Unitary invariance of Connes spectral distances of quantum states

TL;DR

This paper investigates how Connes spectral distances between quantum states behave under unitary transformations, providing explicit examples where these distances match quantum trace distances.

Contribution

It introduces new properties of Connes spectral distances, constructs spectral triples with specific distance characteristics, and explores their implications in noncommutative geometry.

Findings

Spectral triples with Lipschitz seminorms equal to operator norms identified.

Explicit spectral triples constructed where spectral and trace distances coincide.

Results enhance understanding of geometric structures in finite spectral triples.

Abstract

In this paper, we study the properties of Connes spectral distances between quantum states under unitary transformations. We mainly focus on spectral triples with matrix algebras acting on finite dimensional Hilbert spaces via some linear representations. We derive some elementary properties of the Connes spectral distances and optimal elements. We prove that there are some finite spectral triples in which the Lipschitz seminorms are equal to the operator norms. We also explicitly construct some spectral triples in which the Connes spectral distances between quantum states are exactly the quantum trace distances. These results and concrete examples are significant for studies of geometric structures of finite spectral triples and mathematical relations of qubits and other quantum states in the framework of noncommutative geometry.