A general approach to quantum mechanics as a statistical theory
arXiv:1708.03814 · doi:10.1103/PhysRevA.99.012115
Abstract
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 and Phys. Rev. A 96, 022117] 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. Firstly 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 Schrödinger'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.
Major update - removal of unnecessary Fourier transform from definition of the Weyl function which allowed generalisation of the work to a complete framework of phase-space quantum mechanics. This manuscript is completely re-written and improved to reflect this. 2 figures, 16 pages
References in corpus (2)
Cited by in corpus (27)
- Generalized spin mapping for quantum-classical dynamics
- 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
- New Phase Space Formulations and Quantum Dynamics Approaches
- Quantum persistent tennis racket dynamics of nanorotors
- Nonadiabatic Field: A Conceptually Novel Approach for Nonadiabatic Quantum Molecular Dynamics
- Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics
- On detailed balance in nonadiabatic dynamics: From spin spheres to equilibrium ellipsoids
- Orientational order parameters for arbitrary quantum systems
- Fast computation of spherical phase-space functions of quantum many-body states
- Visualizing entanglement in atoms and molecules
- Kinetic theory for spin-1/2 particles in ultra-strong magnetic fields
- Quantum speed limits in arbitrary phase spaces
- Phase Spaces, Parity Operators, and the Born-Jordan Distribution
- Symmetry-adapted decomposition of tensor operators and the visualization of coupled spin systems
- Dynamics of free time-dependent effective mass
- On quantum invariants and the graph isomorphism problem
- Generalized Phase-Space Techniques to Explore Quantum Phase Transitions in Critical Quantum Spin Systems
- Emergence of a Renormalized Expansion in Quenched Critical Many-Body Systems
- Navigating the phase diagram of quantum many-body systems in phase space
- The ambiguity function and the displacement operator basis in quantum mechanics
- Extended Wigner function for the harmonic oscillator in the phase space
- Correspondence rules for Wigner functions over SU(3)/U(2)
- Visualization of correlations in hybrid quantum systems
- Master equations for Wigner functions with spontaneous collapse and their relation to thermodynamic irreversibility
- Wigner functional theory for quantum optics
- Efficient verification and fidelity estimation of discrete bipartite squeezed states