A Radon-Nikodym Theorem for Completely Positive Maps

Transcription

1 A Radon-Nody Theore for Copletely Postve Maps V P Belavn School of Matheatcal Scences, Unversty of Nottngha, Nottngha NG7 RD E-al: vpb@aths.nott.ac.u and P Staszews Insttute of Physcs, Ncholas Coperncus Unversty, Toruń, Poland Orgnally publshed n: Report on Matheatcal Physcs, 4 (1986) Abstract The a of ths paper s to generalze a noncoutatve Radon- Nody theore to the case of copletely postve (CP) ap. By only assung absolute contnuty wth respect to another CP ap the exstence of a Hertan-postve densty as the unque Radon-Nody dervatve s proved n the coutant of the Stensprng representaton of the reference CP ap. 1 Prenares and de ntons Let A be a C*-nored algebra, and let B(h) denote the algebra of all bounded operators on a Hlbert space h. In ths paper we wll obtan a postve selfadont densty operator % for a copletely postve ap fro A nto B(h) strongly absolutely contnuous wth respect to another such ap gven, say, by a fathful weght or trace ' as = 1'. It wll be unquely de ned as a noncoutatve generalzaton of Radon-Nody dervatve n the Hlbert space H of Stensprng representaton of. To ths end, we rst recall the de nton of copete postvty. If A and B are C*-algebras, M(n) (n 1) the algebra of n n coplex atrces and s a lnear ap fro A to B, we shall say that s n-postve f the ap n : A M(n)! B M(n); n (a ) = (a) ; a A; M(n); s postve. The ap s called copletely postve f t s n-postve for all ntegers n. 1

2 The copletely postve aps play an portant role n the descrpton of quantu channels and te evolutons of open quantu systes []. Let us consder two quantu systes descrbed n ters of C*-algebras A and B. It can be easly shown that f the Hesenberg dynacs of the copound syste s descrbed by a -endoorphs of AB, then the reduced dynacs as condtonal expectaton of correspondng to an ndependent state on B s descrbed by a copletely postve dentty preservng aps : A! A (such = s usually called a dynacal ap on A). The coplete postvty of a reduced dynacs was rst ponted out by Kraus [4] n the context of state changes produced by quantu easureents. If a C*-algebra A descrbes an open physcal syste subect to copletely postve dynacs, then any dynacal ap of ths syste, consdered n a representaton, s a copletely postve ap of nor one : A! B(h), where =. Let us recall that the condton of coplete postvty of can be wrtten [5] n the for (a a ) 0; 8 h; 8a A; = 1; : : : ; n; 8n N: The condton of noralzaton of can be expressed n the for (1) = 1 f 1 A and 1 stand for denttes n A and B(h), respectvely. Accordng to the faous results of Stnesprng [5] any (noralzed) copletely postve ap : A! B(h) can be represented n the for (a) = F (a)f ; where : A! B(H ) s a representaton of A on a Hlbert space H and F s a bounded (soetrc) lnear operator fro h nto H. Such a representaton of a copletely postve ap wll be called spatal. The noralzaton condton for a dynacal ap ples the soetrcty F F = 1. Let and denote copletely postve aps fro A nto B(h) and let (a ) ; = 1; : : : ; n be a faly of sequences n A. Such a faly wll be called a (; ) faly of sequences f for any n N = (a a ) (a a r ) (a a r ) = 0 (1.1) 8 h; = 1; : : : ; n: Now we generalze varous fors and strengthened fors of the concept of absolute contnuty [3] n the case of copletely postve aps. De nton 1 A copletely postve ap s called

3 (1) copletely absolutely contnuous wth respect to a copletely postve ap f for any n N nf (a a ) = 0 for any ncreasng faly fa g of atrces A = [a a ] ples nf (a a ) = 0; 8 h; = 1; : : : ; n; () strongly copletely absolutely contnuous wth respect to f for any (; ) faly of sequences (a ) ; = 1; : : : ; n we have for any n N (a a ) = 0; 8 h; = 1; : : : ; n; (3) copletely donated by f there exsts a > 0 such that for any n N (a a ) n (a a ) ; 8 h; 8a A; = 1; : : : ; n: It s rather obvous that (3) ) () ) (1). In the partcular case (a) = '(a)1, where ' : A! C denotes a postve functonal on A (e.g. a reference state, or trace), we shall say that s copletely absolutely contnuous or strongly copletely absolutely contnuous or copletely donated by the functonal '. If copletely postve aps are of the for (a) = '(a)1, (a) = {(a)1, where '; { are postve functonals on A, then one can easly verfy that our fors of absolute contnuty (1)-(3) ply that { s (1 0 ) '-absolutely contnuous, ( 0 ) strongly '-absolutely contnuous, (3 0 ) '-donated, respectvely n the sense of Gudder [3]. A Radon-Nody theore for copletely postve aps Theore Let and be a bounded copletely postve aps for A nto B(h) and let H be a Hlbert space of a representaton : A! B (H) n whch s spatal, that s (a) = F (a)f; 8a A; (.1) where F s assued to be bounded operator h! H. Then 3

