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arXiv:gr-qc/9304025 · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Information-theoretic measure of uncertainty due to quantum and thermal fluctuations

Arlen Anderson, J. J. Halliwell

Published Sep 15, 1993• 130 citations
arXivHTMLPDF

Abstract

We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information I of the phase-space probability distribution 〈z\ensuremath{\Vert}\ensuremath{\rho}\ensuremath{\Vert}z〉, where \ensuremath{\Vert}z〉 are coherent states and \ensuremath{\rho} is the density matrix. As shown by Lieb I\ensuremath{\ge}1, and this bound represents a strengthened version of the uncertainty principle. For a harmonic oscillator in a thermal state, I coincides with von Neumann entropy, -Tr(\ensuremath{\rho}ln\ensuremath{\rho}), in the high-temperature regime, but unlike entropy, it is nonzero (and equal to the Lieb bound) at zero temperature. It therefore supplies a nontrivial measure of uncertainty due to both quantum and thermal fluctuations. We study I as a function of time for a class of nonequilibrium quantum systems consisting of a distinguished system coupled to a heat bath. We derive an evolution equation for I. For the harmonic oscillator, in the Fokker-Planck regime, we show that I increases monotonically, if the width of the coherent states is chosen to be the same as the width of the harmonic oscillator ground state. For other choices of the width, and for more general Hamiltonians, I settles down to a monotonic increase in the long run, but may suffer an initial decrease for certain initial states that undergo ``reassembly'' (the opposite of quantum spreading). Our main result is to prove, for linear systems, that I at each moment of time has a lower bound ${\mathit{I}}_{\mathit{t}}^{\mathrm{min}}$, over all possible initial states.This bound is a generalization of the uncertainty principle to include thermal fluctuations in nonequilibrium systems, and represents the least amount of uncertainty the system must suffer after evolution in the presence of an environment for time t. ${\mathit{I}}_{\mathit{t}}^{\mathrm{min}}$ is an envelope, equal for each time t, to the time evolution of I for a certain initial state, which we calculate to be a nonminimal Gaussian. ${\mathit{I}}_{\mathit{t}}^{\mathrm{min}}$ coincides with the Lieb bound in the absence of an environment, and is related to von Neumann entropy in the long-time limit. The form of ${\mathit{I}}_{\mathit{t}}^{\mathrm{min}}$ indicates that the thermal fluctuations become comparable with the quantum fluctuations on a time scale equal to the decoherence time scale, in agreement with earlier work of Hu and Zhang. Our results are also related to those of Zurek, Habib, and Paz, who looked for the set of initial states generating the least amount of von Neumann entropy after a fixed period of nonunitary evolution.

