e., F 0 and F = I. Noe ha an inpu even { a } is considered as a PVM c i i

2 3 Basic nropies for Posiive-Definie Marices (proecion-valued measure = { }, ha is, a decomposiion of he ideniy. Suppose ha A= n n n is a specral decomposiion and Φ ( A = nnφ( n where { Φ ( i } is a POVM. Then he Ohya muual enropy I( A; Φ is defined by he Umegai relaive enropy as s ( A B = Tr A( A B U I( A; Φ = sup s ( Φ( A Φ( A U n n n n ([6] which is an exension of (. Thereby, if a relaive marix enropy corresponding o su ( A B is realized, hen we migh exend informaion enropies o marix ones. Simply one migh hin of A( A BA as an observable, bu i is probably no suiable since we hin a relaive enropy should be an iniial vecor of a geodesic for a cerain geomery. In fac, he geodesic of one of he Hiai-Pez geomeries [3] is M (, = exp(( A B and hence is iniial angen vecor is expressed by dm ( A, B SU ( A B d = 0 = U U*( B A U U* [] ( d d where U is a uniary wih diag ( d i = U AU and he funcion f [] is he divided difference f [] ( xy, =, see [6, ]. We hin i is a f ( x f ( y x y marix version of he Umegai enropy. In fac, Tr SU ( A B = Tr A( B A = s ( A B. Also Pez [] defined a quanum condiional enropy h( ρ B = s( ρ sb ( (where a composie marix ρ = W will be U discussed laer and i is relaed o he Umegai enropy: h( ρ B = dim H s ( ρ τ ρ A U A B where τ A is a racial sae. Bu h( ρ B is no always posiive unforunaely. Since he quanum condiional enropy is no posiive hough i is a numerical quaniy and he marix enropy S ( A U B is a somewha awward ool, here we do no use S ( A U B while we fully use he above idea, in paricular, Ohya's consrucion. In his paper, we use he relaive operaor enropy S( A B = A ( A BA A inroduced in [9] and ry o exend informaion enropies o marix ones.. Relaive Operaor nropy To begin wih, we review he relaive operaor enropy for posiive (bounded linear operaors on a Hilber space, see [9, 0,, 6]. Le A# B be a weighed geomeric operaor mean in he sense of Kubo-Ando [5]; A# B = A ( A BA A for inverible A. If A is no inverible, we may define as A# B= lim n ( A # n B in he srong operaor opoy. In [0, 9], we inroduced he relaive operaor enropy S( A B as a derivaive for a differeniable pah of geomeric operaor means A# B if he following srong-limi exiss as a bounded operaor; A# B A lim 0 If B is inverible, hen η S( A B = B ( B B where η is he enropy funcion: η( x = x x if x> 0, η(0 = 0. In addiion, if A is inverible, hen

3 Basic nropies for Posiive-Definie Marices 33 S( A B = A ( A BA A So he exisence condiion is; Lemma. If A is maorized by B, i.e., A α B for some α > 0, hen S( A B exiss. In fac, by Douglas' maorizaion heorem [5], we have A = DB for some operaor D and SA ( B = Bη( D* DB To discuss marix enropies based on S( A B, we summarize is properies [8, 9, 0]: Lemma. The relaive operaor enropy has he following properies if i exiss: ( If B B, hen S( A B S( A B. ( T S( A B T S( T AT T BT (he equaliy holds for inverible T. ( S( αa αb = αs( A B for α > 0. S( A B S( A B S( A A B B. (3 ( S( A B S( A B (3 S(( A A ( B B (4 ( S A B = S( A B. (4 S( A B. for all [0,]. = S( A B for a PVM{ }. (5 S( A B B A. (6 S( A αb = ( α A S( A B for α > 0. The properies (, (', (3, (3' and (4 are called ransformer inequaliy, homogeneiy, subaddiiviy, oin concaviy and orhogonaliy respecively. 3. Marix Probabiliy Space From now on, we assume ha he space is finie dimensional, ha is, operaors are marices. Assume ha A n, he n n posiive-definie marices and B, he m m ones. Le { } be he m (fixed decomposiion of he ideniy, ha is, each be a proecion and = I n. A se { } is considered as elemenary probabiliy evens. Le A= be a specral decomposiion of of an inverible densiy marix, ha is, is posiive-definie and Tr A =. Then, we can observe ha he probabiliy p ( is given by Tr( = Tr(. Le Φ be a quanum channel from n o m. Then each F = Φ ( is considered as an elemenary even, bu i is no longer a proecion. So we ae a fixed POVM { F }, and consider a densiy marix B= sf. Assume s > 0. Then noe ha B is inverible since B min { s} F = min { s} Im. In his siuaion, we define a composie marix W for A and B by w F, where w 0, w Tr = s, w Tr F In fac, for he parial races Tr and Tr for each erm, we have Tr ( W = Tr w F = = A and similarly Tr ( W = B, so ha A and B are he marginal disribuions. The composie probabiliy is p ( F = wtr Tr F, and hence he condiional ones are as follows: wtr p ( F F =, s w Tr F p ( F A ypical example for W is, s F. In his case, A and B are considered independen. In fac, in his case, p ( F = p ( pf ( holds for each,. The following example is no independen: xample. Le ( w i be given by a nonnegaive

