Program

Schedule

Talks

Kenny De Commer

Title: Quantization of semisimple Lie groups and their representation categories

Abstract: Let L be a connected linear real semisimple Lie group. Let L^ be its tensor W*-category of unitary representations. We propose a definition of a tensor C*-category L_q^ which should play the rôle of a quantization of L^. We then show that L_q^ is indeed the representation category of a C*-algebra which, at least conceptually, quantizes the universal group C*-algebra of L. Our construction builds on Letzter's theory of coideal subalgebras of quantized enveloping algebras, coming from the classical theory of symmetric spaces (of compact type). Partly based on joint work with J.R. Dzokou Talla.

Paweł Kasprzak

Title: : Quantum games on quantum spaces: strategies and their values

Abstract: Recently the framework of quantum games has been formulated in the case where:

• Questions and answers are given by matricial quantum states
• non-signaling strategies are given by certain class of matricial quantum channels
• the rule of the game is given by a join preserving map between projection lattices of matrix algebras.

We propose an alternative framework for quantum games where

• the spaces of questions and answers are arbitrary finite quantum spaces
• the rule of the game is given by a single self adjoint projection

and we give the formula for the value of a strategy in such a quantum game.

In my talk I will describe and compare these two approaches and use the second one that we are developing with I. Chełstowski and D. Jasiński to define the value of the strategy in  the first approach (which was proposed by  I. G. Todorov and L. Turowska).

Jacek Krajczok

Title: On the approximation property of locally compact quantum groups

Abstract: One of the most widely studied properties of groups is the notion of amenability - in one of its many formulations, it gives us a way of approximation the constant function by functions in the Fourier algebra.

The notion of amenability was relaxed in various directions: a very weak form of amenability, called the approximation property (AP), was introduced by Haagerup and Kraus in 1994. It still gives us a way of approximating the constant function by functions in the Fourier algebra, but in much weaker sense. During the talk I'll introduce AP for locally compact quantum groups and discuss some of its permanence properties.

The talk is based on a joint work with Matthew Daws and Christian Voigt.

Simon Schmidt

Title: Quantum isomorphic strongly regular graphs from the E₈ root system

Abstract: In this talk, we give a first example of a pair of quantum isomorphic, non-isomorphic strongly regular graphs. The pair consists of the orthogonality graph of the 120 lines spanned by the E₈ root system and a rank 4 graph whose complement was first discovered by Brouwer, Ivanov and Klin. Both graphs are strongly regular with parameters (120, 63, 30, 36). We will see that one can obtain more quantum isomorphic, non-isomorphic strongly regular graphs using Godsil-McKay switching.

Adam Skalski

Title: Gaussian states and Gaussian parts of compact quantum groups

Abstract: I will  explain the notion of a Gaussian state on a compact quantum group 𝔾̂, as introduced by Michael Schürmann in the framework of his theory of quantum Lévy processes. This concept leads to the idea of the Gaussian part of 𝔾̂, understood as the smallest quantum subgroup of 𝔾̂ which supports all the Gaussian states of 𝔾̂. I will discuss properties of Gaussian states and compute Gaussian parts for several examples. This turns out to be related to quantum connectedness and certain topological generation questions for quantum subgroups. The talk will be based on joint work with Uwe Franz and Amaury Freslon, and also with Guillaume Cebron.

Piotr Soltan

Title: Why B(ℓ²) is not L^∞(𝔾)

Abstract: I will discuss the reason the algebra B(ℓ²) does not arise as the von Neumann algebra associated with a compact quantum group. Later I will discuss other examples of injective factors which do arise in this way.