DOI
10.1103/physrevd.48.2753

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We study an information-theoretic measure of uncertainty for quantum systems. It is the Shannon information I of the phase-space probability distribution 〈z\ensuremath{\Vert}\ensuremath{\rho}\ensuremath{\Vert}z〉, where \ensuremath{\Vert}z〉 are coherent states and \ensuremath{\rho} is the density matrix. As shown by Lieb I\ensuremath{\ge}1, and this bound represents a strengthened version of the uncertainty principle. For a harmonic oscillator in a thermal state, I coincides with von Neumann entropy, -Tr(\ensuremath{\rho}ln\ensuremath{\rho}), in the high-temperature regime, but unlike entropy, it is nonzero (and equal to the Lieb bound) at zero temperature. It therefore supplies a nontrivial measure of uncertainty due to both quantum and thermal fluctuations. We study I as a function of time for a class of nonequilibrium quantum systems consisting of a distinguished system coupled to a heat bath. We derive an evolution equation for I. For the harmonic oscillator, in the Fokker-Planck regime, we show that I increases monotonically, if the width of the coherent states is chosen to be the same as the width of the harmonic oscillator ground state. For other choices of the width, and for more general Hamiltonians, I settles down to a monotonic increase in the long run, but may suffer an initial decrease for certain initial states that undergo ``reassembly'' (the opposite of quantum spreading). Our main result is to prove, for linear systems, that I at each moment of time has a lower bound Itmin{\mathit{I}}_{\mathit{t}}^{\mathrm{min}}Itmin​, over all possible initial states.This bound is a generalization of the uncertainty principle to include thermal fluctuations in nonequilibrium systems, and represents the least amount of uncertainty the system must suffer after evolution in the presence of an environment for time t. Itmin{\mathit{I}}_{\mathit{t}}^{\mathrm{min}}Itmin​ is an envelope, equal for each time t, to the time evolution of I for a certain initial state, which we calculate to be a nonminimal Gaussian. Itmin{\mathit{I}}_{\mathit{t}}^{\mathrm{min}}Itmin​ coincides with the Lieb bound in the absence of an environment, and is related to von Neumann entropy in the long-time limit. The form of Itmin{\mathit{I}}_{\mathit{t}}^{\mathrm{min}}Itmin​ indicates that the thermal fluctuations become comparable with the quantum fluctuations on a time scale equal to the decoherence time scale, in agreement with earlier work of Hu and Zhang. Our results are also related to those of Zurek, Habib, and Paz, who looked for the set of initial states generating the least amount of von Neumann entropy after a fixed period of nonunitary evolution.
In the framework of the Lindblad theory for open quantum systems, expressions for the density operator, von Neumann entropy and effective temperature of the damped harmonic oscillator are obtained. The entropy for a state characterized by a Wigner distribution function which is Gaussian in form is found to depend only on the variance of the distribution function. We give a series of inequalities, relating uncertainty to von Neumann entropy and linear entropy. We analyze the conditions for purity of states and show that for a special choice of the diffusion coefficients, the correlated coherent states (squeezed coherent states) are the only states which remain pure all the time during the evolution of the considered system. These states are also the most stable under evolution in the presence of the environment and play an important role in the description of environment induced decoherence.
We study the time evolution of the reduced density operator for a class of quantum Brownian motion models consisting of a particle moving in a potential V(x) and coupled to an environment of harmonic oscillators in a thermal state. Our principal tool is the Wigner function of the reduced density operator and for linear systems we derive an explicit expression for the Wigner function propagator. We use it to derive two generalized uncertainty relations. The first consists of a sharp lower bound on the uncertainty function U=(\ensuremath{\Delta}p)2{)}^{2})2(\ensuremath{\Delta}q)2{)}^{2})2 after evolution for time t in the presence of an environment. The second, a stronger and simpler result, consists of a lower bound at time t on the quantity scrA2{\mathit{scrA}}^{2}scrA2=U-Cpq2{\mathit{C}}_{\mathit{p}\mathit{q}}^{2}Cpq2​, where Cpq{\mathit{C}}_{\mathit{p}\mathit{q}}Cpq​=1/2〈\ensuremath{\Delta}p^\ensuremath{\Delta}q^+\ensuremath{\Delta}q^\ensuremath{\Delta}p^〉. ($scrA--- is essentially the area enclosed by the 1-\ensuremath{\sigma} contour of the Wigner function.) In both cases the minimizing initial state is a correlated coherent state (a nonminimal Gaussian pure state), and in the first case the lower bound is only an envelope. These generalized uncertainty relations supply a measure of the comparative size of quantum and thermal fluctuations. We prove two simple inequalities, relating uncertainty to von Neumann entropy, -Tr(\ensuremath{\rho} ln\ensuremath{\rho}), and the von Neumann entropy to linear entropy, 1-Tr${\mathrm{\ensuremath{\rho}}}^{2}$. We also prove some results on the long-time limit of the Wigner function for arbitrary initial states. For the harmonic oscillator the Wigner function for all initial states becomes a Gaussian at large times (often, but not always, a thermal state). We derive the explicit forms of the long-time limit for the free particle (which does not in general go to a Gaussian), and also for more general potentials in the approximation of high temperature. We discuss connections with previous work by Hu and Zhang and by Paz and Zurek.
