Quantum Symmetries Studied Through the Lens of Non-Local Games
Abstract
Non-local games provide a useful framework for exhibiting the power of quantum entanglement, and we will focus our study on the graph isomorphism game and the metric isometry game. Work in quantum information theory has led to quantum versions of many concepts in classical mathematics, including quantum graphs and quantum metric spaces. 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 prove that the non-commutative algebraic notion of a quantum isomorphism between two finite, classical objects (either graphs or metric spaces) is the same as the more physically motivated one arising from the existence of a perfect quantum strategy for the corresponding game. This is achieved by showing that every algebraic quantum isomorphism between a pair of (quantum) objects X and Y arises from a certain measured bigalois extension for the quantum symmetry groups GX and GY of X and Y . In particular, this implies that the quantum groups GX and GY are monoidally equivalent.
For the case of the graph isomorphism game, we also establish a converse to this result, which says that a compact quantum group G is monoidally equivalent to the quantum automorphism group GX of a given quantum graph X if and only if G is the quantum automorphism group of a quantum graph that is algebraically quantum isomorphic to X.
Citation
Eifler, Kari (2021). Quantum Symmetries Studied Through the Lens of Non-Local Games. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /195815.