Lyudmila Turowska

Title: No-signaling quantum correlations and quantum graph homomorphisms

Abstract: We will discuss quantum no-signaling correlations introduced by Duan and Winter with focus on its different subclasses (quantum commuting, quantum and local). They will appear as strategies of quantum-to-quantum non-local games. We will then discuss concurrent quantum non-local games,
as quantum versions of synchronous/bisynchronous non-local games focusing on quantum graph homomorphism/isomorphism games, and provide tracial characterisations of their perfect strategies belonging to  various correlation classes; e.g. the perfect strategies for quantum graph isomorphism game are  given by tracial states on the universal C*-algebra of the projective free unitary quantum group.
This is a joint work with Michael Brannan, Sam Harris and Ivan Todorov.

Dominic Verdon

Title: Compact quantum group symmetry in quantum information theory

Abstract: Historically, quantum information theory has focused on the study of finite-dimensional C*-algebras and completely positive trace-preserving maps. I will discuss some ways in which compact quantum group symmetry arises in this setting. Examples include quantum teleportation and dense coding; entanglement-assisted strategies for nonlocal games; and classification of mutually unbiased bases. The mathematical notion that unites all these examples is that of a pseudonatural transformation between fibre functors on a rigid C*-tensor category. These pseudonatural transformations, which generalise monoidal natural transformations, can be conceived as finite-dimensional elements of noncommutative torsors for a compact quantum group.

Simeng Wang

Title: Rigidity of quantum group actions on classical compact spaces

Abstract: I will present the recent result that the free quantum permutation group S⁺_N cannot act ergodically on any nontrivial compact connected space, which was an open problem dating back to the related work of Goswami around 2010s. The proof is based on a combinatorial approach to the Tannaka-Krein duality for quantum group actions, and the method also applies to many other easy quantum groups. I will also discuss the non-ergodic and non-connected cases if time permits. This is joint work with Amaury Freslon and Frank Taipe.

Sang-Gyun Youn

Title: Additivity violation of the regularized minimum output entropy in the commuting-operator framework

Abstract: The additivity question of the minimum output entropy (MOE) was a long-standing open question in quantum information theory (QIT), and it was disproved by Hastings in 2009. The result suggests that regularization process of MOE is unavoidable in QIT and would be highly complicated to compute. Indeed, very few results are known for the regularized MOE, and the associated additivity question is still open. We introduce a class of quantum channels whose regularized MOE is computable thanks to a generalized Haagerup inequality. Moreover, we exhibit an additivity violation of the regularized MOE in the commuting-operator setup though it is unclear in the tensor-product setup. This talk is based on a recent joint work with Benoît Collins.

Benjamin Anderson-Sackaney

Title: Tracial and G-invariant States on Quantum Groups

Abstract: For a discrete group G, the tracial states on its reduced group C*-algebra C*_r(G) are exactly the conjugation invariant states. This makes the traces on C*_r(G) amenable to group dynamical techniques. In the setting of a discrete quantum group 𝔾, there is a quantum analogue of the conjugation action of G on C*_r(G). Recent work of Kalantar, Kasprzak, Skalski, and Vergnioux shows that 𝔾-invariant states on the quantum group reduced C*-algebra C*_r(𝔾̂) are in one-to-one correspondence with certain KMS-states, exhibiting a disparity between tracial states and 𝔾-invariant states unless 𝔾 is unimodular. We will show there is still enough of a connection between traciality and 𝔾-invariance to say interesting things about the tracial states of C*_r(𝔾̂).

Alfons Van Daele

Title: From invariant integrals to Haar weights

Abstract: For the operator algebraic approach to quantum groups, the existence of the analogue of the Haar measure on a locally compact group, the Haar weight, is crucial.
A first observation is that, up till now, there is no general theory of locally compact quantum groups where the existence of the Haar weight is a theorem, rather than part of the axioms.

Fortunately, this is not a big problem, because, among other reasons, the Haar weights can be shown to exist when needed. Still, from a theoretical point of view, it would nice to have such results.

Existence results are available for finite quantum groups, compact quantum groups, discrete quantum groups and for the dual of a locally compact quantum group. Moreover Haar weights can be constructed following general ideas for most of the also more complicated examples.

