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.
Starting from Wigner's definition of the function named now after him we systematically develop different representation of this quasiprobability with emphasis on symmetric representations concerning the canonical variables ( ) , q p of phase space and using the known relation to the parity operator.One of the representations is by means of the Laguerre 2D polynomials which is particularly effective in quantum optics.For the coherent states we show that their Fourier transforms are again coherent states.We calculate the Wigner quasiprobability to the eigenstates of a particle in a square well with infinitely high impenetrable walls which is not smooth in the spatial coordinate and vanishes outside the wall boundaries.It is not well suited for the calculation of expectation values.A great place takes on the calculation of the Wigner quasiprobability for coherent phase states in quantum optics which is essentially new.We show that an unorthodox entire function plays there a role in most formulae which makes all calculations difficult.The Wigner quasiprobability for coherent phase states is calculated and graphically represented but due to the involved unorthodox function it may be considered only as illustration and is not suited for the calculation of expectation values.By another approach via the number representation of the states and using the recently developed summation formula by means of Generalized Eulerian numbers it becomes possible to calculate in approximations with good convergence the basic expectation values, in particular, the basic uncertainties which are additionally represented in graphics.Both considered examples, the square well and the coherent phase states, belong to systems with ( ) 1,1 SU symmetry with the same index 1 2 k = of unitary irreducible representations.
Quadrature operators with arbitrary phase are studied from a point of view of the phase-space representation of quantum states, and the results are applied to simultaneous measurement and quantum communication. The Wigner function of arbitrary phase quadrature variables is introduced, which is a generalization of the usual Wigner function of position and momentum. The Kirkwood distribution is also extended for arbitrary phase quadrature variables. The simultaneous measurement of two quadrature operators is investigated using a beam splitter model and a generalized version of the Arthurs-Kelley model. The quantum teleportation of continuous variables is considered in terms of arbitrary phase quadrature variables. A general formula is derived that provides the quantum teleportation channel. The fidelity of the quantum teleportation with an uncontrollable phase is calculated for a coherent state. Furthermore, the mutual information of the quantum dense coding of continuous variables is obtained when classical information is encoded on arbitrary phase quadrature variables. The result is compared with that of the communication system, where information is transmitted using coherent and squeezed states.
We propose the reconstruction of a smoothed Wigner function from quadrature distributions instead of the Wigner function itself to avoid the problems with the inverse Radon transformation. Starting from the well known relation between s -parametrized quasiprobability distributions W ( x , p , s ) and quadrature component distributions w ( x , ), we show that the sampling function needed to determine a smoothed Wigner function W ( x , p ,-| s |) is basically a shifted and scaled version of the well known pattern function necessary to reconstruct the probability of finding no photon in the signal field. An analogous result has been recently found in the literature. Here we obtain this result in a very concise and alternative way.
In this article, we introduce a quasiprobability distribution of work that is based on the Wigner function. This proposal rests on the idea that the work conducted on an isolated system can be coherently measured by coupling the system to a quantum measurement apparatus. In this way, a quasiprobability distribution of work can be defined in terms of the Wigner function of the apparatus. This quasidistribution contains the information of the work statistics and also holds a clear operational definition that can be directly measured in a real experiment. Moreover, it is shown that the presence of quantum coherence in the energy eigenbasis is related with the appearance of features related to non-classicality in the Wigner function such as negativity and interference fringes. On the other hand, from this quasiprobability distribution, it is straightforward to obtain the standard two-point measurement probability distribution of work and also the difference in average energy for initial states with coherences.