Nonlocal games and quantum permutation groups
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Nonlocal games and quantum permutation groups. / Lupini, Martino; Mančinska, Laura; Roberson, David E.
I: Journal of Functional Analysis, Bind 279, Nr. 5, 108592, 2020.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Nonlocal games and quantum permutation groups
AU - Lupini, Martino
AU - Mančinska, Laura
AU - Roberson, David E.
PY - 2020
Y1 - 2020
N2 - We present a strong connection between quantum information and the theory of quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs X and Y are quantum isomorphic if and only if there exists x∈V(X) and y∈V(Y) that are in the same orbit of the quantum automorphism group of the disjoint union of X and Y. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. In this work, we exploit this by using ideas and results from quantum information in order to prove new results about quantum automorphism groups of graphs, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum permutation group literature. Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other. We hope that this work will encourage new collaborations among the communities of quantum information, quantum groups, and algebraic graph theory.
AB - We present a strong connection between quantum information and the theory of quantum permutation groups. Specifically, we define a notion of quantum isomorphisms of graphs based on quantum automorphisms from the theory of quantum groups, and then show that this is equivalent to the previously defined notion of quantum isomorphism corresponding to perfect quantum strategies to the isomorphism game. Moreover, we show that two connected graphs X and Y are quantum isomorphic if and only if there exists x∈V(X) and y∈V(Y) that are in the same orbit of the quantum automorphism group of the disjoint union of X and Y. This connection links quantum groups to the more concrete notion of nonlocal games and physically observable quantum behaviours. In this work, we exploit this by using ideas and results from quantum information in order to prove new results about quantum automorphism groups of graphs, and about quantum permutation groups more generally. In particular, we show that asymptotically almost surely all graphs have trivial quantum automorphism group. Furthermore, we use examples of quantum isomorphic graphs from previous work to construct an infinite family of graphs which are quantum vertex transitive but fail to be vertex transitive, answering a question from the quantum permutation group literature. Our main tool for proving these results is the introduction of orbits and orbitals (orbits on ordered pairs) of quantum permutation groups. We show that the orbitals of a quantum permutation group form a coherent configuration/algebra, a notion from the field of algebraic graph theory. We then prove that the elements of this quantum orbital algebra are exactly the matrices that commute with the magic unitary defining the quantum group. We furthermore show that quantum isomorphic graphs admit an isomorphism of their quantum orbital algebras which maps the adjacency matrix of one graph to that of the other. We hope that this work will encourage new collaborations among the communities of quantum information, quantum groups, and algebraic graph theory.
KW - Quantum automorphism group
KW - Quantum graph isomorphism
KW - Quantum group
KW - Quantum information theory
UR - http://www.scopus.com/inward/record.url?scp=85083782818&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2020.108592
DO - 10.1016/j.jfa.2020.108592
M3 - Journal article
AN - SCOPUS:85083782818
VL - 279
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
SN - 0022-1236
IS - 5
M1 - 108592
ER -
ID: 240640278