Abstract We present an infinite sequence of finite graphs with trivial automorphism group and non‐trivial quantum automorphism group. These are the first known examples of graphs with this property. Moreover, to the best of our knowledge, these are the first examples of any asymmetric classical space that has non‐trivial quantum symmetries. Our construction is based on solution groups to (binary) linear systems, as defined by Cleve, Liu, and Slofstra in the context of non‐local games. We first show that the dual quantum group of every solution group occurs as the quantum automorphism group of some graph, and then construct an infinite sequence of systems whose solution groups are non‐trivial perfect groups. This leads to the desired sequence of graphs. In addition to our main result, we prove a number of related results that allow us to answer several open problems from the literature. We prove a weak quantum analog of Frucht's theorem, namely that every finite classical group occurs as the quantum automorphism group of a finite graph. Combined with our main result, this shows that, for every finite group , there are graphs and that both have classical automorphism group isomorphic to but one of them has quantum symmetry and the other does not. Therefore, the quantum automorphism group of a graph is never determined by its classical automorphism group, and there do not exist any “quantum excluding groups.”
We introduce the notion of nonlocal symmetry of a graph
G, defined as a winning quantum correlation for the
G-automorphism game that cannot be produced classically. Recent connections between quantum group theory and quantum information show that quantum correlations for this game correspond to tracial states on
C(Qut(G)) -- the algebra of functions on the quantum automorphism group of
G. This allows us to also define nonlocal symmetry for any quantum permutation group. We investigate the differences and similarities between this and the notion of quantum symmetry, defined as non-commutativity of
C(Qut(G)). Roughly speaking, quantum symmetry vs nonlocal symmetry can be viewed respectively as non-classicality of our model of reality vs non-classicality of our observation of reality. We show that quantum symmetry is necessary but not sufficient for nonlocal symmetry. In particular, we show that the complete graph on five vertices is the only connected graph on five or fewer vertices with nonlocal symmetry, despite a dozen others having quantum symmetry. In particular this shows that the quantum symmetric group on four points,
S4+, does not exhibit nonlocal symmetry, answering a question from the literature. In contrast to quantum symmetry, we show that two disjoint classical automorphisms do not guarantee nonlocal symmetry. However, three disjoint automorphisms do suffice. We also give a construction of quantum permutation matrices built from a finite abelian group
Γ and a permutation
π on
∣Γ∣ elements. Computational evidence suggests that for cyclic groups of increasing size almost all permutations
π result in nonlocal symmetry. Surprisingly, the construction never results in nonlocal symmetry when
Z23 is used. We also investigate under what conditions nonlocal symmetry arises when taking unions or products of graphs.
Compact graphs are graphs for which the fractional automorphism polytope has no genuinely fractional vertices. This paper proposes a quantum analogue of this idea by evaluating the fundamental magic unitary of the quantum automorphism group on states, which we show to produce a closed convex set of doubly stochastic matrices sitting between the classical automorphism polytope and the full fractional automorphism polytope. Our main result is that the natural quantum analogue of compactness is classical, that is, a quantum compact graph is classically compact.
We also relate this set to the quantum orbital algebra and obtain a hierarchy of classical and quantum compactness pseudo notions. The framework recovers familiar consequences of compactness through commutants and suggests quantum analogues of generous transitivity and distance-transitivity. We also isolate examples and open problems indicating where quantum symmetries may strictly refine the classical compactness theory.
In 2019, Aterias et al. constructed pairs of quantum isomorphic, non-isomorphic graphs from linear constraint systems. This article deals with quantum automorphisms and quantum isomorphisms of colored versions of those graphs. We show that the quantum automorphism group of such a colored graph is the dual of the homogeneous solution group of the underlying linear constraint system. Given a vertex- and edge-colored graph with certain properties, we construct an uncolored graph that has the same quantum automorphism group as the colored graph we started with. Using those results, we obtain the first known example of a graph that has quantum symmetry and finite quantum automorphism group. Furthermore, we construct a pair of quantum isomorphic, non-isomorphic graphs that both have no quantum symmetry.
UDC 519.17, 517.98, 519.8 Let
G be a simple finite graph and let
UG be the corresponding quantum graph. We study the game algebra
C(Qut(UG)) of the quantum automorphisms of
UG. It is shown that, for the complete graph
Kn, the algebra
C(Qut(UKn)) is not commutative even for all
n≥3, unlike
C(Qut(Kn))=C(Sn+). Moreover, we prove that, for any graph
G with
∣V(G)∣≥3, the quantum graph
UG admits nonlocal symmetry, which means that there exists a perfect quantum no-signaling correlation for the quantum automorphism game for
UG, which is not local.
We classify instances of quantum pseudo-telepathy in the graph isomorphism game, exploiting the recently discovered connection between quantum information and the theory of quantum automorphism groups. Specifically, we show that graphs quantum isomorphic to a given graph are in bijective correspondence with Morita equivalence classes of certain Frobenius algebras in the category of finite-dimensional representations of the quantum automorphism algebra of that graph. We show that such a Frobenius algebra may be constructed from a central type subgroup of the classical automorphism group, whose action on the graph has coisotropic vertex stabilisers. In particular, if the original graph has no quantum symmetries, quantum isomorphic graphs are classified by such subgroups. We show that all quantum isomorphic graph pairs corresponding to a well-known family of binary constraint systems arise from this group-theoretical construction. We use our classification to show that, of the small order vertex-transitive graphs with no quantum symmetry, none is quantum isomorphic to a non-isomorphic graph. We show that this is in fact asymptotically almost surely true of all graphs.