General approach to quantum mechanics as a statistical theory

R. P. Rundle, Todd Tilma, J. H. Samson, V. M. Dwyer, R. F. Bishop, and M. J. Everitt

Phys. Rev. A 99, 012115 – Published 16 January 2019

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 (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 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.

Received 7 August 2018

DOI:https://doi.org/10.1103/PhysRevA.99.012115

Physics Subject Headings (PhySH)

Hybrid quantum systems Quantum correlations, foundations & formalism Quantum formalism Quantum foundations

Phase space methods

Quantum Information, Science & TechnologyAtomic, Molecular & OpticalStatistical Physics & Thermodynamics

Authors & Affiliations

R. P. Rundle^1,2, Todd Tilma^1,3, J. H. Samson¹, V. M. Dwyer^1,2, R. F. Bishop^1,4, and M. J. Everitt^1,*

¹Quantum Systems Engineering Research Group, Department of Physics, Loughborough University, Leicestershire LE11 3TU, United Kingdom
²The Wolfson School of Mechanical, Electrical and Manufacturing Engineering, Loughborough University, Loughborough, Leicestershire LE11 3TU, United Kingdom
³Tokyo Institute of Technology, 2-12-1 Ōokayama, Meguro-ku, Tokyo 152-8550, Japan
⁴School of Physics and Astronomy, Schuster Building, The University of Manchester, Manchester, M13 9PL, United Kingdom

^*m.j.everitt@physics.org

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Issue

Vol. 99, Iss. 1 — January 2019

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Images

Figure 1
Here we show in the left column [(a) and (c)] the coherent superposition (Schrödinger cat) state of three macroscopically distinct coherent states: (a) is the Wigner function and (c) is the Weyl function. Each of the coherent states are generated from the displacement operator in Eq. (6), such that $|α〉 = \hat{D} (α) |0〉$ . The state shown in (a) and (c) is explicitly $|ψ〉 = (|- 3〉 + |- 3 exp (2 i π / 3)〉 + |- 3 exp (4 i π / 3)〉) / \sqrt{3}$ . In the right column [(b) and (d)] we show a spin coherent state version of the state shown in the left column. These are a macroscopically distinct coherent superposition of spin coherent states (a spin Schrödinger cat) on the sphere where $j = 40$ . Each of the “cats” in this state has been created by applying the operator in Eq. (43) to the lowest state $|j; - j〉$ , such that a spin coherent state is given by $|j; ϕ, θ〉 = \hat{R} (ϕ, θ) |j; - j〉$ . The position of each spin coherent state with relation to the south pole is determined by the $θ$ rotation. Here $θ = π / 10$ ; as $j$ increases, the value of $θ$ will need to decrease to form the same analog of a cat state seen in a continuous system, and thus in the stereographic projection, the spin coherent Schrödinger cat states at $θ = π / 10, ϕ = π n / 3$ ( $n = 0, 1, 2$ for the three cats), will appear to get further away from each other. The state is explicitly given by $|ψ〉 = (|j; 0, π / 10〉 + |j; π / 3, π / 10〉 + |j; 2 π / 3, π / 10〉) / \sqrt{3}$ . The inset next to each sphere in (b) and (d) is the corresponding stereographic (Riemann) projection of the lower hemisphere onto a circle in Euclidean space, with the boundary at the equator. Here (b) shows the spin Wigner function and (d) shows the spin Weyl function. Both (c) and (d) contain both magnitude (intensity) and phase (color) information for the complex valued Weyl functions as shown by the inset color wheel.
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Figure 2
Here we show in (a) and (c) the superposition state for a spin- $\frac{5}{2}$ spin coherent Schrödinger cat state [42], given by $(|- \frac{5}{2}〉 + |\frac{5}{2}〉) / \sqrt{2}$ and in (b)and (d) the five-qubit GHZ state $(|00000〉 + |11111〉) / \sqrt{2}$ . (a) and (b) The spherical plot for the spin Wigner functions where we have taken the equal angle slice, $ϕ_{i} = ϕ$ and $θ_{i} = θ$ , and where blue is positive and red in negative. (c) and (d) The spin Weyl function spherical plots for the slice $\tilde{Φ} = - \tilde{ϕ}$ , and where we have again taken the equal angle slice ${\tilde{ϕ}}_{i} = \tilde{ϕ}, {\tilde{θ}}_{i} = \tilde{θ}$ . The phase for the spin Weyl functions is given by color according to the color wheel in the center of the figure. The absolute value is shown by saturation, so that the Weyl function is white when the value at that point is zero. Note that we have extended the range when mapping the function onto the sphere, so that the range of $\tilde{θ}$ is doubled.
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