Quantum renewal processes

Abstract

We introduce a general construction of master equations with memory kernel whose solutions are given by completely positive trace preserving maps. These dynamics going beyond the Lindblad paradigm are obtained with reference to classical renewal processes, so that they are termed quantum renewal processes. They can be described by means of semigroup dynamics interrupted by jumps, separated by independently distributed time intervals, following suitable waiting time distributions. In this framework one can further introduce modified processes, in which the first few events follow different distributions. A crucial role, marking an important difference with respect to the classical case, is played by operator ordering. Indeed, for the same choice of basic quantum transformations different quantum dynamics arise. In particular for the case of modified processes it is natural to consider the time inverted operator ordering, in which the last few events are distributed differently.

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

“…It is important to notice that this novel approach leads to no improvements in the detection of quantum correlations. In fact, on the one hand, both witnesses (17) and (18) are able to detect initial correlations if and only if ρ S (t) = ρ S (t) for some t > 0. Thus, the use of the trace norm of the Helstrom matrix does not improve the capability of detecting quantum correlations.…”

“…However, the condition of initially factorized states between the system and the environment is rather limiting and a lot of efforts have been put forth to introduce reduced dynamical maps in the presence of initial correlations [13][14][15][16][17][18]. Nonetheless, such an extension is non trivial and severe limitations are encountered, so that a general and satisfactory treatment still lags behind.…”

Evolution Equations for Quantum Semi-Markov Dynamics

“…For simplicity we have assumed that , and the waiting time distributions are the same at each step, even though a more general description can be considered [ 13 , 17 , 50 ]. In particular, the intermediate time evolution can be fixed to be of the exponential type, thus focusing on the role of different waiting times and jump operators, and giving rise to a class of non-Markovian dynamics which has been termed quantum renewal processes [ 17 , 51 ]. A particular subset of quantum semi-Markov processes can be obtained by assuming , so that where is the probability density for having n jumps in a time t .…”

“…As said before, quantum semi-Markov processes provide a generalisation of the classical concept [ 48 , 49 ]. Nonetheless, the quantum class is more complex, due to the non-trivial role played by the operator ordering [ 16 , 17 , 51 ]; quantum semi-Markov processes provide a further instance of how the notion of Markovianity cannot be naively transferred from the realm of classical stochastic processes to the one of open quantum system dynamics [ 18 ].…”

“…More precisely the approximation describes a CP divisible dynamics, since on general grounds . This is true independently of the waiting time distribution , since the integral over the memory kernel can be identified with the renewal density, also called sprinkling distribution and obeying the renewal equation [ 51 , 57 , 58 ] …”