Non-local games and quantum symmetries of quantum metric spaces

TL;DR

This paper extends the concept of quantum isometry groups to quantum metric spaces, introduces quantum isometries, and connects these ideas to a new non-local game that characterizes metric space isometries.

Contribution

It generalizes Banica's quantum isometry construction to quantum metric spaces and introduces the metric isometry game linking quantum strategies to isometries.

Findings

Quantum isometry groups of quantum metric spaces are monoidally equivalent if the spaces are algebraically quantum isometric.

The metric isometry game characterizes classical and quantum isometries of metric spaces.

Winning quantum strategies correspond to quantum isometries of the spaces.

Abstract

We generalize Banica's construction of the quantum isometry group of a metric space to the class of quantum metric spaces in the sense of Kuperberg and Weaver. We also introduce quantum isometries between two quantum metric spaces, and we show that if a pair of quantum metric spaces are algebraically quantum isometric, then their quantum isometry groups are monoidally equivalent. Motivated by the recent work on the graph isomorphism game, we introduce a new two-player non-local game called the metric isometry game, where players can win classically if and only if the metric spaces are isometric. Winning quantum strategies of this game align with quantum isometries of the metric spaces.