The notion of entropy was introduced by Clausius in order to discuss the thermal behavior of physical systems. After the work of Clausius, Boltzmann, Gibbs and others, the beginning of the exact treatment of entropy was due to von Neumann and Shannon. Their starting point was very di erent, von Neumann was motivated by quantum mechanics and Shannon founded communication theory. The entropy of a state describing a physical system is a quantity expressing the uncertainty or randomness of the system. Shannon regarded this uncertainty attached to a physical system as the amount of information carried by the system, so that the entropy of a state can be read as the information carried by the state. His idea comes from the following consideration: If a physical system has a large uncertainty and one receives information on the system by some procedure, then the so obtained information is more valuable than that received from a system having less uncertainty. It is understood in probability theory that the notion of (Shannon or measure theoretic) entropy has successful applications in a variety of subjects because it determines the asymptotic behavior of certain probabilities in the course of independent trials. First we will discuss this phenomena for nite quantum systems. By a nite quantum system we mean an algebra of matrices which is stable under taking adjoint. (In other words, a nite quantum system is a nite dimensional C*algebra.) If A is such an algebra then there is a linear functional Tr which takes the value 1 at each minimal projection. It is "tracial " in the sense that Tr ab = Tr ba (a; b 2 A) Every functional! on A is determined by a density operator D! 2 A in the form!(a) = Tr D! a (a 2 A): The entropy S(!) of a functional! is de ned by means of its density operator as S(!) = Tr (D!): This notion was introduced by von Neumann in 1927 and we term it von Neumann's entropy or shortly entropy (cf. [1]). In fact, von Neumann de ned the entropy such
We are concerned with an information theoretic measure of uncertainty for quantum systems. Precisely, the Wehrl entropy of the phase space probability (z,m I rho Iz,m) which is known as Husimi function, where rho is a density operator and Iz,m) are coherent states attached to an Euclidean mth Landau level. We obtain the Husimi function Qmbeta of the thermal density operator rhobeta of the harmonic oscillator, which leads, by duality, to the Laguerre probability distribution of the mixed light. We discuss some basic properties of Qmbeta such as its characteristic function and its limiting logarithmic moment generating function from which we derive the rate function of the sequence of probability distributions Qmbeta, m=0,1,... For m=1,2,..., we establish an exact expression for the Wehrl entropy of the density operator rhobeta and we discuss the behavior of this entropy with respect to the temperature parameter T = 1/beta.
This paper is concerned with two questions in the decoherent histories approach to quantum mechanics: the emergence of approximate classical predictability, and the fluctuations about it necessitated by the uncertainty principle. It is in part a continuation of an earlier paper on the uncertainty principle for quantum-mechanical histories. We consider histories characterized by position samplings at n moments of time. This includes the case of position sampling continuous in time in the limit n\ensuremath{\rightarrow}\ensuremath{\infty}. We use this to construct a probability distribution on the value of (discrete approximations to) the field equations, F=mx\ifmmode\ddot\else\textasciidieresis\fi{}+V'(x), at n-2 times. We find that it is peaked around F=0; thus classical correlations are exhibited. We show that the width of the peak \ensuremath{\Delta}F is largely independent of the initial state. We show that the uncertainty principle takes the form 2${\mathrm{\ensuremath{\sigma}}}^{2}$(\ensuremath{\Delta}F)2{)}^{2})2\ensuremath{\ge}${\mathrm{\ensuremath{\Elzxh}}}^{2}$/t2{\mathit{t}}^{2}t2, where \ensuremath{\sigma} is the width of the position samplings, and t is the time scale between projections. We determine the modifications to this result when the system is coupled to a thermal environment, thus obtaining a measure of the comparative sizes of quantum and thermal fluctuations. We show that the thermal fluctuations become comparable to the quantum fluctuations under the same conditions that decoherence effects come into play, in agreement with earlier work. We also study an alternative measure of classical correlations, namely, the conditional probability of finding a sequence of position samplings, given that particular initial phase space data have occurred.We use these results to address the issue of the formal interpretation of the probabilities for sequences of position samplings in the decoherent histories approach to quantum mechanics. Under appropriate conditions, which we describe, they admit an interpretation as a statistical ensemble of classical solutions, with the probability of each individual classical solution given by a smeared Wigner function of its initial data. We study the decoherence properties of histories characterized by the value of the field equations, F, at a sequence of times. We argue that they will be decoherent if their initial data are decoherent.
We derive the uncertainty relation for a quantum open system consisting of a Brownian particle interacting with a bath of quantum oscillators at finite temperature. We examine how the quantum and thermal fluctuations of the environment contribute to the uncertainty in the canonical variables of the system. We show that upon contact with the bath (assumed to be ohmic in this paper) the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time in similar models in the discussion of quantum to classical transition. This offers some insight into the physical mechanisms involved in the environment-induced decoherence process. We obtain closed analytic expressions for this generalized uncertainty relation under the conditions of high temperature and weak damping, separately. We also consider under these conditions an arbitrarily squeezed initial state and show how the squeeze parameter enters in the generalized uncertainty relation. Using these results we examine the transition of the system from a quantum pure state to a nonequilibrium quantum statistical state and to an equilibrium quantum statistical state. The three stages are marked by the decoherence time and the relaxation time, respectively. With these observations we explicate the physical conditions under which the two basic postulates of quantum statistical mechanics become valid. We also comment on the inappropriate usage of the word “classicality” in many decoherence studies of quantum to classical transition.