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Physical Review A

Minimum uncertainty products from the principle of maximum entropy

A. K. Rajagopal, S. Teitler

Published Jul 1, 1989• 2 citations
Publisher

Abstract

The maximum-entropy method is here generalized to obtain many possible extrema of the uncertainty product corresponding to the generalized minimum uncertainty products recently discussed by Lahiri and Menon (LM) [Phys. Rev. A 38, 5412 (1988)]. Unlike the LM work, the present work applies to mixed states and leads to a new annealing algorithm for obtaining the extrema of the entropy functional.

DOI
10.1103/physreva.40.515

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The maximum-entropy method is here generalized to obtain many possible extrema of the uncertainty product corresponding to the generalized minimum uncertainty products recently discussed by Lahiri and Menon (LM) [Phys. Rev. A 38, 5412 (1988)]. Unlike the LM work, the present work applies to mixed states and leads to a new annealing algorithm for obtaining the extrema of the entropy functional.
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We show how to determine the maximum and minimum possible values of one measure of entropy for a given value of another measure of entropy. These maximum and minimum values are obtained for two standard forms of probability distribution (or quantum state) independent of the entropy measures, provided the entropy measures satisfy a concavity/convexity relation. These results may be applied to entropies for classical probability distributions, entropies of mixed quantum states and measures of entanglement for pure states.
Local minimum solutions to Heisenberg's uncertainty principle are studied in detail. Of particular concern is obtaining an optimal procedure to squeeze the uncertainty in one canonical operator while introducing, in a well-defined mathematical sense, the least possible increase in the uncertainty of the relevant conjugate operator. The result is a hierarchy of ``minimum uncertainty $(\ensuremath{\mu}\ensuremath{-})$ wavelets,'' which are intimately associated with the ``Hermite distributed approximating functionals'' [D. K. Hoffman and D. J. Kouri, Phys. Rev. Lett. 85, 5263 (2000)]. The $\ensuremath{\mu}$ wavelets can be used with any quantum state to squeeze the uncertainty of a particular observable. Several specific examples are given and some possible applications are discussed.
In the framework of the recent quest for well-behaved nonlinear extensions of the traditional Schrödinger-von Neumann unitary dynamics that could provide fundamental explanations of recent experimental evidence of loss of quantum coherence at the microscopic level, a recent paper [Gheorghiu-Svirschevski 2001 Phys. Rev. A 63 054102] reproposes the nonlinear equation of motion proposed by the present author [see Beretta G P 1987 Found. Phys. 17 365 and references therein] for quantum (thermo)dynamics of a single isolated indivisible constituent system, such as a single particle, qubit, qudit, spin or atomic system, or a Bose-Einstein or Fermi-Dirac field. As already proved, such nonlinear dynamics entails a fundamental unifying microscopic proof and extension of Onsager's reciprocity and Callen's fluctuation-dissipation relations to all nonequilibrium states, close and far from thermodynamic equilibrium. In this paper we propose a brief but self-contained review of the main results already proved, including the explicit geometrical construction of the equation of motion from the steepest-entropy-ascent ansatz and its exact mathematical and conceptual equivalence with the maximal-entropy-generation variational-principle formulation presented in Gheorghiu-Svirschevski S 2001 Phys. Rev. A 63 022105. Moreover, we show how it can be extended to the case of a composite system to obtain the general form of the equation of motion, consistent with the demanding requirements of strong separability and of compatibility with general thermodynamics principles. The irreversible term in the equation of motion describes the spontaneous attraction of the state operator in the direction of steepest entropy ascent, thus implementing the maximum entropy production principle in quantum theory. The time rate at which the path of steepest entropy ascent is followed has so far been left unspecified. As a step towards the identification of such rate, here we propose a possible, well-behaved and intriguing, general closure of the dynamics, compatible with the nontrivial requirements of strong separability. Based on the time–energy Heisenberg uncertainty relation, we derive a lower bound to the internal-relaxation-time functionals that determine the rate of entropy generation. This bound entails an upper bound to the rate of entropy generation. By this extreme maximal-entropy-generation-rate ansatz, each indivisible subsystem follows the direction of steepest locally perceived entropy ascent at the highest rate compatible with the time– energy uncertainty principle.