On the 'Polarized distances between quantum states and observables'

Abstract

The scheme for construction of distances, presented in the 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_0^2 (1 - g(a,b)), which includes the known Bures-Uhlmann and Hilbert-Schmidt distances between quantum states, is established.