Employing the extended phase states, we show that six conditions analogous to Wigner's original ones cannot lead uniquely to the number-phase Wigner function. To show this fact explicitly, we propose a new example of the Wigner function satisfying all these conditions. The nonuniqueness of the number-phase Wigner function results from the 2π-periodicity of the phase. It is also shown that the two number-phase Wigner functions obtained by several authors have an endpoint problem, which leads to an ambiguous result. Their correct integral forms are derived from those defined in the extended Fock space with negative number states.
We propose a unified method for obtaining the number-phase distribution functions in the extended Fock space and apply it to various distribution functions. A physical number-phase distribution function is easily obtained; it is the expectation value of an extended Wigner operator in a physical state. This method corresponds to Cohen's method for the position-momentum Wigner functions. The properties of the number-phase distribution functions and their relations are presented in a unified manner. Also, through the distribution functions, the correspondence between classical functions and quantum operators is obtained explicitly.
It is shown that the number-phase (or rotational) Wigner function is obtained from the Weyl symmetrization rule for the correspondence between classical functions and quantum operators. In spite of the complicated form of the commutator for the number and phase operators, the Weyl symmetrization rule is similar to that in the case of the position and momentum operators. In addition, it is found that the ordering of the number and phase operators also has the same structure as that for the position-momentum pair.
Various quasiprobability distributions have been developed in the past using the Hilbert space of the single-mode light field. The development of a quasiprobability distribution associated with a phase operator has previously been impossible because of the absence of a unique Hermitian phase operator defined on the Hilbert space. Recently, however, Pegg and Barnett [Europhys. Lett. 6, 483 (1988); Phys. Rev. A 39, 1665 (1989)] and Barnett and Pegg [J. Mod. Optics 36, 7 (1989)] introduced a new formalism that does allow the construction of a Hermitian phase operator and associated phase eigenstates. In this paper we develop a quasiprobability distribution associated with the number and phase operators of the single-mode light field in the new formalism. The new distribution, which we call the number-phase Wigner function, has properties analogous to the Wigner function. We also derive the number-phase Wigner representation of number states, phase states, general physical states, coherent states, and the squeezed vacuum. We find this new representation has features that are related to the number and phase properties of states. For example, the number-phase Wigner representation of a number state is nonzero only on a circle, while the representation of a phase state is only nonzero along a radial line.
We present further progress, in the form of analytical results, on the Wigner entropy conjecture set forth by Van Herstraeten and Cerf [Phys. Rev. A 104, 042211 (2021)] and Hertz et al. [J. Phys. A: Math. Theor. 50, 385301 (2017)]. Said conjecture asserts that the differential entropy defined for non-negative, yet physical, Wigner functions is minimized by pure Gaussian states while the minimum entropy is equal to $1+ln\ensuremath{\pi}$. We prove this conjecture for the qubits formed by Fock states $|0\ensuremath{\rangle}$ and $|1\ensuremath{\rangle}$ that correspond to non-negative Wigner functions. In particular, we derive an explicit form of the Wigner entropy for those states lying on the boundary of the set of Wigner non-negative qubits. We then consider general mixed states and derive a sufficient condition for the conjecture's validity. Lastly, we elaborate on the states which are in accordance with our condition.
We define a quasiprobability distribution
SNP(n,\ensuremath{\theta}) which describes the quantum statistics of the photon number and phase observables of a single-mode field (or, equivalently, a harmonic oscillator). The properties of
SNP(n,\ensuremath{\theta}) are the photon number and phase analogies of the properties of Wigner's original function, which describes the position and momentum observables. For example, the marginals of
SNP(n,\ensuremath{\theta}) are the continuous phase and the discrete photon-number probability distributions. We give examples of the
SNP(n,\ensuremath{\theta}) representation of various states and show, in particular, that
SNP(n,\ensuremath{\theta}) displays the quantum interference associated with Schr\"odinger cat states. We also describe how
SNP(n,\ensuremath{\theta}) can be determined from quantities that are, in principle, measurable.