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arXiv:1801.03988 · Journal of Physics A Mathematical and Theoretical

Asymptotic properties for Markovian dynamics in quantum theory and general probabilistic theories

Yuuya Yoshida, Masahito Hayashi

Published Apr 2, 2020• 8 citations
arXivHTMLPDF

Abstract

We address asymptotic decoupling in the context of Markovian quantum dynamics. Asymptotic decoupling is an asymptotic property on a bipartite quantum system, and asserts that any correlation between two quantum systems is broken after a sufficiently long time passes. In this paper, we show that asymptotic decoupling is equivalent to local mixing which asserts the convergence to a unique stationary state on at least one quantum system. If dynamics is asymptotically decoupling, any correlation between two quantum systems is broken exponentially. Also, we give a criterion of mixing that is a system of linear equations. All results in this paper are proved in the framework of general probabilistic theories, but we also summarize them in quantum theory.

DOI
10.1088/1751-8121/ab8599

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In the dynamics of open quantum systems, information may propagate in time through either the system or the environment, giving rise to Markovian and non-Markovian temporal correlations, respectively. However, despite their notable coexistence in most physical situations, it is not yet clear how these two quantities may limit the existence of one another. Here, we address this issue by deriving several inequalities relating the temporal correlations of general multitime quantum processes. The dynamics are described by process tensors, and the correlations are quantified by the mutual information between subsystems of their Choi states. First, we prove a set of upper bounds to the non-Markovianity of a process given the degree of Markovianity in each of its steps. This immediately implies a nontrivial maximum value for the non-Markovianity of any process, independent of its Markovianity. Finally, we determine how the non-Markovianity limits the amount of total temporal correlations that could be present in a given process. These results show that, although any multitime process must pay a price in total correlations to have a given amount of non-Markovianity, this price vanishes exponentially with the number of steps of the process, while the maximum non-Markovianity grows only linearly. This implies that even a highly non-Markovian process might be arbitrarily close to having the maximum amount of total correlations if it has a sufficiently large number of steps.
The noise decoupling problem is investigated for general NNN-level Markovian open quantum systems. First, the concept of Cartan decomposition of the Lie algebra su(N)\mathrm{su}(N)su(N) is introduced as a tool of designing control Hamiltonians. Next, under certain assumptions, it is shown that a part of variables of the coherence vector of the system density matrix can be asymptotically decoupled from the environmental noises. The resulting noise decoupling scheme is applied to one-qubit, qutrit, and two-qubit quantum systems, by which the coherence evolution of the one-qubit and qutrit systems can always be asymptotically preserved, while, for two-qubit systems, our findings indicate that evolution of some variables can be preserved only for some initial states.
In the framework of the theory of open systems based on completely positive quantum dynamical semigroups, the master equation for two independent harmonic oscillators interacting with an environment is solved in the asymptotic long‐time regime. Using the Peres–Simon necessary and sufficient condition for separability of two‐mode Gaussian states, we show that the two non‐interacting systems become asymptotically entangled for certain environments, so that in the long‐time regime they manifest non‐local quantum correlations. We calculate also the logarithmic negativity characterizing the degree of entanglement of the asymptotic state.