The Wigner distribution function is one of the pillars of the phase space formulation of quantum mechanics. Its original formulation may be cast in terms of the unitary representations of the Weyl–Heisenberg group. Following the construction proposed by Wolf and coworkers in constructing the Wigner functions for general Lie groups using the irreducible unitary representations of the groups, we develop here the Wigner functions and Weyl operators on the Euclidean motion group of rank three. We give complete derivations and proofs of their important properties.
In 1981, Agarwal proposed a Wigner quasiprobability distribution function on the group SU(2) that serves to analyze two-particle spin states on a sphere. Recent work by our group has included the definition of an apparently distinct Wigner function on generic Lie groups whose natural range has the dimension of the group and serves for all square-integrable representations; for the SO(3) case this entails a three-dimensional ``meta-phase'' space. Both have the fundamental properties covariance and completeness. Here we show how the former is obtained as a restriction of the latter.
The general problem of describing quantum states not only by wave functions and density operators but also by probability distribution functions is discussed in this paper. The qubit-state density-matrix elements expressed in terms of probability distributions are connected with the irreducible representation of group SU(2). The transform of the probability distributions corresponding to the unitary transform of the qubit state of the spin-1/2 system is obtained and expressed in terms of matrix elements of the group SU(2). The possibility to extend the introduced formalism to other quantum states is suggested. For qubit systems, the notion of density operators algebra and the relation with the probability representation of quantum states are discussed. The Schrödinger equation for qubit states is obtained as an equation for probabilities.
The Weyl-Wigner transformation enables one to construct a representation of quantum motion equations using functions in phase space. The states of the system are determined bv the quasi-probability distribution in phase space and the motion is described by an orthogonal integral operator. This formalism is employed in the study of the classical meaning of the discrete symmetry groups, in the problem of defining the current probability and in the proving of the expression of cross section from the S-matrix. (GE)
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.
This paper aims to explore the inherent connection between Heisenberg groups, quantum Fourier transform (QFT) and (quasi-probability) distribution functions. Distribution functions for continuous and finite quantum systems are examined from three perspectives and all of them lead to Weyl–Gabor–Heisenberg groups. The QFT appears as the intertwining operator of two equivalent representations arising out of an automorphism of the group. Distribution functions correspond to certain distinguished sets in the group algebra. The marginal properties of a particular class of distribution functions (Wigner distributions) arise from a class of automorphisms of the group algebra of the Heisenberg group. We then study the reconstruction of the Wigner function from the marginal distributions via inverse Radon transform giving explicit formulae. We consider some applications of our approach to quantum information processing and quantum process tomography.