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arXiv:quant-ph/0701015

Classical States and Their Quantum Correspondence

Itay Hen, Amir Kalev

Published Jan 3, 2007• 1 citation
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Abstract

We point out a correspondence between classical and quantum states, by showing that for every classical distribution over phase--space, one can construct a corresponding quantum state, such that in the classical limit of $\hbar\to 0$ the latter converges to the former with respect to all measurable quantities.

DOI
10.48550/arxiv.quant-ph/0701015

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We point out a correspondence between classical and quantum states, by showing that for every classical distribution over phase--space, one can construct a corresponding quantum state, such that in the classical limit of ℏ→0\hbar\to 0ℏ→0 the latter converges to the former with respect to all measurable quantities.
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We study the problem of correspondence between classical and quantum statistical models. We show that (contrary to a rather common opinion) it is possible to construct a natural pre-quantum classical statistical model. The crucial point is that such a pre-quantum classical statistical model is not the conventional classical statistical mechanics on the phase space R 2 n , but its infinite-dimensional analogue. Here the phase space Ω = H × H , where H is the (real separable) Hilbert space. The classical → quantum correspondence is based on the Taylor expansion of classical physical variables—maps f :Ω → R . The space of classical statistical states consists of Gaussian measures on Ω having zero mean value and dispersion ≈ h . The quantum statistical model is obtained as the lim h →0 of the classical one. All quantum states including so-called ‘pure states’ (wavefunctions) are simply Gaussian fluctuations of the ‘vacuum field’, ω = 0 ∊ Ω, having dispersions of the Planck magnitude.
The information provided by a classical measurement is unambiguously determined by the mutual information between the output results and the measured quantity. However, quantum mechanically there are at least two notions of information gathering which can be considered, one characterizing the information provided about the initial preparation, useful in communication, and the other characterizing the information about the final state, useful in state-preparation and control. Here we are interested in understanding the properties of these measures, and the information gathering capacities of quantum and classical measurements. We provide a partial answer to the question `in what sense does information gain increase with initial uncertainty?' by showing that, for classical and quantum measurements which are symmetric with respect to reversible transformations of the state space, the information gain regarding the initial state always increases with the observer's initial uncertainty. In addition, we calculate the capacity of all unitarily covariant and commutative permutation-symmetric measurements for obtaining classical information. While it is the von Neumann entropy of the effects which appears in the latter capacity, it is the subentropy which appears in the expression for the former.
We formulate various semiclassical propagators for the Wigner phase space representation from a unified point of view. As is shown in several studies, the Moyal equation, which is an equation of motion for the Wigner distribution function, can be regarded as the Schrödinger equation of an extended Hamiltonian system where its "position" and "momentum" correspond to the middle point of two points of the original phase space and the difference between them, respectively. Then we show that various phase-space semiclassical propagators can be formulated just by applying existing semiclassical propagators to the extended system. As a result, a phase space version of the Van Vleck propagator, the initial-value Van Vleck propagator, the Herman-Kluk propagator, and the thawed Gaussian approximation are obtained. In addition, we numerically compare the initial-value phase-space Van Vleck propagator, the phase-space Herman-Kluk propagator, and the classical mechanical propagation as approximation methods for the time propagation of the Wigner distribution function in terms of both accuracy and convergence speed. As a result, we find that the convergence speed of the Van Vleck propagator is far slower than others as is the case of the Hilbert space, and the Herman-Kluk propagator keeps its accuracy for a long period compared with the classical mechanical propagation while the convergence speed of the latter is faster than the former.