C. Fuchs and C. M. Caves, \Mathematical techniques for quantum communication theory," Journal of Open Systems and Information Dynamics, vol. 3, pp. 1, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....by fixing a single quantum measurement. Each density operator will generate a unique probability distribution for the possible outcomes of that measurement (although other density operators may generate the same distribution) Even more interesting is the distance introduced by Fuchs and Caves [29] and defined by maxi53 mizing the preceding measure over quantum measurements. That is, d F (ae 1 ; ae 2 ) min fF i g (1 Gamma ( X i q trae 1 F i q trae 2 F i ) 2 ) 3.3) where F Gamma i 0, P i F i = I. A priori, this would appear to differ from the geodesic distance in that it ....

....and for the optimal measurement at every point along that geodesic. Perhaps surprisingly, the situation turns out to resemble that in the pure case: the geodesic distance is the same as the Bhattacharya Wooters distance calculated with the optimal quantum measurement (found by Fuchs and Caves [29]) the optimal measurement for the geodesic distance is the same as for the BhattacharyaWooters distance, and does not vary along a geodesic. 3.3 The geodesics of the statistical geometry We want to investigate the geometry of density operators which act on a Hilbert space A. To do this, we will ....

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C. Fuchs and C. M. Caves, "Mathematical techniques for quantum communication theory," Journal of Open Systems and Information Dynamics, vol. 3, pp. 1, 1995. 209

....x) Pr(J b 2 A x ) So we have obtained the desired results. Now, let us consider the Bhattacharyya Wootters distance BW = X j2A Pr(J 0 = j) 1 2 Pr(J 1 = j) 1 2 It is explained in [10] that (1 Gamma BW ) K=2. Therefore, we have BW (1 Gamma 4 Theta 2 Gammaffn ) Furthermore, in [4, 12] it is shown that the minimum of BW over all possible measurement is the fidelity F between ae 0 and ae 1 . So, we have F (1 Gamma 4 Theta 2 Gammaffn ) A theorem due to Uhlmann [6, 11] says that the fidelity between two mixed states ae 0 and ae 1 on a system H is given by F = max jhOE 0 ....

C. A. Fuchs and C. M. Caves, "Mathematical techniques for quantum communication theory", Open Systems and Information Dynamics, vol. 3(3), pp. 1 -- 12, to be published.

....65 that it is multiplicative over tensor products: overlap(ae 0 Omega ae 1 ; ae 2 Omega ae 3 ) overlap(ae 0 ; ae 2 ) overlap(ae 1 ; ae 3 ) 3.16) The following proposition establishes the connection between the Bhattacharyya coefficient and the overlap. 3. 13 Proposition (Fuchs and Caves [FC95]) Overlap is equivalent to the quantum Bhattacharyya coefficient: overlap(ae 0 ; ae 1 ) B(ae 0 ; ae 1 ) 3.17) Apart from the geometric interpretation of the overlap mentioned above, there also exists a physical interpretation for the square of this value. The following discussion is based on ....

FUCHS, C. A. AND C. M. CAVES, "Mathematical techniques for quantum communication theory", Open Systems and Information Dynamics 3, 3 (1995), pp. 345--356.

....; 2 ) 1 )G( 1 ; 2 ) 33) The proof uses a representation of the quantum delity in terms of measurement probabilities. Given a measurement described by a positive operator valued measure (POVM) with POVM elements E i , the probability for outcome i is p i = trace E i . Fuchs and Caves [29] showed that the quantum delity of 1 and 2 is the classical delity of the measurement probabilities for the measurement that, according to the classical delity, best distinguishes the two density operators, i.e. F ( 1 ; 2 ) min fE i g F cl (p 1 ; p 2 ) 34) 21 Here the minimum ....

C. Fuchs and C. M. Caves, \Mathematical techniques for quantum communication theory," Journal of Open Systems and Information Dynamics, vol. 3, pp. 1, 1995.

....(1 Gamma )G(oe 1 ; oe 2 ) 33) The proof uses a representation of the quantum fidelity in terms of measurement probabilities. Given a measurement described by a positive operator valued measure (POVM) with POVM elements E i , the probability for outcome i is p i = trace aeE i . Fuchs and Caves [29] showed that the quantum fidelity of ae 1 and ae 2 is the classical fidelity of the measurement probabilities for the measurement that, according to the classical fidelity, best distinguishes the two density operators, i.e. F (ae 1 ; ae 2 ) min fE i g F cl (p 1 ; p 2 ) 34) Here the ....

C. A. Fuchs and C. M. Caves, "Mathematical techniques for quantum communication theory," Journal of Open Systems and Information Dynamics, vol. 3, pp. 1, 1995.

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