# Séminaire du LPTMS : Viktor Eisler & Zdzislaw Burda

## Entanglement in mixed states: the negativity

### Dr. Viktor Eisler (TU Graz)

In pure states of many-body systems, entanglement is routinely studied via the Renyi entropies, which give a complete characterization of the bipartite case. The situation becomes more complicated, if the system is composed of more than two parts, and one is interested in the entanglement between two non-complementary pieces. Such a scenario can be studied by introducing a new entanglement measure, the so-called negativity, which has been the focus of recent interest. In this talk I would like to give an overview about the available methods to calculate the negativity, and present some new results on mixed-state entanglement.

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## Quaternionic R transform and non-hermitian random matrices

### Dr. Zdzislaw Burda (AGH University of Science and Technology)

Using the Cayley-Dickson construction we rephrase and review the non-hermitian diagrammatic formalism [R. A. Janik, M. A. Nowak, G. Papp and I. Zahed, Nucl.Phys. B 501, 603 (1997)], that generalizes the free probability calculus to asymptotically large non-hermitian random matrices. The main object in this generalization is a quaternionic extension of the R transform which is a generating function for planar (non-crossing) cumulants. We demonstrate that the quaternionic R transform generates all connected averages of all distinct powers of X and its hermitian conjugate X:  <<(1/N) Tr [Xa Xb Xc …] >>  for N → ∞. We show that the R transform for Gaussian elliptic laws is given by a simple linear quaternionic map R(z+wj) = x + σ² (μ e^{2iφ} z + w j) where (z,w) is the Cayley-Dickson pair of complex numbers forming a quaternion q=(z,w)≡ z+ wj. This map has five real parameters Re[x], Im[x], φ, σ and μ. We use the R transform to calculate the limiting eigenvalue densities of several products of Gaussian random matrices.

Date/Time : 16/10/2015 - 10:30 - 12:00