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arXiv:1004.5405 · Physical Review Letters

Sufficient and Necessary Condition for Zero Quantum Entropy Rates under any Coupling to the Environment

César A. Rodríguez-Rosario, Gen Kimura, Hideki Imai, Alán Aspuru‐Guzik

Published Feb 2, 2011• 28 citations
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Abstract

We find the necessary and sufficient conditions for the entropy rate of the system to be zero under any system-environment Hamiltonian interaction. We call the class of system-environment states that satisfy this condition lazy states. They are a generalization of classically correlated states defined by quantum discord, but based on projective measurements of any rank. The concept of lazy states permits the construction of a protocol for detecting global quantum correlations using only local dynamical information. We show how quantum correlations to the environment provide bounds to the entropy rate, and how to estimate dissipation rates for general non-Markovian open quantum systems.

DOI
10.1103/physrevlett.106.050403

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We find the necessary and sufficient conditions for the entropy rate of the system to be zero under any system-environment Hamiltonian interaction. We call the class of system-environment states that satisfy this condition lazy states. They are a generalization of classically correlated states defined by quantum discord, but based on projective measurements of any rank. The concept of lazy states permits the construction of a protocol for detecting global quantum correlations using only local dynamical information. We show how quantum correlations to the environment provide bounds to the entropy rate, and how to estimate dissipation rates for general non-Markovian open quantum systems.
We establish the necessary and sufficient conditions for a quantum system to be stable under any general system-environment interaction. Quantum systems are stable when the time-derivative of their purity is zero. This stability provides a dynamical explanation of the classicality of measurement apparatus. We also propose a protocol to detect global quantum correlations using only local dynamical information. We show how quantum correlations to the environment provide bounds to the purity rate, which in turn can be used to estimate dissipation rates for general non-Markovian open quantum systems.
The joint state of a system that is in contact with an environment is called lazy, if the entropy rate of the system under any coupling to the environment is zero. Necessary and sufficient conditions have recently been established for a state to be lazy [Phys. Rev. Lett. 106, 050403 (2011)], and it was shown that almost all states of the system and the environment do not have this property [Phys. Rev. A 81, 052318 (2010)]. At first glance, this may lead us to believe that low entropy rates themselves form an exception, in the sense that most states are far from being lazy and have high entropy rates. Here, we show that in fact the opposite is true if the environment is sufficiently large. Almost all states of the system and the environment are pretty lazy-their entropy rates are low for any coupling to the environment.
Quantum discord measures the total non-classical correlations in a composite quantum system. As a possible resource for the classically non-achievable tasks, amount of quantum discord is crucial for efficient performance of the quantum technology tasks. Zero discord means the absence of the required resources. For a wide class of the bipartite open quantum systems, initial nonzero discord is known practically never goes to zero, except asymptotically. However, if initial discord equals zero, there remains the possibility to remain zero for the finite time intervals for certain Markovian models of open systems-that we call Markovian classicality. In this paper we search for the models that allow for Markovian classicality. We point out a general model that represents a matter-of-principle formal proof, i.e. a sufficient condition for the, otherwise not obvious, existence of Markovian classicality. Physical relevance of the model is twofold. First, the model is in intimate relation to the topics of quantum information locality, quantum discord saturation and quantum decorrelation. Second, the model is of the general physical interest. It pertains to a specific structure (decomposition into parts/subsystems) of a composite system, not to a special kind of composite system. Being a characteristic of a structure, by definition, the model of Markovian classicality is not a model of sudden death of discord. We emphasize wide-range implications of our results.
A general state of an n m system is a classical-quantum state if and only if its associated left-correlation matrix has rank no larger than m-1. Based on this condition, a computable measure of quantum discord is presented, which coincides with the tight lower bound on the geometric measure of discord. Therefore such obtained tight lower bound fully captures the quantum correlation of a bipartite system, so it can be used as a measure of discord in its own right. Accordingly, a vanishing tight lower bound on the geometric discord is a necessary and sufficient condition for a state to be zero-discord.
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.
Using Tsallis-qqq entropy, we introduce the generalized concept of global quantum discord, namely the qqq-global quantum discord, and provide its analytic evaluation for two classes of multiqubit states. We also provide a sufficient condition, for which the pairwise quantum correlations in terms of qqq-global quantum discord is monogamous in multiparty quantum systems.