In a previous paper, with a similar title, the authors showed that for bijective (i.e. one-to-one onto) canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of Wigner distribution functions and recover the classical expectation for this kernel in the limit to 0. For non-bijective canonical transformations the situation is more complex as the phase space acquires either a Riemann surface structure or requires the introduction of the concept of an ambiguity group. By discussing the non-bijective canonical transformation taking them from an oscillator of frequency kappa -1 , where kappa is an integer, to one of unit frequency, they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group, i.e. the ambiguity spins. They obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the Riemann surface picture of their phase space. While they illustrate their analysis through the representation in Wigner distribution phase space of a specific non-bijective canonical transformation, the procedure is clearly extendable to the representations of general canonical transformations of this type.
A method for constructing all admissible unitary non-equivalent Wigner quasiprobability distributions providing the Stratonovic-h-Weyl correspondence for an arbitrary N-level quantum system is proposed. The method is based on the reformulation of the Stratonovich–Weyl correspondence in the form of algebraic “master equations” for the spectrum of the Stratonovich–Weyl kernel. The later implements a map between the operators in the Hilbert space and the functions in the phase space identified by the complex flag manifold. The non-uniqueness of the solutions to the master equations leads to diversity among the Wigner quasiprobability distributions. It is shown that among all possible Stratonovich–Weyl kernels for a N=(2j+1)-level system, one can always identify the representative that realizes the so-called SU(2)-symmetric spin-j symbol correspondence. The method is exemplified by considering the Wigner functions of a single qubit and a single qutrit.
A set of quasi-probability distribution functions which give the correct quantum mechanical marginal distributions of position and momentum is studied. The phase-space distribution does not have to be bilinear in the state function. The Wigner distribution is a special case. A general relationship between the phase-space distribution functions and the rule of associating classical quantities to quantum mechanical operators is derived. This allows the writing of correspondence rules at will,of which the ones presently known are particular cases. The dynamics and other properties of the generalized phase-space distribution are considered.
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.