On the ‘ Polarized distances between quantum states and observables ’ D .

The scheme for construction of distances, presented in our previous paper quant-ph/0005087, v.1 (Ref. 1) is amended. The formulation of Proposition 1 of Ref. 1 does not ensure the triangle inequality, therefore some of the functionals D(a, b) in Ref. 1 are in fact quasi-distances. In this note we formulate sufficient conditions for a functional D(a, b) of the (squared) form D(a, b) 2 = f (a) 2 + f (b) 2 − 2f (a)f (b)g(a, b) to be a distance and provide some examples of such distances. A one parameter generalization of a bounded distance of the (squared) form D(a, b) 2 = D 2 0 (1 − g(a, b)), which includes the known Bures-Uhlmann and Hilbert-Schmidt distances between quantum states, is established. In the scheme of paper quant-ph/0005087 [1] (to be cited also as Ref. 1) two functionals, f (a) and g(a, b) on a set A (a, b ∈ A) are involved. f (a) was required to be positive, g(a, b) – symmetric, g(a, b) = g(b, a), with values in the interval [−1, 1]. In addition it was supposed that a = b is equivalent to g(a, b) = 1 and f (a) = f (b) (i.e. g(a, b) = 1, f (a) = f (b) ←→ a = b). Then the expression D[a, b], D[a, b] = f (a) 2 + f (b) 2 − 2f (a)f (b) g(a, b) 1/2 (1) was proposed [1] as distance between elements of A. The functional g(a, b) is a cosine-type functional, and f (a) plays the role of polarization. If f (a) = const. then the distance (1) is not polarized. In somewhat different form the notion of polarized distance was introduced in [2]. Before proceed further let us recall the defining properties (d1)-(d4) of the distance D(a, b) between elements a, b, c of a given set A: D(a, b) ≥ 0 (nonnegativity) , (d1) D(a, b) = 0 iff a = b (Euclidean property), (d2) D(a, b) = D(b, a) (symmetricity) , (d3) D(a, b) + D(b, c) ≥ D(a, c) (triangle inequality). (d4) Trivial distance D(a, a) = 0, D(a, b = a) = 1 always exists. If all but (d2) are valid then D(a, b) is called pseudo-distance. If all but the triangle inequality (d4) are valid then D(a, b) would be called quasi-distance.