In this talk, I will discuss some aspects of this. The results are available in the literature, but not so easy accessible, and often with more complicated arguments then necessary.

Slides

Nicolas Fabre

Title: Wigner distribution on a double-cylinder phase space for studying quantum error-correction protocols

Abstract: 𝔼(2) symmetry group naturally deals with quantum system possessing a discrete integer variable canonically conjugate to its angular position. Single photons state with an orbital angular momentum (OAM) degree of freedom is an example of such quantum system. The natural manifold for representing such a state is a discrete cylinder. We present in this talk another type of system which is also described by such a 𝔼(2) symmetry group. We show that by defining two lattices along the position and momentum variables with periodicity l and 2π/l, we can define modular variables (MVs). They are bounded variables equivalent to two angles variables, and thus
canonically conjugated to two discrete variables. We are then lead to introduce a quasiprobability phase space distribution with two pairs of azimuthal-angular coordinates, which then takes values into a double cylinder. This representation is well adapted to describe quantum systems with discrete symmetry, in a sense that it there is an avoidance of redundancy of information compared to a plane phase space. Quantum error correction of states encoded in continuous variables using translationally invariant states, called the Gottesman-Kitaev-Preskill state is studied as an example of application. We also propose an experimental scheme for measuring such distribution.

[1] N. Fabre, A. Keller, and P. Milman, Wigner Distribution on a Double-Cylinder Phase Space for Studying Quantum Error-Correction Protocols, Phys. Rev. A 102, 022411 (2020).

Lucas Hataishi

Title: On factorization homology,  finite gauge theory and C*-categories.

Abstract: Factorization homology can be thought of as a machinery producing homology theories for manifolds, assigning to them categorical invariants and giving rise to topological field theories. We explain how it can be used to recover the finite gauge theory of Daniel Freed and Constantin Teleman as a particular instance of a correspondence which associates to an algebraic field theory a topological one. To obtain such a correspondence in full generality, we prove that C*-categories are amenable to factorization homology.

Ting Lu

Title: Computing the Haar state of the quantum group SL_q(3)

Abstract: I will discuss the general strategy of solving the Haar state on SL_q(n), n>2, by constructing linear systems consisting of special monomials which serves as linear basis of SL_q(n) and then focus on an alternative approach to compute the Haar state on SL_q(3) without solving large scaled matrices.

Lukas Rollier

Title: Quantum symmetries of connected locally finite graphs

Abstract: Using combinatorial machinery developed by Laura Mančinska and David Roberson, one may construct, for any connected locally finite graph Π a unitary 2-category which encodes the quantum symmetries of this graph. Using a Tannaka-Krein type reconstruction, this category may then be used to construct the locally compact quantum automorphism group of Π as an algebraic quantum group. This talk will give a brief overview of this construction. This has been joint work with Stefaan Vaes.

Arthur Troupel

Title: Free wreath products as fundamental graph C*-algebras

Abstract: The free wreath product of a compact quantum group by the quantum permutation group S_N⁺ has been introduced by Bichon in order to give a quantum counterpart of the classical wreath product. The representation theory of such groups is well-known, but some results about their operator algebras were still open, for example Haagerup property, K-amenability or factoriality of the von Neumann algebra. I will present a joint work with Pierre Fima in which we identify these algebras with the fundamental C*-algebras of certain graphs of C*-algebras, and we deduce these properties from these constructions.

Ami Viselter

Mateusz Wasilewski

Title: Quantum Cayley graphs

Abstract: I will propose a definition of a quantum Cayley graph of a discrete quantum group. I will first focus on the unimodular case and discuss how to extend the notion of quantum graphs to the (highly restricted) infinite dimensional setting. It turns out that quantum Cayley graphs to a large extent do not depend on the "generating set", just like for classical groups. In the last part of the talk I will address the case of non-unimodular discrete quantum groups and present some examples. Work in progress.