We introduce a notion of quantum function, and develop a compositional framework for finite quantum set theory based on a 2-category of quantum sets and quantum functions. We use this framework to formulate a 2-categorical theory of quantum graphs, which captures the quantum graphs and quantum graph homomorphisms recently discovered in the study of nonlocal games and zero-error communication, and relates them to quantum automorphism groups of graphs considered in the setting of compact quantum groups. We show that the 2-categories of quantum sets and quantum graphs are semisimple. We analyse dualisable and invertible 1-morphisms in these 2-categories and show that they correspond precisely to the existing notions of quantum isomorphism and classical isomorphism between sets and graphs.
We present an infinite number of construction schemes involving unitary error bases, Hadamard matrices, quantum Latin squares and controlled families, many of which have not previously been described. Our results rely on biunitary connections, algebraic objects which play a central role in the theory of planar algebras. They have an attractive graphical calculus which allows simple correctness proofs for the constructions we present. We apply these techniques to construct a unitary error basis that cannot be built using any previously known method.
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 vertextransitive 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.
A combinatorial theory of associative n-categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a notion of 'homotopy' with a natural geometrical interpretation. Such a theory has the potential to serve as an attractive foundation for a computer proof assistant for higher category theory, since it allows composites to be uniquely described, and relieves proofs from the bureaucracy of associators, unitors and their coherence. However, this basic theory lacks a high-level way to construct homotopies, which would be intractable to build directly in complex situations; it is not therefore immediately amenable to implementation.We tackle this problem by describing a 'contraction' operation, which algorithmically constructs complex homotopies that reduce the lengths of composite terms. This contraction procedure allows building of nontrivial proofs by repeatedly contracting subterms, and also allows the contraction of those proofs themselves, yielding in some cases single-step witnesses for complex homotopies. We prove correctness of this procedure by showing that it lifts connected colimits from a base category to a category of zigzags, a procedure which is then iterated to yield a contraction mechanism in any dimension. We also present homotopy.io, an online proof assistant that implements the theory of associative n-categories, and use it to construct a range of examples that illustrate this new contraction mechanism. 12 1 arXiv:1902.03831v1 [math.CT]
We prove that every slightly degenerate braided fusion category admits a minimal nondegenerate extension. As a corollary, every pseudounitary super modular tensor category admits a minimal modular extension. This completes the program of characterizing minimal nondegenerate extensions of braided fusion categories.Our proof relies on the new subject of fusion 2-categories. We study in detail the Drinfel'd centre Z(Mod-B) of the fusion 2-category Mod-B of module categories of a braided fusion 1-category B. We show that minimal nondegenerate extensions of B correspond to certain trivializations of Z(Mod-B). In the slightly degenerate case, such trivializations are obstructed by a class in H 5 (K(Z 2 , 2); k × ) and we use a numerical invariant -defined by evaluating a certain two-dimensional topological field theory on a Klein bottle -to prove that this obstruction always vanishes.Along the way, we develop techniques to explicitly compute in braided fusion 2categories which we expect will be of independent interest. In addition to the known model of Z(Mod-B) in terms of braided B-module categories, we introduce a new computationally useful model in terms of certain algebra objects in B. We construct an S-matrix pairing for any braided fusion 2-category, and show that it is nondegenerate for Z(Mod-B). As a corollary, we identify components of Z(Mod-B) with blocks in the annular category of B and with the homomorphisms from the Grothendieck ring of the Müger centre of B to the ground field.
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