Fisher Information and Semiclassical Treatments
Abstract
:1. Introduction
2. Background Notions
2.1. HO’s coherent states
2.2. HO-expressions
2.3. Husimi probability distribution
2.4. Wehrl entropy
3. Fisher’s Information Measure
4. Fisher, Thermodynamics’ Third Law, and Thermodynamic Quantities
5. HO-Semiclassical Fisher’s Measure
5.1. MaxEnt approach
5.2. Delocalization
5.3. Second moment of the Husimi distribution
6. Thermodynamics-Like Relations
- part of it originates from excitation energy and
- the remaining is accounted for by phase space delocalization.
7. On Thermal Uncertainties
- from the excitation energy, that supplies a contribution and
- from the delocalization factor D.
8. Degrees of Purity Relations
8.1. Semiclassical purity
8.2. Quantal purity
9. Conclusions
- a connection between Wehrl’s entropy and (cf. Equation (25)),
- an interpretation of as the HO’s ground state occupation probability (cf. Equation (26)),
- an interpretation of proportional to the HO’s specific heat (cf. Equation (30)),
- the possibility of expressing the HO’s entropy as a sum of two terms, one for each of the above FIM realizations (cf. Equation (31)),
- a new form of Heisenberg’s uncertainty relations in Fisher terms (cf. Equation (73)),
- that efficient -estimation can be achieved with at all temperatures, as the minimum Cramer–Rao value is always reached (cf. Equation (24)).
- established that the semiclassical Fisher measure contains all relevant statistical quantum information,
- shown that the Husimi distributions are MaxEnt ones, with the semiclassical excitation energy as the only constraint,
- complemented the Lieb bound on the Wehrl entropy using ,
- observed in detailed fashion how delocalization becomes the counterpart of energy fluctuations,
- written down the difference between the semiclassical and quantal entropy also in terms,
- provided a relation between energy excitation and degree of delocalization,
- shown that the derivative of twice the uncertainty function with respect to is the Planck constant ℏ,
- established a semiclassical uncertainty relation in terms of the semiclassical purity , and
- expressed both and the quantal degree of purity in terms of .
Acknowledgements
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Pennini, F.; Ferri, G.; Plastino, A. Fisher Information and Semiclassical Treatments. Entropy 2009, 11, 972-992. https://doi.org/10.3390/e11040972
Pennini F, Ferri G, Plastino A. Fisher Information and Semiclassical Treatments. Entropy. 2009; 11(4):972-992. https://doi.org/10.3390/e11040972
Chicago/Turabian StylePennini, Flavia, Gustavo Ferri, and Angelo Plastino. 2009. "Fisher Information and Semiclassical Treatments" Entropy 11, no. 4: 972-992. https://doi.org/10.3390/e11040972
APA StylePennini, F., Ferri, G., & Plastino, A. (2009). Fisher Information and Semiclassical Treatments. Entropy, 11(4), 972-992. https://doi.org/10.3390/e11040972