TY - JOUR
T1 - Bigalois Extensions and the Graph Isomorphism Game
AU - Brannan, Michael
AU - Chirvasitu, Alexandru
AU - Eifler, Kari
AU - Harris, Samuel
AU - Paulsen, Vern
AU - Su, Xiaoyu
AU - Wasilewski, Mateusz
N1 - Publisher Copyright:
© 2019, Springer-Verlag GmbH Germany, part of Springer Nature.
PY - 2020/5/1
Y1 - 2020/5/1
N2 - We study the graph isomorphism game that arises in quantum information theory. We prove that the non-commutative algebraic notion of a quantum isomorphism between two graphs is same as the more physically motivated one arising from the existence of a perfect quantum strategy for graph isomorphism game. This is achieved by showing that every algebraic quantum isomorphism between a pair of (quantum) graphs X and Y arises from a certain measured bigalois extension for the quantum automorphism groups GX and GY of X and Y. In particular, this implies that the quantum groups GX and GY are monoidally equivalent. 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. Using the notion of equivalence for non-local games, we apply our results to other synchronous games, including the synBCS game and certain related graph homomorphism games.
AB - We study the graph isomorphism game that arises in quantum information theory. We prove that the non-commutative algebraic notion of a quantum isomorphism between two graphs is same as the more physically motivated one arising from the existence of a perfect quantum strategy for graph isomorphism game. This is achieved by showing that every algebraic quantum isomorphism between a pair of (quantum) graphs X and Y arises from a certain measured bigalois extension for the quantum automorphism groups GX and GY of X and Y. In particular, this implies that the quantum groups GX and GY are monoidally equivalent. 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. Using the notion of equivalence for non-local games, we apply our results to other synchronous games, including the synBCS game and certain related graph homomorphism games.
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U2 - 10.1007/s00220-019-03563-9
DO - 10.1007/s00220-019-03563-9
M3 - Article
AN - SCOPUS:85073998510
SN - 0010-3616
VL - 375
SP - 1777
EP - 1809
JO - Communications in Mathematical Physics
JF - Communications in Mathematical Physics
IS - 3
ER -