Schedule for: 25w5405 - Operator Systems and their Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Meet and Greet at the BIRS Lounge (PDC Second Floor)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Recently, C. Duquet and C. Le Merdy gave a characterisation of the absolutely dilatable Schur multipliers over a measure space extending the Haagerup-Musat characterisation from the special case of the matrix algebras. In this talk, which is based on a joint work with I. G. Todorov and L. Turowska, we generalise Duquet-Le Merdy's theorem to the noncommutative setting replacing the modularity over the maximal abelian von Neumann algebra with modularity over the commutant $ \mathcal{D'}$ of an abritrary von Neumann algebra $\mathcal{ D}$. More precisely, we show that a unital completely positive $\mathcal{D'}$-bimodule map $\Phi : \mathcal{B}(H)\to \mathcal{B}(H)$ is absolutely dilatable if and only if it admits a representation of the form $$\Phi(z) = (\rm id \hspace{-0.05cm}\mbox{}_{\mathcal{B}(H)}\otimes \tau_{ \mathcal{N}})(D^*(z\otimes 1_{\mathcal{ N}})D),$$where $D\in \mathcal{D}\bar\otimes\mathcal{B}(K)$ is a unitary satisfying natural requirements and $\mathcal{N}\subseteq \mathcal{B}(K)$ is a von Neumann algebra equipped with a finite trace $\tau$. Moreover, we exhibit a hierarchy for absolutely dilatable $\mathcal{D'}$-bimodule maps based on the type of the ancilla $ \mathcal{N}$. We define the classes of local, quantum and quantum approximate types of maps, as the maps admitting an ancilla $ \mathcal{N}$ that is abelian, finite dimensional or embeddable in an ultrapower of the hyperfinite II$_1$-factor. We further reformulate Connes' embedding problem as the question of whether all absolutely dilatable maps are quantum approximate.

In their 2012 work, Navescues, Cooney, Perez-Garcia, and Villanueva [NCPGV'12] gave an alternative characterization of the quantum spatial correlations C_qs. In particular, they showed that C_qs was equivalent to the set of correlations produced by certain collections of subnormalized quantum states and a single collection of measurement operators on a Hilbert space. They termed the correlations from this model quantum-causal or "quansal" for short because of their nonsignalling property. We discuss how their characterization can be viewed as an algebraic nonsignalling condition for the algebra of measurement operators with respect to B(H). We then show that a similar characterization exists for the set of (quantum) commuting operator correlations where the algebraic nonsignalling condition for the measurements operators is made with respect to arbitrary C*-algebras. In both cases, we can interpret the correlations as strategies for quantum sequential games, a "nonlocal game" similar to a prepare-and-measure scenario. I will discuss how these correlations emerge in the asymptotic limit of computational nonsignalling strategies for compiled nonlocal games, and discuss how this characterization enabled us to bound the quantum value of all compiled nonlocal games. This is based on joint work with Alexander Kulpe, Giulio Malavolta, Simon Schmidt, and Michael Walter (https://arxiv.org/pdf/2408.06711).

We provide a characterisation of equivariant Fock covariant injective representations for product systems. We square this characterisation with established results concerning Fock covariance, on compactly aligned product systems over right LCM semigroups and on product systems with one-dimensional fibers. Using our characterisation we resolve the reduced Hao-Ng isomorphism problem for generalised gauge actions by discrete groups. This is a joint work with Evgenios Kakariadis.

Over the last decade, several non-commutative generalizations of graphs have been studied in the literature. Three equivalent perspectives emerged: a quantization of the adjacency matrix leading to the definition of quantum Schur-idempotents, a quantization of the edge relation giving an operator system viewpoint, and the quantum edge projection, which also gives a translation between perspectives. A natural question to ask is which properties of graphs generalize to the quantum setting. In this talk, we will focus on the property of regularity. After an overview of the different perspectives, we define regularity for quantum graphs and discuss properties of the regularity constant. This is joint work with Matthew Kennedy and Junichiro Matsuda.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

See https://sites.google.com/view/operator-systems-applications/ for more information.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Does there exist a self-dual tensor product of convex cones? We generalise the framework of operator systems to general conic systems by promoting the underlying structure of cones of positive semidefinite matrices to far more general structures. This allows us to prove the existence of a self-dual tensor product for finite dimensional proper cones and to reveal connections between the many examples that arise as siblings of operator systems, such as cone systems, topological field theories, group representations and mapping cones. This talk is based on arXiv:2312.13983.

