Quasiprobabilities from incomplete and overcomplete measurements

TL;DR

The paper addresses how to construct measurement-based quasiprobabilities and notions of nonclassicality when available measurements are informationally incomplete or overcomplete.It develops a general framework based on a POVM {Π_k}, a metric g, and a pseudo-inverse to define P = g^{-1}Q and its $\sigma$-parametrizations, unifying KD distributions and $s$-parametrized phase-space functions as special cases via $\sigma=(1+s)/2$.The authors illustrate the approach with single-qubit POVMs under noisy, incomplete, and overcomplete conditions, showing how the classical region in Bloch space depends on the measurement set and the chosen $\sigma$.The method provides a device- and system-agnostic toolbox for robust nonclassicality analysis in realistic experiments, enabling partial state characterizations and flexible comparisons across measurement regimes.

Abstract

We discuss the (re-)construction of quasiprobability representations from generic measurements, including noisy ones. Based on the measurement under study, quasiprobabilities and the associated concept of nonclassicality are introduced. A practical concern that we address is the treatment of informationally incomplete and overcomplete measurement scenarios, which can significantly alter the assessment of which states are deemed classical. Notions, such as Kirkwood-Dirac quasiprobabilities and s-parametrized quasiprobabilities in quantum optics, are generalized by our approach. Single-qubit systems are used to exemplify and to compare different measurement schemes, together with the resulting quasiprobabilities and set of nonclassical states.

Figures (1)

• Figure 1: Qubit measurements. Classical states with $\vec{P}_\sigma \geq 0$ are shown as green convex regions embedded in the Bloch sphere. The first column illustrates an informationally complete POVM; the second, an incomplete POVM lacking information about the $z$ axis. The third column depicts an overcomplete POVM, while the last shows a measurement set that is both informationally incomplete and overcomplete. The first row, $\sigma=0.5$, resembles the Wigner function for the measurement-based quasiprobabilities $\vec{P}_\sigma$. The second row, $\sigma=.75$, is closer to the analog of the Glauber--Sudarshan distribution, which itself is depicted in third row and is given by $\sigma=1$. Additional noise in the measurement can be modeled with $\sigma>1$, which is shown in the last row for $\sigma=1.25$. States outside the green set are $\sigma$-nonclassical, $\vec{P}_\sigma\ngeq0$, for the selected $\sigma$ parameter and POVM under consideration.