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A conic approach to entangled-assisted graph parameters

Presented by: 
T Piovesan Centrum voor Wiskunde en Informatica (CWI)
Friday 19th July 2013 - 11:30 to 12:00
INI Seminar Room 1
Graph parameters as the independence and the chromatic number are related to classical (zero-error) communication problems. It is known that allowing the presence of entanglement, one of quantum mechanics most peculiar feature, might increase the efficiency of the (zero-error) communication. However there are still many open problem, for example the maximal possible separation between classical and quantum communication, computational complexity and approximation of the quantum variant of the graph parameters etc. We propose a new framework for studying the quantum parameters, introducing a cone that lies between the completely positive and the double non-negative one. We say that a matrix X is in this cone if there exists a set of positive semidefinite matrices {A_i} such that the i,j-th entry of X is equal to the inner product between A_i and A_j. Testing membership of the dual cone is equivalent to determine whether a particular polynomial is trace positive over all the real symmetric matrices of any dimension. This problem is therefore related to a special case of the Connes’ embedding conjecture. This conic approach allow us to prove better bounds for the quantum variant of the graph parameters, to have a more unified framework and hopefully to build approximation hierarchies.
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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons