Zero-error information theory arose in Shannon's seminal work in the 1950's as a meeting point of Information Theory, Graph Theory and Analysis. Shannon noticed that the zero-error transmission properties of an information channel can be captured by the so-called confusability graph of the channel. In particular, the zero-error capacity of the channel can be defined as an asymptotic parameter involving the independence numbers of the powers of the corresponding graph. Notoriously difficult to compute, the zero-error capacity was unknown even for graphs on few vertices, for example for the 5-cycle, for many years, until Lovasz proposed a highly original approach involving what is known today as the Lovasz number of a graph. This parameter is defined via a semi-definite program and is computable in polynomial time.

Similar development took place only recently for quantum channels. The confusability graphs are in this case replaced by operator systems -- selfadjoint subspaces of the space of all n by n matrices, containing the identity operator. In fact, operator systems acting on finite dimensional spaces turn out to be a genuine quantisation of classical graphs; this viewpoint has reinforced the links between analysis, (quantum) information theory and combinatorics, and has led to the birth of the new area of non-commutative graph theory.

Analysis plays a fundamental role in different - albeit related - aspects of quantum information, providing a bridge between discrete mathematics and quantum physics. The area in question is the theory of non-local games. It is known that the employment of quantum, as opposed to classical, strategies, can lead to genuinely better performance of the players of such a game. The celebrated Bell Theorem models the impact of quantum entanglement through the possible separations between the classes of classical and quantum correlations. Polynomial optimization (in non-commutative variables) offers powerful tools to approximate these separations and, here too, the Lovasz theta number provides efficiently computable bounds for some non-local games such as quantum graph coloring. A number of deep mathematical approaches are needed to make further advances in non-local game theory, including optimisation, operator space theory, representation theory, group theory and others.

The current workshop will be a meeting point of these developments, emphasising the interaction between optimisation, operator algebras, combinatorics and quantum information. It also aims to highlight the infinite dimensional - von Neumann algebraic - approach to Quantum Computing, and the use of the commuting model, which has enjoyed a renewed interest in recent years. We hope that it will provide an opportunity for the different communities to work together on the many open questions and intriguing directions these interactions have to offer.

Below, we have compiled a brief bibliography related to the topics of the workshop.

J.S. Bell, On the Einstein Podolsky Rosen paradox. Physics 1(3), 195–200 (1964)

J. Briet, H. Buhrman, M. Laurent, T. Piovesan and G. Scarpa., Entanglement-assisted zero-error source-channel coding. IEEE Transactions on Information Theory, 61(2):1-15, 2015.

P.J. Cameron, A. Montanaro, M.W. Newman, S. Severini, A. Winter, On the quantum chromatic number of a graph. Electron. J. Comb.14(1), 81 (2007)

A. Doherty, Y.C. Liang, B. Toner, S. Wehner, The quantum moment problem and bounds on entangled multiprover games. In Proceedings of the 2008 IEEE 23rd Annual Conference on Computational Complexity pp. 199–210 (2008)

R. Duan, S. Severini and A. Winter, Zero-error communication via quantum channels, non-commutative graphs and a quantum Lovasz theta function, IEEE Trans. Inf. Theory 59 (2013), no. 2, 1164-1174.

R. Duan and A. Winter, No-signalling-assisted zero-error capacity of quantum channels and an information theoretic interpretation of the Lovasz number, IEEE Trans. Inf. Theory 62 (2016), 891-914.

K. Dykema, V.I. Paulsen and J. Prakash, Non-closure of the set of quantum correlations via graphs, preprint (2017), arXiv:1709.05032.

S. Gribling, D. de Laat and M. Laurent, Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization, preprint (2018), arXiv:1708.09696.

M. Grotschel, L. Lovasz and A. Schrijver, Geometric algorithms and combinatorial optimization, Springer-Verlag, Berlin, 1993.

D.E. Knuth, The Sandwich Theorem, Electronic Journal of Combinatorics, 1 (1994): http://eudml.org/doc/118559.

M. Laurent and T. Piovesan, Conic approach to quantum graph parameters using linear optimization over the completely positive semidefinite cone, SIAM J. Optim. 25 (2015), 2461-2493.

R. Levene, V. I. Pauslen and I. G. Todorov, Complexity and capacity bounds for quantum channels, IEEE Trans. Inform. Theory 64 (2018), no. 10, 6917-6928.

L. Lovasz, On the Shannon capacity of a graph, IEEE Trans. Inf. Theory 25 (1979), no. 1, 1-7.

M. Navascués, S. Pironio, A. Acín, A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations. New J. Phys.10(7), 073,013 (2008)

V. I. Pauslen, Entanglement and non-locality, Lecture Notes, University of Waterloo, 2016.

W. Slofstra, Tsirelson's problem and an embedding theorem for groups arising from non-local games, preprint (2016), arXiv:1606.03140.

J. Watrous, The theory of quantum information, Cambridge University Press, 2018.