Random Matrices: Theory and Applications

Dear Colleagues,

A few years after the E. Wigner’s proposal of the ensemble of random Gaussian matrices, R. Balian showed that it can be obtained by maximizing the Shannon entropy of the joint distribution of their entries. Balian also discussed the introduction of constraints in order to derive other kind of ensembles. Decades later, this scheme was used to generate what has been called the generalized ensemble obtained by maximizing so-called non-additive entropy. As it happens with the Gaussian ensemble, the invariance of the joint distribution is preserved under unitary transformations. The further development of this new ensemble led to the introduction of the concept of disorder in random matrices theories.

Turning now to quantum information, random matrices entered the field by providing models of random pure states to study entropies that describe the entanglement of bipartite systems. In this case, the ensembles used are those with fixed trace matrices.

In another more recent development, PT-symmetry systems aroused the interest in non-Hermitian Hamiltonians or, more precisely, in the class of pseudo-Hermitian operators whose eigenvalues, as it occurs in the PT case, are real or complex conjugates. The question, in this case, is the transition that occurs in the quantum entanglement measured by von Neumann entropy when, as a function of its parameters, a Hamiltonian changes from a regime of real to a regime of complex conjugate eigenvalues.

Dr. Mauricio Porto Pato
Dr. Lu Wei
Guest Editors

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