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arXiv:1111.3080 · Physical Review A

Dependence of a quantum-mechanical system on its own initial state and the initial state of the environment it interacts with

Adrian Hutter, Stephanie Wehner

Published Jan 18, 2013• 15 citations
arXivHTMLPDF

Abstract

We present a unifying framework to the understanding of when and how quantum-mechanical systems become independent of their initial conditions and adapt macroscopic properties (like temperature) of the environment. By viewing this problem from a quantum information theory perspective, we are able to simplify it in a very natural and easy way. We first show that for any interaction between the system and the environment, and almost all initial states of the system, the question of how long the system retains memory of its initial conditions can be answered by studying the temporal evolution of just one special initial state. This special state thereby depends only on our knowledge of macroscopic parameters of the system. We provide a simple entropic inequality for this state that can be used to determine whether most states of the system have or have not become independent of their initial conditions after time $t$. We discuss applications of our entropic criterion to thermalization times in systems with an effective light cone and to quantum memories suffering depolarizing noise. We make a similar statement for almost all initial states of the environment and finally provide a sufficient condition for which a system never thermalizes but remains close to its initial state for all times.

DOI
10.1103/physreva.87.012121

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Abstract In the interactions of electromagnetic radiation with matter which we have considered so far, we have almost invariably known accurately the state of the system before interaction. As a result the notion of an open system with an external environment did not have to be systematically discussed. In the quantum electrodynamics of elementary particles, the scattering events are predominantly affected by only few-body interactions and the concepts of temperature and environment are irrelevant. (The resolution of the problem of infra-red divergences is an exception where the environment of soft photons is crucial.) For very small numbers of atoms it may be possible to know the initial state of the system, but in a macroscopic device this is essentially impossible. Although for transitions at optical frequencies the temperature of the environment may be negligible, this will not necessarily be the case for microwave transitions. We have already seen how atomic linewidths can be produced by interactions with the bath of photons at zero temperature. In the general case a quantum-mechanical system will show not only fluctuations due to the non-commutative nature of quantum theory but also thermal fluctuations and dissipation due to interactions with the environment. For complicated systems it will not be possible to know with certainty the initial state of the system, and so the approach of studying the evolution of a state vector through the Schrodinger equation has to be replaced by a consideration of the time evolution of the density operator introduced in Chapter 4. We will formulate a systematic way of treating the dynamics of a quantum-mechanical open system interacting with a heat bath, which will be illustrated with important examples.

