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IEEE International Symposium on Information Theory (ISIT)

Asymptotically Decoupling and Mixing Properties in Quantum System

Yuuya Yoshida, Masahito Hayashi

Published Jun 1, 2018• 2 citations
Publisher

Abstract

The mixing property is a useful property for quantum dynamics as well as ergodicity. The decoupling property plays an important role in open quantum systems and quantum information. We show that these two properties are equivalent to each other for quantum dynamics. We derive several useful necessary and sufficient conditions for these conditions by using the ergodicity. Additionally, we discuss irreducibility and primitivity which are important properties used in Perron-Frobenius theory.

DOI
10.1109/isit.2018.8437491

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The mixing property is a useful property for quantum dynamics as well as ergodicity. The decoupling property plays an important role in open quantum systems and quantum information. We show that these two properties are equivalent to each other for quantum dynamics. We derive several useful necessary and sufficient conditions for these conditions by using the ergodicity. Additionally, we discuss irreducibility and primitivity which are important properties used in Perron-Frobenius theory.
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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.
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We give a simple and physically intuitive necessary and sufficient condition for a map acting on a compact metric space to be mixing (i.e. infinitely many applications of the map transfer any input into a fixed convergency point). This is a generalization of the 'Lyapunov direct method'. First we prove this theorem in topological spaces and for arbitrary continuous maps. Finally we apply our theorem to maps which are relevant in open quantum systems and quantum information, namely quantum channels. In this context, we also discuss the relations between mixing and ergodicity (i.e. the property that there exists only a single input state which is left invariant by a single application of the map) showing that the two are equivalent when the invariant point of the ergodic map is pure.
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The paper provides a systematic characterization of quantum ergodic and mixing channels in finite dimensions and a discussion of their structural properties. In particular, we discuss ergodicity in the general case where the fixed point of the channel is not a full-rank (faithful) density matrix. Notably, we show that ergodicity is stable under randomizations, namely that every random mixture of an ergodic channel with a generic channel is still ergodic. In addition, we prove several conditions under which ergodicity can be promoted to the stronger property of mixing. Finally, exploiting a suitable correspondence between quantum channels and generators of quantum dynamical semigroups, we extend our results to the realm of continuous-time quantum evolutions, providing a characterization of ergodic Lindblad generators and showing that they are dense in the set of all possible generators.