Quantum and non-signalling graph isomorphisms

Atserias, Albert; Mančinska, Laura; Roberson, David E.; Šámal, Robert; Severini, Simone; Varvitsiotis, Antonios

doi:10.48550/arxiv.1611.09837

Cited by 7 publications

(31 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [2], a nonlocal game is introduced which captures the notion of graph isomorphism. In turn, by allowing entangled strategies, this allows one to define a type of quantum isomorphism in a natural way.…”

Section: The Isomorphism Gamementioning

confidence: 99%

“…In [2], along with others, the second and third authors introduced a family of nonlocal games called isomorphism games, and investigated the classical and quantum strategies that win the game perfectly (with probability 1). They showed that the game can be won perfectly by a classical strategy if and only if the corresponding graphs are isomorphic.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Nonlocal games and quantum permutation groups

Lupini

Mančinska

Roberson

2020

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: The Isomorphism Gamementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Nonlocal games and quantum permutation groups

Lupini

Mančinska

Roberson

2020

Journal of Functional Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

“…, 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.…”

Section: Correlations As Perfect Strategiesmentioning

confidence: 99%

“…, 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, . .…”

Section: Correlations As Perfect Strategiesmentioning

confidence: 99%

Perfect Strategies for Non-Local Games

Lupini

Mančinska

Paulsen

et al. 2020

Math Phys Anal Geom

View full text Add to dashboard Cite

We unify and consolidate various results about non-signalling games, a subclass of non-local two-player one-round games, by introducing and studying several new families of games and establishing general theorems about them, which extend a number of known facts in a variety of special cases. Among these families are reflexive games, which are characterised as the hardest non-signalling games that can be won using a given set of strategies. We introduce imitation games, in which the players display linked behaviour, and which contains as subclasses the classes of variable assignment games, binary constraint system games, synchronous games, many games based on graphs, and unique games. We associate a C*-algebra C * (G) to any imitation game G, and show that the existence of perfect quantum commuting (resp. quantum, local) strategies of G can be characterised in terms of properties of this C*-algebra, extending known results about synchronous games. We single out a subclass of imitation games, which we call mirror games, and provide a characterisation of their quantum commuting strategies that has an algebraic flavour, showing in addition that their approximately quantum perfect strategies arise from amenable traces on the encoding C*-algebra. We describe the main classes of non-signalling correlations in terms of states on operator system tensor products.

show abstract

“…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].…”

Section: Skew Projective Permutation Matricesmentioning

confidence: 99%

Orthogonality for Quantum Latin Isometry Squares

Musto¹,

Vicary²

2019

Electron. Proc. Theor. Comput. Sci.

View full text Add to dashboard Cite

Goyeneche et al recently proposed a notion of orthogonality for quantum Latin squares, and showed that orthogonal quantum Latin squares yield quantum codes. We give a simplified characterization of orthogonality for quantum Latin squares, which we show is equivalent to the existing notion. We use this simplified characterization to give an upper bound for the number of mutually orthogonal quantum Latin squares of a given size, and to give the first examples of orthogonal quantum Latin squares that do not arise from ordinary Latin squares. We then discuss quantum Latin isometry squares, generalizations of quantum Latin squares recently introduced by Benoist and Nechita, and define a new orthogonality property for these objects, showing that it also allows the construction of quantum codes. We give a new characterization of unitary error bases using these structures.

show abstract

Quantum and non-signalling graph isomorphisms

Cited by 7 publications

References 20 publications

Nonlocal games and quantum permutation groups

Nonlocal games and quantum permutation groups

Perfect Strategies for Non-Local Games

Orthogonality for Quantum Latin Isometry Squares

Contact Info

Product

Resources

About