How to calculate decoherence matrices numerically

Abstract

A central object in the interpretation of quantum mechanics of closed systems is the decoherence matrix. But only for a very small number of models one is able to give explicit expressions for its elements. So numerical methods are required. Unfortunately the dimensions of these matrices are usually very high, which makes also a direct numerical calculation impossible. In this paper I will develop an algorithm that combines the numerical calculation of the elements of the decoherence matrix with a permanent estimation. If the chosen histories are almost decoherent, the algorithm will provide the diagonal and maybe some near to diagonal elements only. In this case, within numerical errors, we can assume, that all not calculated elements are smaller than a given threshold. If the histories are coherent, the algorithm has to calculate too much data (there exist too much non-vanishing elements) and the success depends mainly on the available computer time. As an example I apply this procedure to the Caldeira—Leggett-model.