Discrete Wigner Functions from Informationally Complete Quantum Measurements

TL;DR

This paper explores the connection between discrete Wigner functions and informationally complete quantum measurements, revealing how they can be interconverted and highlighting the significance of symmetric informationally complete measurements in quantum representations.

Contribution

It demonstrates that minimal discrete Wigner functions are orthogonalizations of minimal informationally complete measurements, linking phase space and probabilistic quantum representations.

Findings

Wigner bases are orthogonalizations of MICs.

Not imposing a phase space structure reveals new insights into Wigner functions.

Symmetric informationally complete measurements are significant in quasiprobability representations.

Abstract

Wigner functions provide a way to do quantum physics using quasiprobabilities, that is, "probability" distributions that can go negative. Informationally complete POVMs, a much younger subject than phase space formulations of quantum mechanics, are less familiar but provide wholly probabilistic representations of quantum theory. In this paper, we show that the Born Rule links these two classes of structure and discuss the art of interconverting between them. In particular, we demonstrate that the operator bases corresponding to minimal discrete Wigner functions (Wigner bases) are orthogonalizations of minimal informationally complete measurements (MICs). By not imposing a particular discrete phase space structure at the outset, we push Wigner functions to their limits in a suitably quantified sense, revealing a new way in which the symmetric informationally complete measurements (SICs) are significant. Finally, we speculate that astute choices of MICs from the orthogonalization preimages of Wigner bases may in general give quantum measurements conceptually underlying the associated quasiprobability representations.