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arXiv:quant-ph/9801040

Remarks concerning an Entropy-like Quantity $S_q$ especially for Quantum Systems with a Particle Picture; Model with pure Point Spectra

Siegfried Schlieder

Published Jan 20, 1998• 0 citations
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Abstract

For quantum mechanical systems an entropy-like quantity $S_q$ is defined. $S_q$ can differ from the usually defined entropy $S$ and $S_q$ may increase with time for an isolated system. The essential condition for the difference between $S$ and $S_q$ is the assumption that the set {\bf A} of observables which can be represented by a measurement is a proper subset of the set of selfadjoint operators. The underlying idea is made visible in the case of particle systems with non-trivial scattering. The model-character of the reasoning comes from the fact that continuous spectra are replaced by point-spectra. So it seems evident, that no direct connection exists between $S_q$ and the Sinai-Kolmogorov-Entropy at least in this model with pure point-spectra.

DOI
10.48550/arxiv.quant-ph/9801040

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Remarks concerning an Entropy-like Quantity SqS_qSq​ especially for Quantum Systems with a Particle Picture; Model with pure Point Spectra
For quantum mechanical systems an entropy-like quantity SqS_qSq​ is defined. SqS_qSq​ can differ from the usually defined entropy SSS and SqS_qSq​ may increase with time for an isolated system. The essential condition for the difference between SSS and SqS_qSq​ is the assumption that the set {\bf A} of observables which can be represented by a measurement is a proper subset of the set of selfadjoint operators. The underlying idea is made visible in the case of particle systems with non-trivial scattering. The model-character of the reasoning comes from the fact that continuous spectra are replaced by point-spectra. So it seems evident, that no direct connection exists between SqS_qSq​ and the Sinai-Kolmogorov-Entropy at least in this model with pure point-spectra.
A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov-Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as h-+0 is less than or equal to the KS-entropy of the corresponding classical map.
We propose a mechanism by which dynamical randomness may appear in many-body systems without sensitivity to initial conditions. We characterize dynamical randomness using the Kolmogorov-Sinai (KS) entropy per unit time and its 1987 quantum extension by Connes, Narnhofer, and Thirring (CNT). In this respect, quantum systems fall into (at least) two large classes: (1) few-body bounded systems with discrete energy spectra whose entropy per unit time is vanishing; and (2) many-body open systems of statistical mechanics with continuous energy spectra whose entropy per unit time may be positive, like ideal gases and beams of particles. We calculate the classical and quantum entropies per unit time of the ideal gases and show that they are positive and in correspondence. Moreover, we show that the KS entropy per unit time is recovered from the CNT entropy per unit time using a decomposition suggested by the works of Griffiths, Gell-Mann, Hartle, and Omnès on quantum histories. We also discuss about the relationships between the quantum histories, the exponential decay of some n-time correlation functions, repeated measurements, and the CNT entropy per unit time.
We propose a mechanism by which dynamical randomness may appear in many-body systems without sensitivity to initial conditions. We characterize dynamical randomness using the Kolmogorov-Sinai (KS) entropy per unit time and its 1987 quantum extension by Connes, Narnhofer, and Thirring (CNT). In this respect, quantum systems fall into (at least) two large classes: (1) few-body bounded systems with discrete energy spectra whose entropy per unit time is vanishing; and (2) many-body open systems of statistical mechanics with continuous energy spectra whose entropy per unit time may be positive, like ideal gases and beams of particles. We calculate the classical and quantum entropies per unit time of the ideal gases and show that they are positive and in correspondence. Moreover, we show that the KS entropy per unit time is recovered from the CNT entropy per unit time using a decomposition suggested by the works of Griffiths, Gell-Mann, Hartle, and Omnès on quantum histories. We also discuss about the relationships between the quantum histories, the exponential decay of some n-time correlation functions, repeated measurements, and the CNT entropy per unit time.
A new definition of the entropy of a given dynamical system and of an instrument describing the measurement process is proposed within the operational approach to quantum mechanics. It generalizes other definitions of entropy, in both the classical and quantum cases. The Kolmogorov–Sinai (KS) entropy is obtained for a classical system and the sharp measurement instrument. For a quantum system and a coherent states instrument, a new quantity, coherent states entropy, is defined. It may be used to measure chaos in quantum mechanics. The following correspondence principle is proved: the upper limit of the coherent states entropy of a quantum map as ℏ→0 is less than or equal to the KS-entropy of the corresponding classical map. ‘‘Chaos umpire sits, And by decision more imbroils the fray By which he reigns: next him high arbiter Chance governs all.’’ John Milton, Paradise Lost, Book II
It is often stated that the Gibbs ‘‘paradox’’ is resolved or at least alleviated in quantum mechanics because the value of the quantum mechanical entropy of mixing continuously approaches zero if the difference between the particles that are being mixed is made to vanish (i.e., if the value of the inproduct of the respective quantum states goes to one). In this article, the justification of that claim is discussed. It is pointed out that, strictly speaking, an entropy of mixing is not defined if the quantum states of the particles are no longer orthogonal. It seems, therefore, that from a fundamental point of view the Gibbs ‘‘paradox’’ is there no less in quantum mechanics than in classical statistical mechanics.
Generalizing the notion of relative entropy, the difference between a priori and a posteriori relative entropy for quantum systems is drawn. The former, known as quantum relative entropy, is associated with quantum states recognition. The latter -- a posteriori relative quantum entropy is shown to be related with state reconstruction due to the following property: given a density operator ρρρ, ensembles of pure states with Gibbs distribution with respect to the defined distance are proved to represent the initial state ρρρ up to an amount of white noise (completely mixed state) which can be made arbitrary small.