Continuous phase-space representations for finite-dimensional quantum states and their tomography

Abstract

Continuous phase spaces have become a powerful tool for describing, analyzing, and tomographically reconstructing quantum states in quantum optics and beyond. A plethora of these phasespace techniques are known, however a thorough understanding of their relations was still lacking for finite-dimensional quantum states. We present a unified approach to continuous phase-space representations which highlights their relations and tomography. The infinite-dimensional case from quantum optics is then recovered in the large-spin limit. * balint.koczor@materials.ox.ac.uk

Continuous phase spaces and the time evolution of spins: star products and spin-weighted spherical harmonics

“…An important class of infinite-dimensional phase-space representations of a density operator ρ contains s-parametrized phase-space distribution functions (where −1 ≤ s ≤ 1) which can be defined via [3,15,28,43,44]…”

“…The particular case of s = 0 corresponds to the Wigner function. All s-parametrized phase-space distribution functions are related to each other via Gaussian smoothing [15,28], and the convolution of the vacuum-state representation F |0 (Ω, s ) with the distribution function F ρ (Ω, s) results in a distribution function…”

“…In this work, we extend our earlier results in [34] on Wigner functions of coupled spins 1/2 and present the explicit form of the star product for the general class of sparametrized phase spaces which is applicable to single and coupled spins of arbitrary spin number J. We also rely on phase-space techniques for single spins J that have been developed in [28]. We introduce spin-weighted spherical harmonics [39] as an important new technical tool to the theory of phase spaces, even though they have not been considered in this context before.…”

“…Particular cases include Wigner [13], Husimi Q [14], and Glauber P [15] functions. In this work, we are particularly interested in phase-space methods that are applicable to (finite-dimensional) spin systems [16][17][18][19][20][21][22][23][24][25][26][27] and how these methods are related to infinitedimensional phase spaces [28]. Building on earlier results in [17,18,21,22,27,29], we have developed in [28] a unified description for the general class of s-parametrized phase spaces with −1 ≤ s ≤ 1 which is applicable to single spins with integer or half-integer spin number J and which naturally recovers the infinite-dimensional case in the large-J limit.…”

“…This work has the following structure: We start in section 2 by recapitulating elementary properties of infinite-dimensional phase spaces and their star products. In section 3, we recall the structure of phase-space representations for single spins J following the approach of [28]. We introduce spin-weighted spherical harmonics and summarize their main features in section 4.…”