Abstract
We review here the difference between quantum statistical treatments and semiclassical ones, using as the main concomitant tool a semiclassical, shift-invariant Fisher information measure built up with Husimi distributions. Its semiclassical character notwithstanding, this measure also contains abundant information of a purely quantal nature. Such a tool allows us to refine the celebrated Lieb bound for Wehrl entropies and to discover thermodynamic-like relations that involve the degree of delocalization. Fisher-related thermal uncertainty relations are developed and the degree of purity of canonical distributions, regarded as mixed states, is connected to this Fisher measure as well.
Highlights
A quarter of century before Shannon, R.A
In [9] the authors discuss quantum-mechanical phase space distributions expressed in terms of the celebrated coherent states |z⟩ of the harmonic oscillator, eigenstates of the annihilation operator â [14, 15], i.e., â|z⟩ = z|z⟩
Let us consider a system that is specified by a physical parameter θ, while x is a real stochastic variable and fθ (x), which in turn depends on the parameter θ, is the probability density for x
Summary
A quarter of century before Shannon, R.A. Fisher advanced a method to measure the information content of continuous (rather than digital) inputs using not the binary computer codes but the statistical distribution of classical probability theory [1]. FIM has been shown to provide an interesting characterization of the “arrow of time”, alternative to the one associated with Boltzmann’s entropy [7, 8]. The semiclassical approximation (SC) has had a long and distinguished history and remains today a very important weapon in the physics armory. It is indispensable in many areas of scientific endeavor. For the convenience of the reader, we describe some fundamental aspects of the HO canonical-ensemble description from a coherent states’ viewpoint [9], the Husimi probability distribution function, and the Wehrl information measure
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