Phase-space representations of SIC-POVM fiducial states

TL;DR

This paper investigates the phase-space properties of SIC-POVM fiducial states in finite dimensions, revealing their maximal delocalization and exploring the implications of Zauner symmetry for their structure.

Contribution

It provides a detailed phase-space analysis of numerically obtained SIC-POVM fiducial states and links their properties to symmetry considerations, offering new insights into their structure.

Findings

Fiducial states exhibit maximal phase-space delocalization.

SIC-POVM condition corresponds to a maximal delocalization property.

Constructed an operator related to Zauner symmetry with semiclassical eigenfunctions.

Abstract

The problem of finding symmetric informationally complete POVMs (SIC-POVMs) has been solved numerically for all dimensions $d$ up to 67 (A.J. Scott and M. Grassl, {\it J. Math. Phys.} 51:042203, 2010), but a general proof of existence is still lacking. For each dimension, it was shown that it is possible to find a SIC-POVM which is generated from a fiducial state upon application of the operators of the Heisenberg-Weyl group. We draw on the numerically determined fiducial states to study their phase-space features, as displayed by the characteristic function and the Wigner, Bargmann and Husimi representations, adapted to a Hilbert space of finite dimension. We analyze the phase-space localization of fiducial states, and observe that the SIC-POVM condition is equivalent to a maximal delocalization property. Finally, we explore the consequences in phase space of the conjectured Zauner symmetry. In particular, we construct an Hermitian operator commuting with this symmetry that leads to a representation of fiducial states in terms of eigenfunctions with definite semiclassical features.