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arXiv:1607.05963 · Physical review. E

Time evolution of Rényi entropy under the Lindblad equation

Sumiyoshi Abe

Published Aug 4, 2016• 18 citations
arXivHTMLPDF

Abstract

In recent years, the Rényi entropy has repeatedly been discussed for characterization of quantum critical states and entanglement. Here, time evolution of the Rényi entropy is studied. A compact general formula is presented for the lower bound on the entropy rate.
DOI
10.1103/physreve.94.022106

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A brief review of two recent topics on the Lindblad equation is presented. One is concerned with time evolution of the quantum Renyi entropy under the equation. The lower bound of the entropy rate is derived in a compact form. The other is about the concept of weak invariants, which generalize the Lewis-Riesenfeld invariant and are defined in such a way that they are not constant but their expectation values remain invariant in time. The Lindbladian operator describing the time-dependent damped quantum oscillator is identified, and the corresponding weak invariant is explicitly constructed as an example.
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We study a recent conjecture about the behavior of the quantum relative entropy compared to the relative entropy of entanglement in open bipartite systems. The conjecture states that, under a dissipative time evolution, the positive rate of change of the relative entropy will always be larger than that of the relative entropy of entanglement. After explicitly solving a 2-qubit master equation of Lindblad type with separable and entangled stationary states, we show that the conjecture can be violated for initial states with an entangled asymptotic state, while it appears to be confirmed when the asymptotic states are separable.
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We provide an upper bound on the maximal entropy rate at which the entropy of the expected density operator of a given ensemble of two states changes under nonlocal unitary evolution. A large class of entropy measures in considered, which includes Rényi and Tsallis entropies. The result is derived from a general bound on the trace-norm of a commutator, which can be expected to find other implementations. We apply this result to bound the maximal rate at which quantum dynamics can generate entanglement in a bipartite closed system with Rényi and Tsallis entanglement entropies taken as measures of entanglement in the system.
Exploiting the master equations in the Lindblad form, we establish a sufficient and necessary condition for a Markovian dynamics to be unital. Based on this condition, we analyze the dynamical property of quantum R\'enyi entropy and propose a characterization of quantum non-Markovianity for unital dynamics in terms of R\'enyi entropy, which contains the previous criteria of non-Markovianity via von Neumann entropy and linear entropy as particular cases. The effectiveness of this measure in capturing the backflow of information from the environment is illustrated through several typical unital dynamics.