Informationally overcomplete measurements from generalized equiangular tight frames

TL;DR

This paper introduces a new class of informationally overcomplete measurements based on generalized equiangular tight frames, expanding the tools for quantum state estimation with potential advantages over traditional measurement sets.

Contribution

It generalizes equiangular measurements to non-projective POVMs, providing a construction method, analyzing symmetry, and identifying conical 2-designs for quantum tomography.

Findings

Provides a construction method for generalized equiangular POVMs.

Identifies a wide class of conical 2-designs within these measurements.

Shows benefits of using a single overcomplete measurement in quantum tomography.

Abstract

Informationally overcomplete measurements find important applications in quantum tomography and quantum state estimation. The most popular are maximal sets of mutually unbiased bases, for which trace relations between measurement operators are well known. In this paper, we introduce a more general class of informationally overcomplete POVMs that are generated by equiangular tight frames of arbitrary rank. This class provides a generalization of equiangular measurements to non-projective POVMs, which include rescaled mutually unbiased measurements and bases. We provide a method of their construction, analyze their symmetry properties, and provide examples for highly symmetric cases. In particular, we find a wide class of generalized equiangular measurements that are conical 2-designs, which allows us to derive the index of coincidence. Our results show benefits of considering a single informationally overcomplete measurement over informationally complete collections of POVMs.