Inverting non-bijective maps through recovering channels

Abstract

In this dissertation we are interested in the dynamics of open quantum systems in the context of quantum information. First, we investigate the dynamics of one qubit at the regime where an inverse evolution is not well defined. This limit is widely used and explored in this work, called as the limit of non-bijectivity. We demonstrate how to compute a completely positive inverse evolution, based on the theory of recovery maps, here to be the Petz recovery map. The analyzed evolutions are the typical decoherence processes: dephasing, depolarizing and amplitude damping channels in the regime of non-bijectivity. To measure how efficient the Petz can be recovering a random quantum state, we use the fidelity function as the figure of merit. Also, we quantify how non-invertible a dynamics is compared to another. As an application for this formalism, in the second part of this work we show how we can explore recovery maps in the context of non-Markovian evolutions, in order to minimize memory effects. We demonstrate that recovering maps can be useful under certain constraints to simulate a non-Markovian evolution by an “almost” Markovian one. We demonstrate that there exists a strong dependence between the success of this strategy and the initial correlations between system and environment.