# Fisher Information and Semiclassical Treatments

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## Abstract

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## 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

**x**is a real stochastic variable and ${f}_{\theta}(\mathbf{x})$, which in turn depends on the parameter θ, is the probability density for $\mathbf{x}$. An observer makes a measurement of $\mathbf{x}$ and estimates θ from this measurement, represented by $\tilde{\theta}=\tilde{\theta}(\mathbf{x})$. One wonders how well θ can be determined. Estimation theory [26] asserts that the best possible estimator $\tilde{\theta}(\mathbf{x})$, after a very large number of $\mathbf{x}$-samples is examined, suffers a mean-square error ${e}^{2}$ from θ that obeys a relationship involving Fisher’s I, namely, $I{e}^{2}=1$, where the Fisher information measure I is of the form

**Figure 1.**Fisher (${I}_{\tau}$) and Wehrl (W) information measures vs. T (in $\hslash \omega $ units) for HO-Husimi distribution.

## 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

**Figure 2.**Numerical computation results for the HO: changes $\mathrm{d}U$ and $\mathrm{d}{I}_{\tau}$ vs. $\mathrm{d}S$ that ensue after randomly generating variations $\delta {p}_{i}$ in the underlying microscopic canonical probabilities ${p}_{i}$.

- 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 $C/\omega $ contribution and
- from the delocalization factor D.

## 8. Degrees of Purity Relations

#### 8.1. Semiclassical purity

#### 8.2. Quantal purity

## 9. Conclusions

- an interpretation of ${I}_{\tau}$ as the HO’s ground state occupation probability (cf. Equation (26)),
- 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 $\left|z\right|$-estimation can be achieved with ${I}_{\tau}$ at all temperatures, as the minimum Cramer–Rao value is always reached (cf. Equation (24)).

- established that the semiclassical Fisher measure ${I}_{\tau}$ contains all relevant statistical quantum information,
- shown that the Husimi distributions are MaxEnt ones, with the semiclassical excitation energy $\mathcal{E}$ as the only constraint,
- complemented the Lieb bound on the Wehrl entropy using ${I}_{\tau}$,
- observed in detailed fashion how delocalization becomes the counterpart of energy fluctuations,
- written down the difference $W-S$ between the semiclassical and quantal entropy also in ${I}_{\tau}-$terms,
- provided a relation between energy excitation and degree of delocalization,
- shown that the derivative of twice the uncertainty function $F(\beta \omega )=\Delta x\Delta p$ with respect to ${I}_{\tau}^{-1}$ is the Planck constant ℏ,
- established a semiclassical uncertainty relation in terms of the semiclassical purity ${d}_{\mu}$, and
- expressed both ${d}_{\mu}$ and the quantal degree of purity in terms of ${I}_{\tau}$.

## 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

**AMA Style**

Pennini F, Ferri G, Plastino A. Fisher Information and Semiclassical Treatments. *Entropy*. 2009; 11(4):972-992.
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**Chicago/Turabian Style**

Pennini, Flavia, Gustavo Ferri, and Angelo Plastino. 2009. "Fisher Information and Semiclassical Treatments" *Entropy* 11, no. 4: 972-992.
https://doi.org/10.3390/e11040972