On Bures distance over standard form vN-algebras

TL;DR

This paper investigates the properties of Bures distance in standard form von Neumann algebras, exploring when the distance can be realized by implementing vectors and providing counterexamples for nonfinite cases.

Contribution

It analyzes the extent to which Bures distance can be realized by vectors when one vector is fixed, and offers a foundational overview of related C*-algebraic tools.

Findings

Counterexamples for nonfinite algebras where Bures distance is not attained

Conditions under which Bures distance can be realized by vectors

Development of foundational tools for Bures geometry in von Neumann algebras

Abstract

In case of a standard form vN-algebra, the Bures distance is the natural distance between the fibres of implementing vectors at normal positive linear forms. Thereby, it is well-known that to each two normal positive linear forms implementing vectors exist such that the Bures distance is attained by the metric distance of the implementing vectors in question. We discuss to which extend this can remain true if a vector in one of the fibres is considered as fixed. For each nonfinite algebra, classes of counterexamples are given and situations are analyzed where the latter type of result must fail. In the course of the paper, an account of those facts and notions is given, which can be taken as a useful minimum of basic C^*-algebraic tools needed in order to efficiently develop the fundamentals of Bures geometry over standard form vN-algebras.