Operator systems, non-signalling correlations and quantum graph parameters

Ivan Todorov (Queen's University Belfast)

Tuesday 9th February, 2016 16:00-17:00 Maths 522


The chromatic number of a graph is a well-known and widely used parameter, with far reaching applications both within and outside Graph Theory. It is defined as the smallest number of colours that are needed in order to colour the vertices of the graph in such a way that adjacent vertices receive different colours. A quantum version of the chromatic number was defined in 2007 by P. Cameron, A. Montanaro, M. Newman, S. Severini and A. Winter, utilising an entangled state, shared between two players, and it was demonstrated that this new chromatic number can be strictly smaller than the classical one. Since then, it was realised that classes of non-signalling quantum correlations can be used to introduce and study quantum versions of a variety of parameters and notions from Graph Theory. In this talk, based on a joint work with S. Severini, D. Stahlke, V. Paulsen and A. Winter, I will describe how ideas from operator algebra theory, including operator system tensor products and C*-algebras with traces, can be utilised to give descriptions of non-signalling correlations and  to define non-commutative versions of chromatic numbers and orthogonality ranks of graph. I will highlight the connection between Tsirelson's Problem and Connes' Embedding Problem and their link with the introduced chromatic invariants.

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