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We present a unifying framework to the understanding of when and how quantum-mechanical systems become independent of their initial conditions and adapt macroscopic properties (like temperature) of the environment. By viewing this problem from a quantum information theory perspective, we are able to simplify it in a very natural and easy way. We first show that for any interaction between the system and the environment, and almost all initial states of the system, the question of how long the system retains memory of its initial conditions can be answered by studying the temporal evolution of just one special initial state. This special state thereby depends only on our knowledge of macroscopic parameters of the system. We provide a simple entropic inequality for this state that can be used to determine whether most states of the system have or have not become independent of their initial conditions after time ttt. We discuss applications of our entropic criterion to thermalization times in systems with an effective light cone and to quantum memories suffering depolarizing noise. We make a similar statement for almost all initial states of the environment and finally provide a sufficient condition for which a system never thermalizes but remains close to its initial state for all times.
The decoupling technique was originally developed for information-theoretical purposes. It describes the conditions under which the correlations in a bipartite state disappear if one part undergoes an evolution separated from the other. In the past years there has been enormous progress in understanding the foundations of statistical mechanics from first principles of quantum mechanics. By use of the decoupling technique we are able to reproduce and generalize major results of this development and to approach open problems. As a first application of the decoupling technique we generalize the result of [Popescu et al., Nat. Phys. 2, 754-758 (2006)] about the apparent validity of the postulate of equal a priori probabilities to states which may be correlated to a reference. We express it in a form which allows to apply recent results about random two-qubit interactions. We give a criterion for the apparent validity of the postulate which is tight up to differences between different entropy measures. Similarly, we generalize the result of [Linden et al., Phys. Rev. E 79, 061103 (2009)] about the independence of the temporal average of a quantum mechanical system of its initial state to initial states which may be correlated to a reference. We develop a criterion for whether a quantum-mechanical system has “forgotten” about its initial state which is tight up to differences between different entropy measures. We find that comparing two local entropies of just one particular state tells us whether generic initial states of the system have already evolved to the same state or not. After developing new bounds on the times which are necessary for entropy changes, we are able to provide lower bounds on the times which are necessary for a system to become independent of its initial state. We discover an intimate connection between a system becoming independent of its initial state and a loss of an observer’s knowledge about the state of the system. As a further application of the decoupling technique we find sufficient conditions under which a system stays close to its initial state for all times, thereby extending the result of [Gogolin et al., Phys. Rev. Lett. 106, 040401 (2011)].
A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρℐ(t) approaches an equilibrium state ρ eq ℐ , the latter depending only on macroscopic characteristics of the bath (e.g. temperature), but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present contribution we discuss the recently proven general theorems about thermalization and demonstrate how they work in exactly solvable models. In particular, we review a necessary condition for the system initial state independence (ISI) of ρ eq ℐ , which was proven in our previous work, and apply it for two exactly solvable models, the XX spin chain and a central spin model with a special interaction with the environment. In the latter case we are able to prove the absence of the system ISI. Also the Eigenstate Thermalization Hypothesis is discussed. It is pointed out that although it is supposed to be generically true in essentially not-integrable (chaotic) quantum systems, it is how-ever also valid in the integrable XX model.
A small system in contact with a macroscopic environment usually approaches an asymptotic state, determined only by some macroscopic properties of the environment, such as the temperature or the chemical potential. In the long-time limit, the state of the small system is thus expected to be independent of its initial state. In some situations, however, the asymptotic state of the system is influenced by its initial state, and some information about the initial state is kept for all times. Motivated by this finding, we propose a measure to quantify the influence of the initial state of an open system on its dynamics. Using this measure we derive conditions under which the asymptotic state exists and is unique and discuss how this can be used to characterize the asymptotic behavior of an open system. We demonstrate our concepts for the dynamics of the spin-boson model, identify three qualitatively different long-time behaviors, and discuss how they can be distinguished based on the proposed measure.
Simulations are performed of a small quantum system interacting with a quantum environment. The system consists of various initial states of two harmonic oscillators coupled to give normal modes. The environment is "designed" by its level pattern to have a thermodynamic temperature. A random coupling causes the system and environment to become entangled in the course of time evolution. The approach to a Boltzmann distribution is observed, and effective fitted temperatures close to the designed temperature are obtained. All initial pure states of the system are driven to equilibrium at very similar rates, with quick loss of memory of the initial state. The time evolution of the von Neumann entropy is calculated as a measure of equilibration and of quantum coherence. It is pointed out using spatial density distribution plots that quantum interference is eliminated only with maximal entropy, which corresponds thermally to infinite temperature. Implications of our results for the notion of "classicalizing" behavior in the approach to thermal equilibrium are briefly considered.
A system put in contact with a large heat bath normally thermalizes. This means that the state of the system ρS(t) approaches an equilibrium state ρ(eq)(S), the latter depending only on macroscopic characteristics of the bath (e.g., temperature) but not on the initial state of the system. The above statement is the cornerstone of the equilibrium statistical mechanics; its validity and its domain of applicability are central questions in the studies of the foundations of statistical mechanics. In the present paper we concentrate on one aspect of thermalization, namely, on the system initial state independence (ISI) of ρ(eq)(S). A necessary condition for the system ISI is derived in the quantum framework. We use the derived condition to prove the absence of the system ISI in a specific class of models. Namely, we consider a single spin coupled to a large bath, the interaction term commuting with the bath self-Hamiltonian (but not with the system self-Hamiltonian). Although the model under consideration is nontrivial enough to exhibit the decoherence and the approach to equilibrium, the derived necessary condition is not fulfilled and thus ρ(eq)(S) depends on the initial state of the spin.
Abstract In the interactions of electromagnetic radiation with matter which we have considered so far, we have almost invariably known accurately the state of the system before interaction. As a result the notion of an open system with an external environment did not have to be systematically discussed. In the quantum electrodynamics of elementary particles, the scattering events are predominantly affected by only few-body interactions and the concepts of temperature and environment are irrelevant. (The resolution of the problem of infra-red divergences is an exception where the environment of soft photons is crucial.) For very small numbers of atoms it may be possible to know the initial state of the system, but in a macroscopic device this is essentially impossible. Although for transitions at optical frequencies the temperature of the environment may be negligible, this will not necessarily be the case for microwave transitions. We have already seen how atomic linewidths can be produced by interactions with the bath of photons at zero temperature. In the general case a quantum-mechanical system will show not only fluctuations due to the non-commutative nature of quantum theory but also thermal fluctuations and dissipation due to interactions with the environment. For complicated systems it will not be possible to know with certainty the initial state of the system, and so the approach of studying the evolution of a state vector through the Schrodinger equation has to be replaced by a consideration of the time evolution of the density operator introduced in Chapter 4. We will formulate a systematic way of treating the dynamics of a quantum-mechanical open system interacting with a heat bath, which will be illustrated with important examples.