For a symmetric
N-quDit system described by a density matrix
ρ, we construct a one-parameter
s family
Fρ(s) of quasi-probability distributions through generalized Fano multipole operators and Stratonovich-Weyl kernels. The corresponding phase space is the complex projective
CPD−1=U(D)/U(D−1)×U(1), related to fully symmetric irreducible representations of the unitary group
U(D). For the particular cases
D=2 (qubits) and
D=3 (qutrits), we analyze the phase-space structure of Schrödinger
U(D)-spin cat (parity adapted coherent) states and we provide plots of the corresponding Wigner
Fρ(0) function. We examine the connection between non-classical behavior and the negativity of the Wigner function. We also compute the generalized heat kernel relating two quasi-probability distributions
Fρ(s) and
Fρ(s′), with
t=(s′−s)/2 playing the role of ``time'', together with their twisted Moyal product in terms of a trikernel. In the thermodynamic limit
N→∞, we recover the usual Gaussian smoothing for
s′>s. A diagramatic interpretation of the phase-space construction in terms of Young tableaux is also provided.
Husimi distributions [11] and Wigner distributions [20] are well-known quasi-probability distributions which appear in several contexts. In this paper, we show some remarkable aspects of these distribution functions related to geometric structures of generalized coherent state systems [15,16] and operational quantum physics [7], and a scheme of formulating generalized version of quasi-probability distributions. Our scheme gives concrete formulae of quasi-probability distributions in more direct way from the theory of coherent state systems and clarify their operational meanings, especially of Husimi distributions and mutual relation between Husimi distributions and other classes of quasi-probability distributions.
A set of orthonormalized and complete functions of two real variables in complex representation involving Laguerre polynomials as a substantial part is introduced and is referred to as the set of Laguerre two-dimensional (2D) functions. The properties of the set of Laguerre 2D-functions are discussed and the Fourier and Radon transforms of these functions are calculated. The Laguerre 2D-functions form a basis for a realization of the five-dimensional Lie algebra to the Heisenberg-Weyl group for a two-mode system. Real representation of this set of functions by a sum over products of Hermite functions involving Jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between Laguerre and Hermite polynomials in both directions. The set of Laguerre 2D-functions is the most appropriate set of functions for the Fock-state representation of quasi-probabilities in quantum optics. The Wigner quasi-probability in Fock-state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding Laguerre 2D-function. The properties of orthonormality and completeness of the Laguerre 2D-functions provide the Fock-state matrix elements of the density operator directly from the quasi-probabilities. The Perina-Mista representation of the Glauber-Sudarshan quasi-probability can be represented with advantage by the Laguerre 2D-functions. The Fock-state matrix elements of the displacement operator and the scalar product of displaced Fock states are closely related to Laguerre 2D-functions.
Abstract In this paper we address the definition and explicit computation of quasi-probability functions, such as the Wigner function, for the discrete systems with SU(D) symmetry. To do so, we introduce the concept of coherent superstates and super-frames, that gives a natural interpretation to the Stratonovich-Weyl approach to phase-space quantum mechanics. The expressions that we obtain are compatible with previous results while they are more general and give an insight into how this problem might be solved for more complex cases. In addition, time evolution for quasi-probability functions is addressed using the aforementioned formalism and some examples are given.