Nonlocal games and quantum permutation groups

Abstract

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.

On the quantum symmetry of distance-transitive graphs

“…Therefore the Shrikhande graph is a nice example of a graph whose quantum orbital algebra is different from the coherent algebra of the graph. See [12] for more on quantum orbital algebras of graphs.…”

“…Quantum automorphism groups are in close relation to quantum isomorphisms appearing in quantum information theory. This was discovered for example in [12] and [13]. By results in [13,Corollary 3.7,Corollary 4.15], one can classify the quantum graphs and the classical graphs that are quantum isomorphic to a given graph, if this graph has no quantum symmetry.…”

Models of quantum permutations

“…Our results may be interpreted in the sense that Woronowicz's operation ⊥ applied iteratively to the representation R as above is powerful enough to finally reproduce all of C(S + N ), at least in the cases N = 4 and N = 5. Hence, R ⊥ n yields good models of quantum permutation matrices for practical purposes such as in [LMR18], see also Section 3.4. In Section 5, we comment on how to generalize the presented ideas and results in the situation of easy quantum groups, of which the (quantum) permutation groups are special cases.…”

Bigalois Extensions and the Graph Isomorphism Game

“…Universiteit Leuven, mateusz.wasilewski@kuleuven.be 2. Conversely, for any compact quantum group G monoidally equivalent to G X , one can construct from this monoidal equivalence a quantum graph Y , an isomorphism of quantum groups G ∼ = G Y , and an algebraic quantum isomorphism X ∼ = A * Y .Recasting all of the above in the context of the (classical) graph isomorphism game, our results show that the condition A(Iso(X, Y )) = 0 is sufficient to ensure the existence of perfect quantum strategies for this game (Corollary 4.8 and Theorem 4.9):Theorem Two classical graphs X and Y are algebraically quantum isomorphic if and only if the graph isomorphism game has a perfect quantum-commuting (qc)-strategy.We mention that a weaker version of the above theorem (that assumed the existence of a non-zero C * -algebra representation of A(Iso(X, Y ))) was recently proved in [18].…”

“…We mention that a weaker version of the above theorem (that assumed the existence of a non-zero C * -algebra representation of A(Iso(X, Y ))) was recently proved in [18].…”