Quantum and non-signalling graph isomorphisms

Abstract

We introduce a new two-player nonlocal game, called the (G, H)-isomorphism game, where classical players can win with certainty if and only if the graphs G and H are isomorphic. We then define the notions of quantum and non-signalling isomorphism, by considering perfect quantum and non-signalling strategies for the (G, H)-isomorphism game, respectively. First, we prove that non-signalling isomorphism coincides with the well-studied notion of fractional isomorphism, thus giving the latter an operational interpretation. In the quantum case, we consider both the tensor product and commuting frameworks for nonlocal games. Second, we show that, in the tensor product framework, quantum isomorphism is equivalent to the feasibility of two polynomial systems in non-commuting variables, obtained by relaxing the standard integer programming formulations for graph isomorphism to Hermitian variables. Finally, we provide a construction for reducing linear binary constraint system games to isomorphism games. This allows us to determine quantum isomorphic graphs that are not isomorphic. Furthermore, it allows us to show that our two notions of quantum isomorphism, from the tensor product and commuting frameworks, are in fact distinct relations, and that the latter is undecidable. Our proof techniques are related to the Feige, Goldwasser, Lovász, Safra, and Szegedy reduction from the inapproximability literature [

Perfect Strategies for Non-Local Games

“…, k}. In [1], it was shown that there exists a perfect ns-strategy for the (G, H)-isomorphism game if and only if the graphs have common coarsest equitable partitions.…”

“…, Q k respectively. Though it is not explicitly stated in [1], it follows from the proof of Lemma 4.2 therein and the proof of [34, Theorem 2.2], that in any perfect ns-strategy p for the (G, H)isomorphism game, the marginal probability p(h|g) = p(h, h|g, g) vanishes unless g ∈ P i and h ∈ Q i for some i ∈ {1, . .…”

Orthogonality for Quantum Latin Isometry Squares

“…PPMs, also known as magic unitaries and quantum bijections between classical sets have recently appeared in the context of quantum non-local games [1,2,13,14] and the study of compact quantum groups [3,5,17].…”

The Morita Theory of Quantum Graph Isomorphisms

“…For a quantum graph Γ, we classify quantum isomorphic quantum graphs Γ ′ in terms of simple 3 dagger Frobenius monoids in the representation categories QAut(Γ); these are dagger Frobenius monoids X (see Definition 2.2) in QAut(Γ) whose underlying algebra F X is simple, where F : QAut(Γ) − → Hilb is the forgetful functor. In terms of the Hopf C * -algebra A(Γ) such a structure can equivalently be defined as a matrix algebra Mat n (C) with normalised trace inner product A, B = 1 n Tr(A † B), equipped with a *representation ⊲: A(Γ) − → End(Mat n (C)) such that the following holds for all x ∈ A(Γ) and A, B ∈ Mat n (C):…”

“…Such tasks are usually formulated as games, where isolated players Alice and Bob are provided with inputs, and must return outputs satisfying some winning condition. One such game is the graph isomorphism game [3], whose instances correspond to pairs of graphs Γ and Γ ′ , and whose winning classical strategies are exactly graph isomorphisms Γ − → Γ ′ . Winning quantum strategies are called quantum isomorphisms.…”