A comparison of two quantum distances

TL;DR

This paper compares two different notions of distance in quantum metric spaces, revealing that Rieffel's quantum Gromov-Hausdorff distance differs fundamentally from the classical Gromov-Hausdorff distance when applied to state spaces.

Contribution

It demonstrates the non-equivalence of Rieffel's quantum distance and the classical distance on associated state spaces, clarifying their distinct properties.

Findings

Rieffel's quantum Gromov-Hausdorff distance is not equivalent to the classical Gromov-Hausdorff distance.

The difference impacts how quantum metric spaces are compared and understood.

The result highlights fundamental distinctions between quantum and classical metric notions.

Abstract

We show that Rieffel's quantum Gromov-Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov-Hausdorff distance applied to the associated state spaces.