4 (a) s copletely absolutely contnuous wth respect to f and only f t has a spatal representaton (a) = K (a)k wth (a)k = #(a)f, where # s a densely de ned operator n the n nal H, coutng wth P o (A) = f (a) ; a Ag on the lneal D = (a )F. (b) s strongly copletely absolutely contnuous wth respect to f and only f s spatal n (; H) and there exsts a postve self-adont operator %, unquely de ned on D, a lated wth the coutant (A) 0 and such that (a) = F %(a)f = (% 1= F ) (a)(% 1= F ); 8a A; (.) (c) s copletely donated by f and only f (.) holds and % s bounded. Proof. Let us rst setch the prove the part (a) gven n [1]. The condton of absolute contnuty eans that s noral n the nal spatal representaton of wth the support orthoproector P aorsed by the support P of. Therefore t s spatal, wth the operator K : h! H unquely de nng the operator # = 0 (K) on D by 0 (K) (a)f = (a) K; 8A A; h. such that t coutes wth (A). The reverse s obvous. Let us now prove the part (b) of our theore. ()) Let be a representaton of a C*-algebra A n the Hlbert space H generated by the algebrac tensor product A h wth respect to a postve Hertan blnear for D E a a = n (a a ) (.3) de ned by the equalty E (a) a = E aa (.4) (for detals see [5]). Let us denote by F the bounded operator H! h, F : 7! 1 ; (.5) (a canoncal soetry h! H f s noralzed). Then we have [5] (a) = F (a)f : (.6) De ne an operator I n H nto H by the forula I : E (a )F 7! a = (a )F : (.7) 4

5 Ths s a consstent de nton of a lnear operator on the lneal D H because condton () ples (1) fro whch, tang nto account (.1), (.5) and (.6), we obtan the condton (a )F (a )F = 0 ) a a : = 0 Obvously, we have F = I F. nto prove that I s closable let us rst note that any sequence of eleents P of (a o P )F can be expressed n the for (a F be any P sequence such that (a )F! 0 and P a s convergent. Then for any set ; = 1; : : : ; n Moreover, h (a a ) ; = h F (a ) (a )F ; = h(a )F (a )F = ; (a )F = 0: (a a r ) (a a r ) ; D E = (a a r ) (a a r ) = (a a r ) = 0; hence P a s Cauchy by assupton. Hence (a ) ; = 1; : : : ; n for a (; ) faly of sequences. Then fro the strong coplete absolute contnuty of wth respect to we have for any n N 0 = h (a a ) ; D E = a a = a : Ths proves that I s closable. 5

6 Denote byn I ts closure. Then there exsts an adont operator I de ned on the lneal P a o, dense n H, by the equalty D E a a D = (a )F I a E: (.8) s a l- The postve self-adont operator % = I I on the lneal (a )F ated wth (A) 0 because on the doans of I and I we have (a)i = I (a); I (a) = (a)i ; (.9) Let us verfy the rst of the equaltes (.9). Tang nto account (.7) and (.4) we have E (a)i (a )F = (a) E a E = aa Tang nto account that F = I F, we obtan E = I (aa )F (a) = F (a)f = F %(a)f = I (a) (a )F E: = (% 1= F ) (a)(% 1= F ) = K (a)k; where K = % 1= F. (() Let (a ) ; = 1; : : : ; n be a faly of (; ) sequences. Then P (a )F! 0 and oreover 0 = (a a r ) (a a r ) ; = (% 1= F ) (a a r ) (a a r )(% 1= F ) = = ; D % 1= % 1= (a (a )F E a r )F % 1= % 1= (a (a r )F : a r )F E Hence % 1= P (a )F s Cauchy, and snce %1= s closed, % 1= (a )F! 0: 6

7 Then we have ; h (a a ) = = h (% 1= F ) (a ) (a )(% 1= F ) ; % 1= (a )F = 0: Ths eans that s strongly copletely absolutely contnuous wth respect to. Ths copletes the proof of part (a). Let us prove part (c) of our theore. ()) Suppose to be copletely donated by. As the condton (3) ples (), therefore (a) holds. It reans to prove that % s bounded. The boundedness of % follows fro the followng calculatons: % 1= D (a )F = % 1= (a )F E % 1= (a )F = (a a ) (a a ) ; ; D E = (a )F (a )F = (a )F : (() Suppose that % 1= s bounded, then (a a ) = (% 1= F ) (a a )(% 1= F ) ; = = ; D % 1= % 1= = % 1= ; (a )F % E 1= (a )F (a )F % 1= (a )F (a a ) : Hence s copletely donated by. The unqueness of % can be assured by choosng the sallest Hlbert space H n whch (a) has the Stensprng for (a) = F (a)f. Note that, f = 1', H = h H ', (a) = 1 ' (a) and F = 1 f, where H ' 3 f s the space of the cyclc representaton '(a) = f ' (a) f of a postve functonal ' on A. The forulaton of coplete absolute contnuty for CP aps belongs to VPB, and the Man Theore n the forulaton of Parts (a) and (b) was orgnally gven n [1]. 7

8 References [1] V.P. Belavn, In: Wors of the 8 th All-Unon Conference on Codng Theory and Transsson of Inforaton, Kubyshev, June 1981 (n Russan). [] V. Gorn, A. Kossaows and E.C.G. Sundarshan, preprnt, Unversty of Texas at Austn, ORO , CPT 44 (1975). [3] S. Gudder, Pac c J. Math., 80, (1979), 141. [4] K. Kraus, Ann. Phys. 64 (1971), 311. [5] W.F. Stnesprng, Proc. Aer. Math. Soc. 6 (1955), 11. 8