The study of non-local games, rooted in Bell's celebrated work from the '60's, has over the past few decades shown the benefits and limitations of quantum entanglement. Interest in quantum non-local games- where input and output sets of questions and answers are replaced by input and output states- has been particularly growing in the last few years, due to significant potential applications in the study of operator algebras. Utilizing the simulation paradigm in information theory (among other tools), in this talk I introduce the necessary setup for the comparison and transference of strategies of quantum games, which are sufficiently ``similar". I investigate game strategy transport and the existence of strategies for quantum games using an operator system approach. Time permitting, I also will look at applications of strategy transfer to quantum graph isomorphism games. This is partially based on previous joint work with Ivan G. Todorov (in arXiv:2211.04851 and arXiv:2311.06355), and current solo work (in arXiv:2410.09599).

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk introduces multipartite nonlocality by extending two-player nonlocal games to multiplayer settings defined on graph vertices. We identify conditions under which a two-player game can only demonstrate Bell nonlocality for certain finite sets of graphs. Using these results, we show that the CHSH game displays nonlocality only in two cases: the original two-player scenario and a four-player scenario on a path graph. In contrast, several other well-studied games fail to exhibit nonlocal violations in any larger multipartite structures.

Spectral truncations arise from a finite resolution approach to noncommutative geometry and fit well into the framework of compact quantum metric spaces, which are modeled on operator systems. Compact quantum metric spaces can be compared in terms of complete Gromov--Hausdorff distance and this allows to make the question precise whether spectral truncations converge as more spectral data is taken into account. We give an overview over the positive answers to this question in the case of tori by using a procedure resembling Berezin quantization, and in the case of compact quantum groups by using the nice symmetry properties of their truncations.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Given a classical simple graph $G$, there is a $\{0,1\}$-matrix $A$ which represents which vertices are connected via a directed edge. This matrix gives rise to the Cuntz--Krieger algebra $\mathcal{O}_{A}$ whose algebraic properties reflect combinatorial properties of the graph $G$, and vice versa. This same graph also another structure, called its edge (or graph) correspondence $E_G$, which gives rise to the Cuntz--Pimsner algebra $\mathcal{O}_{E_G}$. When $G$ has no sources (i.e., every vertex in $G$ receives an edge), the Cuntz—Krieger algebra $\mathcal{O}_{A}$ and the Cuntz—Pimsner algebra $\mathcal{O}_{E_G}$ are isomorphic. In joint work with Mitch Hamidi and Brent Nelson, we examine whether or not two analogous $C^*$-algebras associated to a quantum graph $\mathcal{G}$ are isomorphic. Quantum graphs have different (though equivalent) definitions in recent literature. We define them as triples involving a quantum set, which is a C*-algebra equipped with a special type of state, along with a {\em quantum} adjacency matrix, which generalizes the classical $\{0,1\}$-adjacency matrix acting as a linear operator on a commutative C*-algebra. Analogous to the classical setting, we defined in a previous work the graph’s {\em quantum edge correspondence} $E_\mathcal{G}$. In this talk, we will present two results: the Cuntz--Pimsner algebra of the quantum edge correspondence is precisely the {\em local} quantum Cuntz--Krieger algebra for $\mathcal{G}$, i.e., it is a quotient of the (full) quantum Cuntz—Krieger algebra by ``local relations,’’ however, we can construct an example of a quantum graph whose (full) quantum Cuntz--Krieger algebra is not isomorphic to its local counterpart. This example settles an open question raised in the paper which introduced local quantum Cuntz—Krieger algebras.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.