Donald Bures, "An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w -algebras," Transactions of the American Mathematical Society, vol. 135, pp. 199, 1969.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....term fidelity . We may define it by: F (ae 1 ; ae 2 ) j max jOE 2 i jhOE 1 jOE 2 ij 2 ; 2.31) where jOE 1 i is an arbitrary purification of ae 1 and jOE 2 i ranges over all possible purifications of ae 2 . This quantity was introduced (in the more general w algebra context) 40 by Bures [14], who showed that it was equal to (tr( q ae 1=2 1 ae 2 ae 1=2 1 ) 2 (2.32) Jozsa [15] presents version of the proof which does not use the language of algebra representation theory; Uhlmann [13] also contains a proof. For pure states, the fidelity is just jh 1 j 2 ij 2 ; for one ....

....the basis that diagonalizes the operator ae Gamma1=2 1 q ae 1=2 1 ae 2 ae 1=2 1 ae Gamma1=2 1 . 55 Proof: We begin with the fact that the infinitesimal form of the statistical distance is just twice the Bures distance between the infinitesimally separated density operators. Bures ([14], Prop. 2.3) has shown that tr( q ae 1=2 1 ae 2 ae 1=2 1 ) maxjhOE 1 jOE 2 ij; 3.7) where jOE 1 i is an arbitrary purification of ae 1 and jOE 2 i ranges over all possible purifications of ae 2 . Jozsa [15] presents version of the proof which does not use the language of algebra ....

Donald Bures, "An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w -algebras," Transactions of the American Mathematical Society, vol. 135, pp. 199, 1969.

....) k = 1; N (3) be the reduced state in the k th signal position after coding and decoding; i.e. ae k is the decoded version of the k th transmitted state ae i k . Let F (ae 1 ; ae 2 ) i trace(ae 1=2 1 ae 2 ae 1=2 1 ) 1=2 j 2 (4) denote the Bures Uhlmann fidelity function [15, 16, 17]. The coding decoding scheme has fidelity 1 Gamma ffl if it satisfies the following fidelity requirement: There is an N 0 such that for all N N 0 , X oe N Prob(oe N ) N Y k=1 F (ae i k ; ae k ) 1 Gamma ffl (LOCAL FID) 5) Note that high fidelity according to (LOCAL FID) allows ....

....mixture of j 1 i and j 2 i. Thus to get the greatest benefit from Schumacher compression, the purifications should be chosen so that their ensemble has least von Neumann entropy; i.e. the two purifications should be as parallel as possible. According to Bures and Uhlmann s basic 11 theorem [15, 16, 17], the minimum possible angle min between purifications of ae 1 and ae 2 is given by cos 2 min = F (ae 1 ; ae 2 ) Moreover, a 50 50 mixture of states at angle min has entropy S min = H 1 cos min 2 ; 1 Gamma cos min 2 ; which gives the Schumacher limit of ....

D. Bures, "An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w -algebras," Transactions of the American Mathematical Society, vol. 135, pp. 199, 1969. 22

....necessarily true that repeating the measurement is guaranteed to give the same result when the operation is F 1=2 b : I conjecture there is no operation which can provide this guarantee in the case of nonorthogonal F b . I will use the fidelity F (ae; oe) tr q ae 1=2 oeae 1=2 ) 2 [7], 8] 9] in specifying a measure of disturbance for quantum states. For pure states ae = j ih j, this is just h joej i. It is unity when ae = oe, and zero when their supports are orthogonal. It is therefore a reasonable measure of how similar two quantum states are. We may define 1 Gamma F ....

Donald Bures, "An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite w -algebras," Transactions of the American Mathematical Society, vol. 135, pp. 199, 1969.

....to the geodesics. The observables can be shown to form a Jordan spin factor of type I 2 [26] Representing it as a Jordan subalgebra of a matrix algebra, the spheres and hemispheres in question can be identified with submanifolds of density operators. By the introduction of the Bures metric [18] this identification becomes an isometric embedding onto geodesic submanifolds of the space of density operators. This enables an unambigious way to introduce the parallel transport a la Berry [15] in its extension to density operators [37] Explicit expressions for the parallel transport along ....

....: y 2 n 1 y 2 = 1 4 ; y 0g (12) which is a deformation of the ball (9) to a hemisphere of dimension n 1. In the interior of the ball the metric has been changed. The new metric reflects partly the superposition principle. Below we shall see its close relation to the Bures distance [18]. The metric (12) indicates an extension of the notation of transition probability as introduced by (6) p(x; y) p( x ; y ) 1 2 (1 4 xy 4 x y) x; y 2 E n ; 13) so that the mixed states appear embedded within the pure states of a (n 1) sphere: They are purified. The ....

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D. J. C. Bures, An extension of Kakutani's theorem on infinite product measures to the tensor product of semifinite W -algebras, Trans. Amer. Math. Soc. 135 (1969) 199

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