Generalized trace distance approach to quantum non-Markovianity and detection of initial correlations

Abstract

A measure of quantum non-Markovianity for an open system dynamics, based on revivals of the distinguishability between system states, has been introduced in the literature using the trace distance as quantifier for distinguishability. Recently it has been suggested to use as measure for the distinguishability of quantum states the trace norm of Helstrom matrices, given by weighted differences of statistical operators. Here we show that this new approach, which generalizes the original one, is consistent with the interpretation of information flow between the system and its environment associated to the original definition. To this aim we prove a bound on the growth of the external information, that is information which cannot be accessed by performing measurements on the system only, as quantified by means of the Helstrom matrix. We further demonstrate by means of example that it is of relevance in generalizing schemes for the local detection of initial correlations based on the increase of internal information. Finally we exploit this viewpoint to show the optimality of a previously introduced strategy for the local detection of quantum correlations.

Precursors of non-Markovianity

“…for t>s, and they are interpreted as information backflow. This interpretation is substantiated by the following inequality valid for arbitrary  t s, thus extending previous seminal work on the study of initial correlations [24,25]. It appears that to have a local revival of the trace distance at a time t greater than s, that is a positive contribution on the lhs, at least one of the contributions on the rhs, referring to time s, has to be positive.…”

Markovianity of an emitter coupled to a structured spin-chain bath

“…In this appendix we provide an explicit scheme for evaluating the environment correlation functions of our model (defined in Eqs. (25) and (26)). We express them as a sum over Gaussian operator terms that can be computed efficiently.…”

Revealing Correlations Between a System and an Inaccessible Environment

“…On the one hand, the example studied in this reference indicates that the detection power of the trace distance is significantly larger than that of other well-known distance measures for quantum states, namely the Bures metric, the Hellinger distance and the Jensen-Shannon divergence. On the other hand, in those cases the trace norm of the Helstrom matrix can show an even better performance than the trace distance [27].…”

“…The performance of the local detection scheme based on the trace norm ||∆|| of the Helstrom matrix has been studied in Ref. [27]. Quite interestingly, it turns out that for the method based on the dephasing map the trace distance, corresponding to the unbiased case p = p = 1/2, is optimal in the sense that it shows the largest increase due to the presence of correlations.…”