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.
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.
We present a general theory of quasiprobability distributions on phase spaces of quantum systems whose dynamical symmetry groups are (finite-dimensional) Lie groups. The family of distributions on a phase space is postulated to satisfy the Stratonovich - Weyl correspondence with a generalized traciality condition. The corresponding family of the Stratonovich - Weyl kernels is constructed explicitly. In the presented theory we use the concept of generalized coherent states, that brings physical insight into the mathematical formalism.
In this work, the version of Wigner transforms and Wigner functions on discrete systems are formulated. Using frame quantization, the star-product of phase space functions are also computed for a particular finite group. Specifically, the Wigner transform, Wigner function and star-product are computed for the dicyclic group Dien. Frames are the discrete version of coherent states. In the literature, quasiprobability distributions are essential in the study of the phase space representations of quantum mechanics, and are naturally associated to frame quantizations. For this work, the unitary irreducible representations of a finite group play an essential role in the construction.
A classical simulation scheme of quantum computation given a restricted set of states and measurements may be---occasionally, but only occasionally---interpreted naturally as a statistical simulation of positive quasi-probability distributions on a phase space. In this dissertation, we explore phase space representations for finite-dimensional quantum systems and their negativities beyond the usual analogues of the Wigner function. The first line of study focuses on a characterization tool for valid quasi-probability distributions of (possibly mixed) quantum states. A quantum generalization of Bochner's theorem from classical probability theory simultaneously characterizes both the set of valid Wigner functions and the subset of positive ones. We extend this theorem to discrete phase spaces based on projective representations of abelian groups, including the discrete Wigner functions of Gross (J. Math. Phys. 47, 122107 (2006)) and Gibbons et al. (Phys. Rev. A 70, 062101 (2004)). The rest of the dissertation is then dedicated to phase space representations covariant with respect to a general Lie group. This means, in particular, that operators that lie in the same orbit of the group have the same amount of negativity in their quasi-probabilities---a natural requirement if one would like to interpret negativity as non-classicality, and the group action as a classical dynamics. We show that the construction due to Brif and Mann (Phys. Rev. A 59, 971 (1999)) of group covariant phase space representations satisfying the so-called Stratonovich-Weyl correspondence is essentially unique, given the group and an input state in the relevant representation. As an application, we construct quasi-probability representations on a compact phase space of fermionic Gaussian states tailored to a classically simulatable problem of fermionic linear optics and find that, in the first non-trivial case of four fermionic modes, the quasi-probabilities of the Gaussian states exhibit a considerable amount of negativity.
Since the very early days of quantum theory there have been numerous attempts to interpret quantum mechanics as a statistical theory. This is equivalent to describing quantum states and ensembles together with their dynamics entirely in terms of phase-space distributions. Finite dimensional systems have historically been an issue. In recent works [Phys. Rev. Lett. 117, 180401 (2016) and Phys. Rev. A 96, 022117 (2017)] we presented a framework for representing any quantum state as a complete continuous Wigner function. Here we extend this work to its partner function---the Weyl function. In doing so we complete the phase-space formulation of quantum mechanics---extending work by Wigner, Weyl, Moyal, and others to any quantum system. This work is structured in three parts. First we provide a brief modernized discussion of the general framework of phase-space quantum mechanics. We extend previous work and show how this leads to a framework that can describe any system in phase space---putting it for the first time on a truly equal footing to Schr\"odinger's and Heisenberg's formulation of quantum mechanics. Importantly, we do this in a way that respects the unifying principles of ``parity'' and ``displacement'' in a natural broadening of previously developed phase-space concepts and methods. Secondly we consider how this framework is realized for different quantum systems; in particular we consider the proper construction of Weyl functions for some example finite dimensional systems. Finally we relate the Wigner and Weyl distributions to statistical properties of any quantum system or set of systems.