Metrics on completely positive maps via noncommutative geometry

TL;DR

This paper develops a noncommutative geometric framework to define metrics on unital completely positive maps, extending classical concepts with new algebraic and geometric tools.

Contribution

It introduces an infinite-dimensional $C^*$-algebraic analogue of the Choi-Jamiołkowski isomorphism and constructs metrics satisfying key quantum information properties.

Findings

Metrics satisfy stability and chaining properties

Uses Kasparov products of spectral triples to generate metrics

Provides a noncommutative geometric approach to quantum metrics

Abstract

We study methods of inducing metrics on unital completely positive maps by employing seminorms arising in noncommutative geometry. Our main approach relies on the development of an infinite-dimensional $C^*$-algebraic analogue of the Choi-Jamio\l{}kowski isomorphism. Under suitable conditions, we show that the induced metrics satisfy the quantum information theoretic properties of stability and chaining. Moreover, we show how to generate such metrics using constructions native to noncommutative geometry, by for example using external Kasparov products of spectral triples.