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Physical Review A

Nonclassicality indicator for the real phase-space distribution functions

Parvin Sadeghi, Siamak Khademi, S. Nasiri

Published Jul 12, 2010• 21 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.82.012102

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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.

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Comment on “Nonclassicality indicator for the real phase-space distribution functions”

Anatole Kenfack

We show in this Comment that the nonclassicality indicator proposed in Sadeghi et al. [Phys. Rev. A 82, 012102 (2010)] is controversial. Defined in terms of the interferences displayed by the quasidistribution function of at least two superposition states, this indicator may logically lead to zero for any single quantum state irrespective to its nonclassicality, in contradiction with their results.

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Non-classicality indicators are widely applied to describe the physical properties of quantum states. One category of these indicators is defined in terms of negativities portion of Wigner distribution function. These indicators are not indicating to some quantum properties, e.g.: squeezing. Because the squeezing operations don’t effect on, the amount of negativities of Wigner distribution functions. Therefore the non-classicality indicators, e.g.: Kenfak or Benedict, which are working based on the negativities of distribution functions, cannot recognize the squeezing properties. In this paper a non-classicality indicator is introduced, in terms of the radial and angular distribution of absolute value of Wigner distribution function, which can recognize the squeezing properties, e.g.: squeezing parameter and squeezing angle. This non-classicality indicator is applied for the squeezed photonic states. We show that this indicator can evaluate the number of photons as well as the squeezing parameter and squeezing angle.

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Reply to “Comment on ‘Nonclassicality indicator for the real phase-space distribution functions’ ”

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Sadeghi et al. [Phys. Rev. A 82, 012102 (2010)] presented a new nonclassicality indicator which works properly for Husimi and Rivier distribution functions as well as the Wigner distribution function. Some objections on the Sadeghi nonclassicality indicator are presented by Kenfack (preceding Comment, ``Nonclassicality indicator for the real phase-space distribution functions'', Phys. Rev. A 93, 036101 (2016). In this Reply, some misconceptions on Sadeghi's nonclassicality indicator are discussed and clarified.

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We propose a hierachy of nonclassicality criteria in phase space. Our formalism covers the negativity in phase space as a special case and further adresses nonclassicality for quantum states with positive phase-space distributions. Remarkably, it enables us to detect every nonclassical Gaussian state and every finite dimensional state in Fock basis by looking into only three phase-space points. Furthermore, our approach provides an experimentally accessible lower bound for the nonclassicality measure based on trace distance. We also extend our method to detecting genuine quantum non-Gaussianity of a state with a non-negative Wigner function. We finally establish our formalism by employing generalized quasiprobability distributions to demonstrate its power for a practical test using an on-off detector array.

•May 12, 2020•1 citation
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.
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.
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.
Non-classicality indicators are widely applied to describe the physical properties of quantum states. One category of these indicators is defined in terms of negativities portion of Wigner distribution function. These indicators are not indicating to some quantum properties, e.g.: squeezing. Because the squeezing operations don’t effect on, the amount of negativities of Wigner distribution functions. Therefore the non-classicality indicators, e.g.: Kenfak or Benedict, which are working based on the negativities of distribution functions, cannot recognize the squeezing properties. In this paper a non-classicality indicator is introduced, in terms of the radial and angular distribution of absolute value of Wigner distribution function, which can recognize the squeezing properties, e.g.: squeezing parameter and squeezing angle. This non-classicality indicator is applied for the squeezed photonic states. We show that this indicator can evaluate the number of photons as well as the squeezing parameter and squeezing angle.
Sadeghi et al. [Phys. Rev. A 82, 012102 (2010)] presented a new nonclassicality indicator which works properly for Husimi and Rivier distribution functions as well as the Wigner distribution function. Some objections on the Sadeghi nonclassicality indicator are presented by Kenfack (preceding Comment, ``Nonclassicality indicator for the real phase-space distribution functions'', Phys. Rev. A 93, 036101 (2016). In this Reply, some misconceptions on Sadeghi's nonclassicality indicator are discussed and clarified.
We propose a hierachy of nonclassicality criteria in phase space. Our formalism covers the negativity in phase space as a special case and further adresses nonclassicality for quantum states with positive phase-space distributions. Remarkably, it enables us to detect every nonclassical Gaussian state and every finite dimensional state in Fock basis by looking into only three phase-space points. Furthermore, our approach provides an experimentally accessible lower bound for the nonclassicality measure based on trace distance. We also extend our method to detecting genuine quantum non-Gaussianity of a state with a non-negative Wigner function. We finally establish our formalism by employing generalized quasiprobability distributions to demonstrate its power for a practical test using an on-off detector array.