Amaury Freslon

Title : Quantum groups : what they can do for you, what you can do for them

Abstract : I will review two recent important connections between compact quantum groups and quantum information theory. The first one is the construction of quantum channels with good entanglement properties from representations of quantum orthogonal groups. The second is a correspondance between quantum isomorphisms of graphs and monoidal equivalence of quantum automorphism groups. I will introduce along the way the essential features of quantum groups used without entering abstract theory. As a conclusion I will try to highlight some of the broad questions opened by these works.

Andre Kornell

Title: Interpreting propositions in discrete quantum structures

Abstract: What is a quantum graph coloring? The existence of a quantum graph coloring is definitionally equivalent to the existence of a quantum winning strategy for the graph coloring game. Quantum graph colorings have been identified with certain families of projective measurements, and with certain morphisms in a 2-category of finite quantum sets. We will instead interpret the existence of a quantum graph coloring in the internal logic of the category of quantum sets and quantum relations. This internal logic can be viewed as the quantum predicate logic canonically extending the propositional quantum logic of Birkhoff and von Neumann.

Laura Mancinska

Title: Nonlocal Games and Quantum Permutation Groups

Abstract: I will present a connection between quantum information and quantum permutation groups provided by quantum strategies for the so-called graph isomorphism game. I will start by  defining the isomorphism game and its quantum strategies as prescribed by two potentially different models: the tensor product model and the commuting model. We will see that the existence of a commuting strategy can naturally be expressed in the language of quantum groups. Specifically, we will see that two connected graphs X and Y are quantum (commuting) isomorphic if and only if there exists x \in V(X) and y \in 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 behaviors. Finally, we will see that this connection not only allows us to leverage quantum group theoretic results to learn more about quantum strategies but the methods used to investigate quantum strategies for the isomorphism game can also be applied to get new results about quantum automorphism groups of graphs.

This talk is based on arXiv:1712.01820 which is a joint work with Martino Lupini and David E. Roberson.

David Reutter

Title: The Morita theory of quantum graph isomorphisms

Abstract: In this talk, I will describe a classification of (commutative and non-commutative) graphs quantum isomorphic to a given graph G in terms of Morita equivalence classes of certain matrix algebras in the representation category of the quantum automorphism algebra of G. If G has no quantum symmetry, this classification can be expressed in purely group theoretical terms. As an example, I will use this classification to show that the Petersen graph is quantum isomorphic to precisely two non-commutative quantum graphs. Moreover, I will show how all known pairs of quantum-but-not-classically isomorphic graphs arise via this group-theoretical construction from abelian groups.

This talk is based on arXiv: 1801.09705 which is joint work with Ben Musto and Dominic Verdon.

David Roberson

Title: Quantum isomorphism and counting homomorphisms

Abstract: In 1967, Lovasz showed that two graphs are isomorphic if and only if they admit the same number of homomorphisms from any graph. More recently in 2018, Dell, Grohe, and Rattan showed that two graphs are not distinguished by the k-dimensional Weisfeiler-Leman algorithm if and only if they admit the same number of homomorphisms from any graph of treewidth at most k. Just this year we have shown that two graphs are quantum isomorphic (in the commuting operator framework) if and only if they admit the same number of homomorphisms from any planar graph. To prove this we first provide a combinatorial characterization of the intertwiners of the quantum automorphism group of a graph based on counting rooted homomorphisms from planar graphs. I will discuss this combinatorial description of intertwiners and the resulting characterization of quantum isomorphism.

This is joint work with Laura Mancinska.

Simon Schmidt

Title: On the quantum symmetry of distance-transitive graphs

Abstract: An important task in the theory of quantum automorphism groups of finite graphs is to see whether or not a graph has quantum symmetry, i.e. whether or not its quantum automorphism group is commutative.

Focussing on distance-transitive graphs, we will discuss tools for proving that the generators of the quantum automorphism group commute.

Then, using those tools, we will show that certain families of distance-transitive graphs have no quantum symmetry.

