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Mardi 13 mars 2012-14:00

Matrix Product States, Random Matrix Theory and the Principle of Maximum Entropy

Carlos González Guillén (Madrid)

par Bertrand Georgeot - 13 mars 2012

Jaynes’ principle of maximum entropy states that among all possible probability distributions compatible with our prior information, the best choice is the one which maximizes the entropy. To which extent the principle of maximum entropy can be extended to more and more general situations has been a very active field in the last half century. Very recently a series of theoretical and experimental works seem to validate the principle in relaxation processes of quantum systems when focusing on a particular small subsystem, which is the most relevant situation for experimental reasons. Prior information is usually the knowledge of the particular interactions in the model. But what if one wants to consider the problem with less standard prior information ? For instance, that the interactions in our model are local and homogeneous and that we work at zero temperature. In this talk, I will investigate this problem in 1D spin systems. It will be shown that reduced density matrices of small subsystems of translational invariant random Matrix Product States have generically maximum entropy. The proof of this statement will rely on recent developments of random matrix theory, in particular on the graphical Weingarten calculus, and on a novel estimate of the Weingarten function.