W. K. Wootters. Statistical distance and Hilbert space. Physical Review, D23:357-362, 1981. 6  Home/Search   Document Not in Database   Summary   APS   Related Articles   Check

.... 0 = j) Gamma Pr(J 1 = j) X j2A 1 Pr(J 1 = j) Gamma Pr(J 0 = j) Pr(J 0 2 A 0 ) Gamma Pr(J 1 2 A 0 ) Pr(J 1 2 A 1 ) Gamma Pr(J 0 2 A 1 ) j(Pr(X 0 = 0) Gamma Pr(X 1 = 0) j j(Pr(X 1 = 1) Gamma Pr(X 0 = 1) j 4ffl Now, let us consider the Bhattacharyya Wootters distance [7,15,18] BW = X j2A Pr(J 0 = j) 1 2 Pr(J 1 = j) 1 2 : It is explained in [15] that (1 Gamma BW ) K=2. Therefore, we have BW (1 Gamma 2ffl) Furthermore, in [7,18] 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 1 F (1 ....

....0) Gamma Pr(X 1 = 0) j j(Pr(X 1 = 1) Gamma Pr(X 0 = 1) j 4ffl Now, let us consider the Bhattacharyya Wootters distance [7,15,18] BW = X j2A Pr(J 0 = j) 1 2 Pr(J 1 = j) 1 2 : It is explained in [15] that (1 Gamma BW ) K=2. Therefore, we have BW (1 Gamma 2ffl) Furthermore, in [7,18] 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 1 F (1 Gamma 2ffl) A purification of ae b is simply a pure state of the overall system that has ae b for density matrix on Bob s side. A theorem due to Uhlmann [9,16] says ....

W. K. Wootters, "Statistical distance and Hilbert space", Physical Review D, vol. 23, pp. 357 -- 362, 1981.

....distance which is a Riemannian metric on the space of quantum mechanical density operators. This metric may be defined in terms of the structure of correlations between measurements on systems described by such density operators [26, 27] or it may be defined in a way originally due to Wootters [28] (who considered only pure states) in terms of the number of distinguishable states lying between two states described by density operators. The form of this metric for infinitesimally separated states is known, and is given by [25] ds 2 S = 8[1 Gamma tr q ae 1=2 (ae dae)ae 1=2 ] ....

....( 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 requires us to use the same measurement to distinguish states everywhere along the path between the two density operators. Wootters [28] considered the maximal Bhattacharyya Wootters distance for pure states, and showed that it coincided with the geodesic distance. In this chapter I find explicit expressions for the distance along geodesics of the Braunstein Caves statistical metric in density operator space, for the geodesic in ....

W. K. Wootters, "Statistical distance and Hilbert space," Physical Review A, vol. 23(2), pp. 357--362, 1981.

....of quantum distinguishability. Unfortunately the search for an explicit expression for this quantity remains a subject for future research. On the brighter side, however, there is a closely related upper bound on the Cherno bound that is of interest in its own right the statistical overlap [8]. If a measurement generates probabilities p 0 (b) and p 1 (b) for its outcomes, then the statistical overlap between these distributions is de ned to be F (p 0 ; p 1 ) X b q p 0 (b) q p 1 (b) 1.14) This quantity, as stated, gives a more simply expressed upper bound on . This is quite ....

....0 (b) q p 1 (b) 2.40) We shall dub this measure of distinguishability the statistical overlap or delity. It has had a long and varied history, being rediscovered in di erent contexts at di erent times by Bhattacharyya [41, 42] Je reys [43] Rao [44] R enyi [36] Csisz ar [39] and Wootters [45, 8, 46]. Perhaps its most compelling foundational signi cance is that the quantity D(p 0 =p 1 ) cos 1 n X b=1 q p 0 (b) q p 1 (b) 2.41) corresponds to the geodesic distance between p 0 (b) and p 1 (b) on the probability simplex when its geometry is speci ed by the Riemannian metric ds ....

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W. K. Wootters, \Statistical distance and Hilbert space," Physical Review D, vol. 23, pp. 357{ 362, 1981.

.... application of the so called dualistic geometry, a generalization of the Riemannian geometry [1] It is natural to ask whether some geometrical methods are also useful in quantum estimation theory [2] 3] Indeed, many authors have tried to find geometrical aspects of quantum estimation theory [4][5] 6] 7] 8] We should notice, however, that there exist a variety of manners to define quantum counterparts of geometrical notions which played essential roles in the classical estimation theory, such as logarithmic derivative, Fisher information, Cram er Rao inequality, etc. Moreover, only a ....

W. K. Wootters, "Statistical distance and Hilbert space," Phys. Rev. D23, 357--362 (1981).

....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 ....

W. K. Wootters, "Statistical distance and Hilbert space", Physical Review D, vol. 23, pp. 357 -- 362, 1981.

....h j j k i h j j ih j k i] which is identical to the Fubini Study metric [8] 9] 4 The Fubini Study metric is known as a gauge invariant metric on a projective Hilbert space [10] Theorem 1 gives another meaning of the Fubini Study metric, i.e. the statistical distance. Wootters [11] also investigated from a statistical viewpoint the distance between tworays, and obtained d( cos 01 jh j ij. This is identical, up to a constant factor, to the geodesic distance as measured by the Fubini Study metric [12] Theorem 1, together with the following Theorem 2, reveals a ....

W. K. Wootters, "Statistical Distance and Hilbert space," Phys. Rev. D23, 357--362 (1981).

.... Such constructions have abstract mathematical properties beyond statistics (Lauritzen, 1987) The 1=2 geometry is important in quantum mechanics: With a complex scalar field, the space L 2 is exactly the Hilbert space used in quantum mechanics (the space of waves or probability amplitudes) (Wootters, 1981; Brody and Hughston, 1996) The 1 and 0geometries are important in the theory of communication (Shannon, 1948; Kullback and Leibler, 1951; R enyi, 1961; Csisz ar, 1967a; Cencov, 1982) See also (Acz el and Dar oczy, 1975) and references therein. The Pythagorean theorems rely substantially on ....

Wootters, W. K. (1981). Statistical distance and Hilbert space. Phys. Rev., D, 23(2):357--362.

....unravelings of a master equation. Both have the desirable property of being invariant under the transformation (9. 23) The optimal SSE has jumps which are maximally infrequent and large, in the sense that they take a state to an orthogonal state (which is maximally distant in Hilbert space [154]) The Gisin and Percival SSE has infinitely frequent, infinitesimally small jumps, which are represented by diffusion. Thus, if one wishes to investigate the stochastic quantum dynamics of an open system with no particular measurement scheme in mind, then these two limits suggest themselves as ....

W.K. Wootters, "Statistical distance and Hilbert space" Phys. Rev. D 23, 357 (1981).

....coefficient between two density matrices ae 0 and ae 1 is defined by B(ae 0 ; ae 1 ) def = min E2M B(p 0 (E) p 1 (E) 3.13) 64 where the POVM E ranges over the set of all possible measurements M. Surprisingly, it turns out that B is equivalent to another measure, studied by Wootters [WO81] and Jozsa [JO94] among others. Suppose j 0 i and j 1 i are pure states. When we think of these two state vectors geometrically, a natural notion of distinguishability is the angle between j 0 i and j 1 i, or any simple function of this angle. In particular, using the formula jh 0 j 1 ij = ....

WOOTTERS, W., "Statistical distance and Hilbert space", Physical Review D 23 (1981), pp. 357--362.

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W. K. Wootters. Statistical distance and Hilbert space. Physical Review, D23:357-362, 1981. 6

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