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arXiv:quant-ph/0406015 · Journal of Optics B Quantum and Semiclassical Optics

Negativity of the Wigner function as an indicator of non-classicality

Anatole Kenfack, Karol yczkowski

Published Aug 24, 2004• 778 citations
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Abstract

A measure of non-classicality of quantum states based on the volume of the negative part of the Wigner function is proposed. We analyse this quantity for Fock states, squeezed displaced Fock states and cat-like states defined as coherent superposition of two Gaussian wavepackets.
DOI
10.1088/1464-4266/6/10/003

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Two measures of nonclassicality, the entanglement potential and the negativity of the Wigner distribution function defined by the volume of its negative domains, are compared based on an investigation of the nonclassicality for Fock states and Schrödinger cat states in a decoherence process. Both the entanglement potential and the total negative probability are reduced in the linear loss process and the partial negative distribution of the Wigner function is wiped out for large losses while the entanglement potential is always positive. We give a bound condition and find that, though not yet mathematically proven in general, the upper bound of 50% is the maximum allowed loss for the survival of the negative distribution of the Wigner function.
We introduce an operational criterion to identify Wigner function (WF) negativity for an arbitrary quantum state within the framework of quantum non-demolition measurements. This criterion corresponds to experimentally accessible schemes that enable a direct measurement of the WF, and establishes the coherent-state basis as a privileged basis for determining when the WF exhibits negative regions. We show that the absence (presence) of coherent superpositions in the coherent-state basis provides direct information about the positivity (negativity) of the WF. In particular, the absence of such superpositions constitutes a sufficient condition for WF positivity. Although a general proof of necessity remains elusive, we demonstrate that this condition is also necessary in two relevant cases: Schrödinger-cat states and higher-order cat states on a circle. More precisely, for Schrödinger-cat states we establish a necessary and sufficient condition for the positivity of the WF in full generality, whereas for high-order cat states on a circle we derive an analogous condition in the limit of a large number of densely packed coherent states.
In this paper, a quantitative investigation of the non-classical and quantum non-Gaussian characters of the photon-subtracted displaced Fock state [Formula: see text], where k is the number of photons subtracted and n is Fock parameter, is performed by using a collection of measures like Wigner logarithmic negativity, linear entropy potential, skew information-based measure, and relative entropy of quantum non-Gaussianity. It is noticed that the number of photons subtracted ( k) changes the nonclassicality and quantum non-Gaussianity in a significant amount in the regime of small values of the displacement parameter, whereas the Fock parameter ( n) presents a notable change in the large regime of the displacement parameter. In this respect, the role of the Fock parameter is found to be stronger as compared to the photon subtraction number. Finally, the Wigner function dynamics considering the effects of photon loss channel is used to show that the Wigner negativity can only be detected by highly efficient detectors.
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.