Random density matrices versus random evolution of open system

Abstract

We present and compare two families of ensembles of random density matrices. The first, static ensemble, is obtained foliating an unbiased ensemble of density matrices. As criterion we use fixed purity as the simplest example of a useful convex function. The second, dynamic ensemble, is inspired in random matrix models for decoherence where one evolves a separable pure state with a random Hamiltonian until a given value of purity in the central system is achieved. Several families of Hamiltonians, adequate for different physical situations, are studied. We focus on a two qubit central system, and obtain exact expressions for the static case. The ensemble displays a peak around Werner-like states, modulated by nodes on the degeneracies of the density matrices. For moderate and strong interactions good agreement between the static and the dynamic ensembles is found. Even in a model where one qubit does not interact with the environment excellent agreement is found, but only if there is maximal entanglement with the interacting one. The discussion is started recalling similar considerations for scattering theory. At the end, we comment on the reach of the results for other convex functions of the density matrix, and exemplify the situation with the von Neumann entropy.