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arXiv:2006.02793 · Journal of Physics A Mathematical and Theoretical

Quantum evolution with a large number of negative decoherence rates

Katarzyna Siudzińska, Dariusz Chruściński

Published Jul 21, 2020• 19 citations
arXivHTMLPDF

Abstract

Abstract We analyze the evolution of open quantum systems governed by time-local master equations beyond the Markovian semigroup. Non-Markovian effects are usually attributed to the negativity of some decoherence rates in the time-dependent generator. For the qubit dynamics, it is well known that there can be one permanently negative rate in the so-called eternally non-Markovian evolution. We show that for qudits one can have (d − 1)2 out of the total d 2 − 1 rates that are always negative, and the evolution is still physically legitimate—that is, represented by a completely positive, trace-preserving dynamical map.

DOI
10.1088/1751-8121/aba7f2

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Abstract We analyze the evolution of open quantum systems governed by time-local master equations beyond the Markovian semigroup. Non-Markovian effects are usually attributed to the negativity of some decoherence rates in the time-dependent generator. For the qubit dynamics, it is well known that there can be one permanently negative rate in the so-called eternally non-Markovian evolution. We show that for qudits one can have (d − 1)2 out of the total d 2 − 1 rates that are always negative, and the evolution is still physically legitimate—that is, represented by a completely positive, trace-preserving dynamical map.
We construct a large class of completely positive and trace-preserving non-Markovian dynamical maps for an open quantum system. These maps arise from a piecewise dynamics for statistical operators characterized by a continuous time evolution interrupted by jumps, randomly distributed in time and described by a quantum channel. The state of the open system is shown to obey a closed evolution equation, given by a master equation with a memory kernel and an inhomogeneous term. The non-Markovianity of the obtained dynamics is explicitly assessed studying the behavior of the distinguishability of two different initial system's states with elapsing time.
We study the role of correlations with the environment as the source of non-Markovian quantum evolutions. We first focus on the impact that correlations with the environment can have on the dynamical map that evolve the system. We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord. The induced maps can be not completely positive when quantum correlations including, but not limited to, entanglement are present. We discuss the implications and limitations of the Markov approximation necessary to derive the Kossakowski-Lindblad master equation. A generalized non-Markovian master equation is derived from the dynamical map of systems correlated with their environment. The physical meaning of not completely positive maps is studied to obtain a consistent theory of non-Markovian quantum dynamics. These are associated to inverse maps necessary to establish correlations and they give rise to a canonical embedding map that is local in time. This master equation goes beyond the Kossakowski-Lindblad master equation. Non-equilibrium quantum thermodynamics can be be studied within this theory. Through out this discussion, the general dynamics of two interacting qubits is used as an example for illustrations.
We analyze a local approach to the non-Markovian evolution of open quantum systems. It turns out that any dynamical map representing evolution of such a system may be described either by non-local master equation with memory kernel or equivalently by equation which is local in time. The price one pays for the local approach is that the corresponding generator might be highly singular and it keeps the memory about the starting point "t0". This is the very essence of non-Markovianity. We illustrate a local approach by simple examples.