We present an entropic description of quantum-mechanical states based on an operational approach to a phase-space measurement. We give a simple phase-space interpretation of sampling entropies in terms of which we derive very general entropic uncertainty relations reflecting the degree of the phase-space uncertainty of the quantum-mechanical state in the given measurement (i.e., for a given ``quantum-ruler'' state). We relate the sampling entropy to the von Neumann and Shannon entropy and show that the Wehrl entropy represents a particular example of a sampling entropy when the quantum ruler is represented by coherent states.
Contents 1 Introduction 2 2 Quantum Interference and the Construction of Nonclassical States 3 2.1 Nonclassical Light, Quadratures and Phase--Sensitive Observables 3 2.2 Phase--Space and Quasi--Probabilities . . . . . . . . . . . . . . . . 13 2.3 Quantum Noise as a Stochastic Process . . . . . . . . . . . . . . 15 2.4 Marginal distributions and reconstruction of Wigner functions . . 18 2.5 Joint Measurements of p, q in Phase Space . . . . . . . . . . . . . 20 2.6 Phase--Space Sampling Entropies . . . . . . . . . . . . . . . . . . 27 2.7 Mutual Information and Entropic Uncertainty . . . . . . . . . . . 28 2.8 Building Nonclassical States by Quantum Interference . . . . . . 30 3 Dissipation Versus Coherent Evolution: Decoherence 36 4 Unravelling the Master Equation 37 5 Quantum Jumps 43 5.1 Intrinsic or Extrinsic Jumps . . . . . . . . . . . . . . . . . . . . . 43 5.2 Observation of Quantum Jumps . . . . . . . . . . . . . .
From a classical perspective, a quantum state is a surprising mixture of information and ignorance. If a system is sitting in an eigenstate of an observable, we can predict with confidence that a measurement result will yield the appropriate eigenvalue. With a superposition, certainty is not granted to us. Unlike the closest possible classical case, where a superposition represents a genuine blending of classical states, the quantum superposition is a strange overlapping of states simultaneously occupied by the system. On measurement, this ontologically curious animal collapses into one of the overlapping states, a process that is not well understood to this day. The collapse is random, a-causal, but governed by quantum amplitudes. Heading into a measurement, we do not know what we are going to get out. Given the evident link between entropy and information, the probabilities and uncertainties baked into a quantum state may well be a fertile ground for another form of entropy. The density matrix gives us a flexible tool for studying the properties of quantum ensembles of any kind. In this chapter, we will see how von Neumann helped develop and use the density matrix to study the problem of measurement and the information/entropy content of quantum states .
Much of the discussion of decoherence has been in terms of a particle moving in one dimension that is placed in an initial superposition state (a Schrödinger "cat" state) corresponding to two widely separated wave packets. Decoherence refers to the destruction of the interference term in the quantum probability function. Here, we stress that a quantitative measure of decoherence depends not only on the specific system being studied but also on whether one is considering coordinate, momentum or phase space. We show that this is best illustrated by considering Wigner phase space where the measure is again different. Analytic results for the time development of the Wigner distribution function for a two-Gaussian Schrodinger "cat" state have been obtained in the high-temperature limit (where decoherence can occur even for negligible dissipation) which facilitates a simple demonstration of our remarks.
A density of the information-theoretic Wehrl entropy in terms of the Hu- simi
Q -function is defined.The entropic density is then applied to describe the quantum properties of some optical fields including Fock states, cohe- rent and squeezed states, superposition of chaotic and coherent states, and Schr\"odinger cat and cat-like states.The entropic density is compared with both the conventional Wehrl entropy and the Husimi phase distribution.It is shown that the entropic density is a good measure of the phase-space uncertainty (noise), and the phase locking or phase bifurcation effects.The advantages of the entropic description of the superposition principle are presented.It is also demonstrated that the entropic density properly de- scribes phase randomization processes, thus can be used as a measure of decoherence.I. INTRODUCTION Quantum entropy is one of the most fundamental concepts of quantum physics [1], parti- cularly useful in quantum information [2].Quantum entropy has been applied, e.g., as a measure of quantum entanglement, qualltum decoherence, photocount statistics, quantum 数理解析研究所講究録 1142 巻 2000 年 1-22 1 optical correlations, purity of states, quantum noise or accessible information in quantum measurement (capacity of quantum channel).Quantum entropy, as a natural generalization of classical Boltzmann entropy, was propo- sed by von Neumann [3].A classical-like entropy associated with quantum fields was intro- duced by Wehrl [4] in terms of the Glauber coherent states and the Husimi
Q -function [5].A rigorous proof that the von Neumann entropy tends to the Wehrl entropy in the clas- sical limit $\hslasharrow 0$ was given by Beretta [6].The quantum von Neumann entropy can be expanded in power series of classical entropies.As was shown explicitly by Pe\v{r}inov\'a et al. [7], the first term of this expansion is the Wehrl entropy.Bu\v{z}ek et al. [8] related the von Neumann entropy with the sampling entropies based on operational approach to a phase-space measurement.They also identified the Wehrl entropy as a particular example of the sampling entropy, when the quantum ruler is represented by coherent states.Other aspects of the relation between classical and quantum-mechanical entropies have also been extensively studied (see, e.g., excellent surveys by Wehrl [9], and Ohya and Petz [1] with references included therein).The Wehrl entropy has been successfully applied in a description of different properties of quantum optical fields.In particular, it has been shown explicitly that the Wehrl entropy is a useful measure of phase-space uncertainty (quantum noise, phase-space localization, wave-packet spreading) [8,10-14], quantum interference [10], decoherence [11,15,16], ioni- zation [14], squeezing [16-18], Schr\"odinger cat formation
[13,19] or, in general, splitting of
Q -function [15].The von Neumann entropy becomes zero for all pure states, thus cannot be used in discriminating them.Paradoxically, the classical Wehrl entropy is more sensitive in di- stinguishing states than the quantum von Neumann entropy, since it is dependent on the choice of pure states.However, there are properties of quantum fields (including phase decoherence or unique description of superposition states), which are not enough precisely described by the conventional Wehrl entropy.Therefore, we propose a new entropic me-
For a system with canonical variables x and p, [x,p]=i\ensuremath{\Elzxh}, we associate with each density operator \ensuremath{\rho} an x coherence \ensuremath{\delta}x and a p coherence \ensuremath{\delta}p which specify, respectively, the characteristic distance and momentum range over which the underlying quantum-mechanical nature of the ensemble cannot be neglected. These coherences are less than or equal to the associated uncertainties \ensuremath{\delta}x\ensuremath{\le}\ensuremath{\Delta}x, \ensuremath{\delta}p\ensuremath{\le}\ensuremath{\Delta}p; so the ratios
cx\ensuremath{\equiv}\ensuremath{\delta}x/\ensuremath{\Delta}x,
cp\ensuremath{\equiv}\ensuremath{\delta}p/\ensuremath{\Delta}p give measures of the quantum character of the ensemble with respect to x and p. Examples are presented.