4 34 Basic nropies for Posiive-Definie Marices marix and all 6 ( w = 3 and F proecions of ran one. Then we have he marginal probabiliies 5 =, =, s =, s 3 3 The are no independen since, for insance, w = = / 4 = s 36. For posiive semidefinie marices of ran one =, =, F =, F =, we have he marginal densiy marices 5 0 A= =, B= F F =, and he composie marix W 4. Marix nropies 6 F F 0 = 3 0 F F = The composie marix enropy HW ( is defined by HW ( = η( W. Also he muual marix enropy by I( ; and he condiional enropies HW ( A and HW ( B defined by I( ; = SW ( A B, H( W A = S( W A I, H( W B = S( W I B. Noe ha hese marices can be defined since A and B are inverible (see Lemma exis. Immediaely we have HW ( 0 and HW ( B 0, while I( ; is no always posiive-semidefinie (see xample ex in he below since he individual muual informaion may be negaive. Bu is race is posiive: Lemma 3. Tr I( ; 0. By Lemma (5, we have Tr I( ; Tr (Tr W Tr A B = Tr A (Tr A(Tr B = = 0. Remar. Conrary o he classical case, anoher muual marix enropy IBA ( ;, even if i is defined well, is no equal o I( ; since A B= / B A and he usual produc also is no commuing. Then, by Lemma (4', we express hese enropies as follows: Lemma 4. Marix enropies have he following decomposiions: HW ( = η( wf. l l l I( ; = S( wf B l l l = l, wf S( lwf B l l l HW ( A = S( l wf l l I w = ( l η l Fl. HW ( A = S( wf B. l l l In addiion, all F are (muually orhogonal proecions, hen HW ( = η( w F., I w F s ( ;, = w. HW A = w F (, w. HW ( B = w F s, w. Thus, he laer case, where ( { F } is a PVM,

5 Basic nropies for Posiive-Definie Marices 35 shows he enropy values in he classical (commuaive case. So we confirm easily ha I( ; is no always posiive even for diagonal marices corresponding o he classical ones: xample. We simplify xample ; F =, F = 0 and B =. By he above lemma 3 0 (, we have only o show S( w F B is neiher posiive nor negaive: S = S Since 8 < 0 and 3 > 0, we have he required resul. Also (' shows his; w 9 w 3 = > 0 and = < 0. s s In he conex for he composie elemenary evens { F }, he enropy η ( A, η ( B should be exended o H ( A = ( w F, and F, H ( B = ( s w F., In fac, we obain Tr ( H ( A = Tr( w F ( F = ( he following relaions similar o he classical cases: Theorem 5. The following equaliies hold: ( HW ( B I( ; = HF ( A ( H ( A HW ( A = HW (. F Proof. By Lemma 4 and Lemma (6, we have ( as follows: HW ( B I( ; = S wf B l S wf B l S w F B l = ( ( wf S wf B l l = ( w F = H ( A. l, F Similarly ( follows from HF ( A HW ( B = ( w F S wf I l, l = η wf l = η w Fl l = η w Fl = η( W = HW (. l, Remar. As in he above, HW ( A I( ; is no equal o H ( B in general. = η( = η( A xample 3. In xample, we exchange { f } o some POVM. Then he following marix enropies and similarly Tr( H ( B = η( B. Then we have similar o he classical ones as in xample : HW ( (6 F ( F 0 =, 0 ( 6 F (3 4 F

6 36 Basic nropies for Posiive-Definie Marices I( ; 9 3 (6 F ( F 0 =, ( F 5 (3 F 5 HW ( A (6 F ( F 0 6 = ( F (3 F 3 HW ( B (6 F ( 4 F = 0 ( 4 F (3 F HF ( A (6 F ( F 0 0 ( F 5 (3 F 5,, Similar ideniies hold lie classical cases for a PVM { F }: Corollary 6. If { F } is a PVM, hen ( HW ( A I( ; = H ( B, and ( H ( B HW ( B = HW (. In he above corollary, he assumpion are necessary as in he following example: xample 4. In he siuaion of xample 3, consider he case ha each F is no a proecion: Le P =, P =, 3 F = P = and F = P = Then 6 5P P B= F F = 3 3 = 6 and H ( B 6 F 3F 3 = 3 F 3 3F P P By we have 6F F 9 9 w F = = P, F 3F 9 wf = = P, S w lf I = P P, l S wl F I = P P l 9 and hence HW ( A P = P Since have I( ; B= and 35 49P 44 B=, we 5 35P P =, P we have HW ( A I( ; P =, 9 5 P

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