Piotr Soltan

Title: Quantum families of maps – examples and additional structure

Abstract: I will define quantum spaces of maps and quantum families of maps and show a number of interesting examples. I will concentrate on emergence of additional structure on some quantum spaces of maps. Such objects arise in various contexts, including synchronous games, and sometimes exhibit rather surprising properties.

Dan Stahlke

Title: Homomorphisms on noncommutative graphs

Abstract: Shannon 1956 analyzed the zero-error communication capacity of a noisy channel by associating a graph to the channel.  Lovasz 1979 derived a bound on the asymptotic capacity via semidefinite programming.  Duan, Severini, Winter 2013 generalized this to the quantum setting, in the process generalizing the notion of graphs to so-called non-commutative graphs, a concept interesting in its own right.  I extend the notion of graph homomorphism (a map between vertex sets of two graphs that preserves adjacency) to non-commutative graphs, by studying a quantum version of zero-error source-channel coding.  This extends a number of graph invariants to non-commutative graphs, namely the chromatic number, entanglement-assisted clique and chromatic numbers, and orthogonal rank.

Dominic Verdon

Title: Quantum symmetry transformations of systems and channels

Abstract:I will introduce various forthcoming results. Firstly, finite-dimensional (f.d.) quantum bijections between finite quantum sets have a physical interpretation as entanglement-invertible channels between f.d. C*-algebras; these generalise tight teleportation and dense coding schemes, which are quantum bijections between a commutative C*-algebra and a matrix algebra.

Each entanglement-invertible channel / quantum bijection P: A -> B, has an associated Hopf image H(P), the smallest Hopf quotient of the quantum permutation algebra of A containing the quantum permutation P* \circ P. We show that every f.d. quantum bijection P: A \to A’  is really part of a fibre functor on CP(Rep(H(P))), the category of f.d. H(P)-C*-algebras and covariant CPTP maps; for f.d. H(P), all fibre functors on CP(Rep(H(P)) are realised by entanglement-invertible transformations. Physically, this means that entanglement-invertible channels P:A->B between f.d. C*-algebras in fact realise a broader global symmetry of H(P)-covariant physics. For example, the classic Pauli teleportation scheme in fact does a larger global symmetry on CP(Rep(Z_2)) swapping the role of the commutative and noncommutative indecomposable 4D Z_2-C*-algebras; the symmetry is activated whenever Alice performs her measurement.

Finally, we can apply this framework to define quantum symmetries of a quantum channel, generalising the quantum symmetry group of a quantum graph. Quantum symmetries of a channel preserve all entanglement-assisted source channel coding schemes and capacities; for example, Pauli teleportation makes use of a  quantum symmetry relating the identity channels on the four dimensional commutative and matrix C*-algebras. This extends to the underlying bipartite quantum graph of the channel encoding its zero-error theory; for example, the quantum isomorphism of bipartite classical graphs arising from the Mermin-Peres square implies equality of zero-error entanglement-assisted capacities for two classical channels with 12 inputs and 12 outputs. At least in the special case of quantum symmetries of classical confusability graphs arising from central type subgroups of the ordinary automorphism group of the graph, these results also extend to confusability graphs, in that quantum isomorphism of confusability graphs really does imply equality of entanglement-assisted zero capacities of all channels with that confusability graph. As a simple example, we show how this may be used to calculate the entanglement-assisted zero-error capacities of the dephasing channel.

This is joint work with David Reutter.

Ivan Todorov

Title: The Lovasz Sandwich Theorem for non-commutative graphs

Abstract:  We define quantum versions of the vertex packing polytope, the fractional vertex packing polytope and the Lovasz theta-body of a graph, and discuss a chain of inclusions between them. This leads to new quantisations of the Lovasz theta number, a new bound on the Shannon capacity of a non-commutative graph, and non-commutative versions of other classical graph parameters. The talk will be based on a joint work with Gareth Boreland and Andreas Winter. 

Andreas Winter