We give a simple proof of Connes implies Tsirelson, and further advertise a hot topic in quantum information: optimal states and robust self-testing. We showcase here how operator algebraic techniques can be quite fruitful.

For this we begin with by recalling quantum strategies in the context of non-local games, and their description in terms of the state space on the full group algebra of certain free groups. With this description at hand, we then directly obtain the main result via an elementary lifting result by Kim, Paulsen and Schafhauser:

The Connes embedding problem implies the synchronous Tsirelson conjecture.

As such the entire proof is elementary, and bypasses all versions of Kirchberg's QWEP conjecture and thelike, as well as any reformulation such as in terms of the micro state conjecture. Moreover, it should be (likely) easier to construct minimal nonlocal games as counterexamples for the synchronous Tsirelson conjecture (which is equivalent to the full Tsirelson conjecture but in a non-trivial way) and so also nonamenable traces for above groups, in other words non-Connes embeddable operator algebras.

After this we continue (as much as time permits) with an advertisement for a topic in quantum information (a hot one):

Device-independent certification of quantum states – more precisely ROBUST SELF-TESTING – 

which has tremendous importance for the coming era of practical quantum computingand we showcase how operator algebraic techniques can be quite fruitful here.More precisely, we illustrate these techniques on the following two prominent classes of nonlocal games:

1) The tilted CHSH game.

We showcase here how to compute the quantum value using operator algebraic techniques, and how to use the same to derive uniqueness for entire optimal states, including all higher moments as opposed to correlations defined on two-moments only, where the latter compares to traditional self-testing.Moreover, we report in this example on previously unknown phase transitions on the uniqueness of optimal states when varying the parameters for the tilted CHSH game.

2) The Mermin–Peres magic square and magic pentagram game. As before, we also note here uniqueness of optimal states, which in these two examples is a basically familiar result.

The first part is based on preprint: https://arxiv.org/abs/2209.07940 

The second part on self-testing (and further robust self-testing) is based on https://arxiv.org/abs/2210.03716 and upcoming joint work with Azin Shahiri.

In the 1950s, Higman introduced the first class of examples of infinite finitely presented groups without any non-trivial finite quotient. We study Higman groups from the viewpoint of measure equivalence -  a notion introduced by Gromov as a measurable counterpart to quasi-isometry. For most Higman groups and some generalizations, we prove a strong measure equivalence rigidity theorem. In this talk, I'll sketch the proof, discuss some of the consequences, and compare to some other measure equivalence rigidity/flexibility results in the literature. This is joint work with Camille Horbez.

We will describe a research project in collaboration with Luca Lionni and Razvan Gurau where we study the joint behavior of iid random elements in tensors of matrix algebras whose law is invariant under local conjugations. We will also mention some results on the norm of tensors of random matrices and explain some applications to QIT.

In this talk I will discuss an approach to study quantum symmetries of infinite sets and graphs. This leads to discrete quantum groups, in analogy to the quantum symmetry groups of Wang and Banica/Bichon. For finite sets and graphs, the resulting quantum groups can in fact be viewed as discretisations of the former (compact) quantum groups. I will discuss a number of examples, and highlight some intriguing open problems as well.

Random matrices are widely used to model quantum systems with chaos and disorder. In such models, the observable is expressed as a quantum operator averaged over an ensemble of random matrices with a given probability measure. In my talk, I demonstrate a general approach, “the random matrix integrable theory”, to the nonperturbative calculation of the random-matrix integrals. With this approach, the internal symmetries of the integration measure, expressed in terms of highly non-trivial nonlinear relations for the original integral (the Toda lattice hierarchy, the Kadomtsev-Petviashvili hierarchy) and the relations following from the deformation of the integration measure (Virasoro constraints), are used to represent the integral as a solution of differential equations, where the differentials are taken over the internal (physical) parameters of the model [1,2]. This method represents a particular implementation of results obtained within a more general theory of $\tau$-functions. In particular, the central theorem of this theory states the existence of the Toda lattice and Kadomtsev-Petviashvili hierarchies for the typical random-matrix integrals.

The particular implementation of the integrable theory will be discussed in the example of the physical problem of quantum transport in chaotic cavities [3,4]. A brief introduction to the physics of the problem and the advantage of the integrable theory method for calculation of the conductance cumulants, and of the shot-noise-conductance joint cumulants are going to be presented. In particular, we demonstrate how the conductance cumulant generation function can be expressed in terms of the solution of the Painleve V transcendent equation. In addition, the results of the integrable theory implementation to the averaged random-matrix characteristic polynomials [1], and also for the problem of the power spectrum of the eigenlevel sequences in the quantum chaotic system [2,5] will be discussed. 

[1] V.Al.Osipov, E.Kanzieper, “Correlations of RMT characteristic polynomials and integrability: Random Hermitian matrices”, Annals of Physics325 (2010) 2251 

[2] R.Riser, V.Al.Osipov, E.Kanzieper, “Nonperturbative theory of power spectrum in complex systems”, Annals of Physics413 (2020) 168065 

[3] V.Al.Osipov, E.Kanzieper, “Integrable theory of quantum transport in chaotic cavities”, Phys.Rev.Let.101 (2008) 176804 

[4] V.Al.Osipov, E.Kanzieper, “Statistics of thermal to shot noise crossover in chaotic cavities”, J.Phys.A:Math.Theor.42 (2009) 475101 

[5] R.Riser, V.Al.Osipov, E.Kanzieper, “Power-spectrum of long eigenlevel sequences in quantum chaology”, Phys.Rev.Let.118 (2017) 204101

Uniform Roe algebras play an important role in noncommutative geometry. It was shown recently by M. Lorentz and R. Willett that all bounded derivations of the uniform Roe algebras of metric spaces of bounded geometry are inner. Here we calculate the space of outer derivations of the uniform Roe algebras with coefficients in uniform Roe bimodules related to various metrics on the two copies of the given space. We also give some results on the higher Hochschild cohomology with coefficients in uniform Roe bimodules.

We explain recent results concerning (quantum) differentials and integrals on the noncommutative (Moyal) plane. We give full characterisation of elements on the noncommutative (Moyal) plane such that their quantum derivatives belong to the weak Schatten class , which means that these derivatives are d-times integrable in the quantum integration sense. We then calculate the quantum integration of these derivatives by adapting Connes' integration formulae to the noncommutative (Moyal) plane. This is a joint work with E. McDonald and X. Xiong.