On Bures-Distance and *-Algebraic Transition Probability between Inner Derived Positive Linear Forms over W*-Algebra

TL;DR

This paper derives a variational formula for the Bures-distance between inner derived positive linear forms over a W*-algebra, extending previous results and connecting to quantum field theory concepts.

Contribution

It introduces a new variational expression for the Bures-distance in the context of W*-algebras, extending earlier non-commutative probability results.

Findings

Derived a variational formula for Bures-distance between positive linear forms.

Extended S. Gudder's results on non-commutative probability.

Connected Bures-distance expression to seminorms in algebraic quantum field theory.

Abstract

On a W*-algebra M, for given two positive linear forms f,g and algebra elements a,b a variational expression for the Bures-distance d_B(f^a,g^b) between the inner derived positive linear forms f^a=f(a* . a) and g^b=g(b* . b) is obtained. Along with the proof of the formula also some earlier result of S.Gudder on non-commutative probability will be slightly extended. Also, the given expression of the Bures-distance nicely relates to some system of seminorms proposed by D.Buchholz and which occured along with the problem of estimating the so-called `weak intertwiners' in algebraic quantum field theory. In the last part some optimization problem will be considered.