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Critical points of the "entropy"-like functional for the quantum distribution functions

E N Evtimova, Vladimir Simeonov Georgiev

Published Jun 10, 1992• 0 citations
Publisher

Abstract

On the analogy of Kandrup's [43] functional of entropy for the Wigner's quantum distribution function a generalised entropy'-like functional for, a larger class than Cohen's one of quantum distribution functions, is introduced. A variation of this functional, using the Gateaux derivatives and taking into account the properties of the distribution functions by means of Lagrange multipliers, leads to the existence of a class of specific distribution functions, the critical points of the functional. One of them is the distribution introduced by Terletsky [4] and Margenau and Hill [6]

DOI
10.5169/seals-116505

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On the analogy of Kandrup's [43] functional of entropy for the Wigner's quantum distribution function a generalised entropy'-like functional for, a larger class than Cohen's one of quantum distribution functions, is introduced. A variation of this functional, using the Gateaux derivatives and taking into account the properties of the distribution functions by means of Lagrange multipliers, leads to the existence of a class of specific distribution functions, the critical points of the functional. One of them is the distribution introduced by Terletsky [4] and Margenau and Hill [6]
Cohen's (see Prentice Hall, Englewood Cliffs, NJ, 1995) class of time-frequency distributions for continuous signals has been to extended to discrete signals using both an axiomatic approach and an operator theory approach. We investigate the formulation of several classical time-frequency distributions (Wigner, Rihaczek, Margenau-Hill, Page, Levin, Born-Jordan, spectrogram) in the discrete Cohen's classes. The main result of this paper concludes that there does not exist a formulation of the Wigner distribution in all of the discrete Cohen's classes.
We examine weak measurements of arbitrary observables where the object is prepared in a mixed state and on which measurements with imperfect detectors are made. The weak value of an observable can be expressed as a conditional expectation value over an infinite class of different generalized Kirkwood quasiprobability distributions. ``Strange'' weak values for which the real part exceeds the eigenvalue spectrum of the observable can only be found if the Terletsky-Margenau-Hill distribution is negative or, equivalently, if the real part of the weak value of the density operator is negative. We find that a classical model of a weak measurement exists whenever the Terletsky-Margenau-Hill representation of the observable equals the classical representation of the observable and the Terletsky-Margenau-Hill distribution is non-negative. Strange weak values alone are not sufficient to obtain a contradiction with classical models. We propose feasible weak measurements of photon number of the radiation field. Negative weak values of energy contradict all classical stochastic models, whereas negative weak values of photon number contradict all classical stochastic models where the energy is bounded from below by the zero-point energy. We examine coherent states in particular and find negative weak values with probabilities of 16% for kinetic energy (or squared field quadrature), 8% for harmonic oscillator energy, and 50% for photon number. These experiments are robust against detector inefficiency and thermal noise.
There is a theorem of Wigner that states that phase-plane distribution functions involving the state bilinearly and having correct marginals must take negative values for certain states. The purpose of this paper is to support the statement that these phase-plane distribution functions are for hardly any state everywhere non-negative. In particular, it is shown that for certain generalized Wigner distribution functions there are no smooth states (except the Gaussians for the Wigner distribution function itself) whose distribution function is everywhere non-negative. This class of Wigner-type distribution functions contains the Margenau–Hill distribution. Furthermore, the argument used in the proof of Wigner’s theorem is augmented to show that under mild conditions one can find for any two states f, g with non-negative distribution functions a linear combination h of f and g whose distribution function takes negative values, unless f and g are proportional.