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Physical Review A

Tsallis entropy in phase-space quantum mechanics

Parvin Sadeghi, Siamak Khademi, Amir H. Darooneh

Published Jul 26, 2012• 19 citations
Publisher

Abstract

In this paper we define the quantum version of the Tsallis entropy in terms of quantum phase space distribution functions. The quantum Tsallis entropy is compared with Kenfack's nonclassicality indicator, for different systems, such as the Schr\"odinger cat state, the thermal state, a superposition of the ground and the first excited number states, and the harmonic oscillator state. These comparisons indicate that the Wigner representation gives us complete information about the state with the nonextensivity parameter $q=1$, while the Husimi representation hides some information with the nonextensivity parameter $q<1$.

DOI
10.1103/physreva.86.012119

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In this paper we define the quantum version of the Tsallis entropy in terms of quantum phase space distribution functions. The quantum Tsallis entropy is compared with Kenfack's nonclassicality indicator, for different systems, such as the Schr"odinger cat state, the thermal state, a superposition of the ground and the first excited number states, and the harmonic oscillator state. These comparisons indicate that the Wigner representation gives us complete information about the state with the nonextensivity parameter q=1q=1q=1, while the Husimi representation hides some information with the nonextensivity parameter q<1q<1q<1.
Benedict et al. and Kenfack et al. advocated nonclassicality indicators based on the measurement of negativity of the Wigner distribution functions. These indicators have some applications in quantum mechanics and quantum optics. In this paper we define a nonclassicality indicator in terms of the interference in phase space, which is applicable to some real distribution functions including those of Wigner. As a special case one may reproduce the previous results using our indicator for the Wigner distribution functions. This indicator is examined for cases of the Schr\"odinger cat state and the thermal states and the results are compared with those obtained by previous methods. It seems that the physical behavior of nonclassicality indicators originates in the uncertainty principle. This is shown by an onto correspondence between these indicators and the uncertainty principle.
The Wehrl information entropy and its phase density, the so-called Wehrl phase distribution, are applied to describe Schr\\"odinger cat and cat-like (kitten) states. The advantages of the Wehrl phase distribution over the Wehrl entropy in a description of the superposition principle are presented. The entropic measures are compared with a conventional phase distribution from the Husimi Q-function. Compact-form formulae for the entropic measures are found for superpositions of well-separated states. Examples of Schr\\"odinger cats (including even, odd and Yurke-Stoler coherent states), as well as the cat-like states generated in Kerr medium are analyzed in detail. It is shown that, in contrast to the Wehrl entropy, the Wehrl phase distribution properly distinguishes between different superpositions of unequally-weighted states in respect to their number and phase-space configuration.
The Wehrl information entropy and its phase density, the so-called Wehrl phase distribution, are applied to describe Schrödinger cat and cat-like (kitten) states. The advantages of the Wehrl phase distribution over the Wehrl entropy in a description of the superposition principle are presented. The entropic measures are compared with a conventional phase distribution from the Husimi Q -function. Compact-form formulae for the entropic measures are found for superpositions of well separated states. Examples of Schrödinger cats (including even, odd and Yurke-Stoler coherent states), as well as the cat-like states generated in the Kerr medium, are analysed in detail. It is shown that, in contrast to the Wehrl entropy, the Wehrl phase distribution properly distinguishes between different superpositions of unequally weighted states with respect to their number and phase-space configuration.
Many non-classicality indicators are used to measure quantum effects of different systems. Kenfack's and Sadeghi's non-classicality indicators are introduced in terms of the amount of Wigner function's negativities and interferences in phase space quantum mechanics, respectively. They are, effectively, applied for some real distribution functions. In this paper, we compared these non-classicality indicators for the entangled state of photonic number states in the Wigner, Husimi and Rivier representations. It is shown that for a two-level entangled state, Sadeghi's indicator has more benefits with respect to the Kenfack's indicator. For the two-level entangled state, we show a correspondence between the Sadeghi's non-classicality indicator and the Von Neumann entropy. It is also shown that for the superposition of squeezed number state the Sadeghi's (and no Kenfack's) non-classicality indicators is sensible with respect to the squeezing parameter for a superposition of squeezed number states.
The local structure present in Wigner and Husimi phase-space distributions and their marginals are studied and quantified via information-theoretic quantities. Shannon, R\'enyi, and cumulative residual entropies of the Wigner and Husimi distributions are examined in the ground and excited states of a harmonic oscillator. The entropies of the Wigner function marginals are lower than the corresponding entropies of the Husimi function marginals, which illustrates how the nodal structure present in the Wigner function is lost upon consideration of the Husimi function. Shannon and cumulative residual entropies of the Wigner function yield entropies which are complex valued. Absolute values and real components of these quantities increase with quantum number, displaying a behavior which is consistent with their real-valued Husimi function counterparts. This comportment is also similar to the real-valued R\'enyi entropies of the Wigner function. The entropies of the Wigner function are observed to be lower than the corresponding Husimi function ones, in agreement with the results for the marginals. The real components of the Wigner function entropies are seen to be closer to the uncertainty relation bound compared to the corresponding Husimi function entropies. These real components are also closer to the bound when contrasted to the entropic sum of the marginal densities. The R\'enyi or collision entropy of the Wigner function sits exactly on the bound. Related statistical correlation measures show that the position-momentum correlation is larger in the Wigner function compared to the Husimi function and increases with quantum number.