Discrete Moyal-type Representations for a Spin

Abstract

In Moyal's formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e. by smooth functions on a two-dimensional sphere. Such prescriptions to associate operators with Wigner functions, P- or Q-symbols, are conveniently expressed in terms of operator kernels satisfying the Stratonovich-Weyl postulates. In analogy to this approach, a discrete Moyal formalism is defined on the basis of a modified set of postulates. It is shown that appropriately modified postulates single out a well-defined set of kernels which give rise to discrete symbols. Now operators are represented by functions taking values on (2s+1)(2s+1) points of the sphere. The discrete symbols contain no redundant information, contrary to the continuous ones. The properties of the resulting discrete Moyal formalism for a quantum spin are worked out in detail and compared to the continuous formalism, and it is illustrated by the example of a spin 1/2.

23 pages, latex2e, 3 figures

References in corpus (3)

• Phase-space formulation of quantum mechanics and quantum state reconstruction for physical systems with Lie-group symmetries
• Quantum Time Evolution in Terms of Nonredundant Expectation Values
• Reconstructing the density matrix of a spin s through Stern-Gerlach measurements (II)

Cited by in corpus (35)

• Negativity of the Wigner function as an indicator of nonclassicality
• Quasi-probability representations of quantum theory with applications to quantum information science
• Frame representations of quantum mechanics and the necessity of negativity in quasi-probability representations
• Framed Hilbert space: hanging the quasi-probability pictures of quantum theory
• Encoding a qubit in a spin
• Overview of the phase space formulation of quantum mechanics with application to quantum technologies
• Continuous phase-space representations for finite-dimensional quantum states and their tomography
• Schrödinger cat states of a nuclear spin qudit in silicon
• Quantum polarization tomography of bright squeezed light
• Discrete coherent and squeezed states of many-qudit systems
• Multipolar hierarchy of efficient quantum polarization measures
• Inverse spin-s portrait and representation of qudit states by single probability vectors
• Sub-Planck phase-space structure and sensitivity for SU(1,1) compass states
• Sub-Planck structures: Analogies between the Heisenberg-Weyl and SU(2) groups
• Simple Minimal Informationally Complete Measurements for Qudits
• Wigner negativity in spin-$j$ systems
• Wigner function for SU(1,1)
• Contracting the Wigner-Kernel of a Spin to the Wigner-Kernel of a Particle
• Quantum versus classical polarization states: when multipoles count
• Nonclassicality of coherent states: Entanglement of joint statistics
• Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics
• Fast computation of spherical phase-space functions of quantum many-body states
• Multi-frequency control and measurement of a spin-7/2 system encoded in a transmon qudit
• Wigner distributions for finite state systems without redundant phase point operators
• Quantum speed limits in arbitrary phase spaces
• Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems
• Phase Spaces, Parity Operators, and the Born-Jordan Distribution
• Measurement Contextuality and Planck's Constant
• Evolution Equation for Joint Tomographic Probability Distribution of Spin-1 Particles
• Continuous phase-space methods on discrete phase spaces
• Polytopes of Absolutely Wigner Bounded Spin States
• Quantum tomography with wavelet transform in Banach space on Homogeneous space
• Grand Unification of All Discrete Wigner Functions on $d\times d$ Phase Space
• Stellar representation of extremal Wigner-negative spin states
• On the stabilizer complexity of Hawking radiation