Quantum symmetries of quantum metric spaces and non-local games. (English) Zbl 07741136
Summary: We generalize T. Banica’s construction of the quantum isometry group of a metric space [Pac. J. Math. 219, No. 1, 27–51 (2005; Zbl 1104.46039)] to the class of quantum metric spaces in the sense of G. Kuperberg and N. Weaver [Mem. Am. Math. Soc. 1010, 1–80 (2012; Zbl 1244.46035)]. We also introduce quantum isometries between two quantum metric spaces, and we show that if a pair of quantum metric spaces is 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 nonlocal 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.
MSC:
| 81P05 | General and philosophical questions in quantum theory |
| 46B04 | Isometric theory of Banach spaces |
| 31E05 | Potential theory on fractals and metric spaces |
| 68T20 | Problem solving in the context of artificial intelligence (heuristics, search strategies, etc.) |
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