Probing the geometry of quantum states with symmetric POVMs

TL;DR

This paper explores how symmetric POVMs can be used to understand the complex geometry of quantum state space, potentially impacting foundational questions in quantum information.

Contribution

It introduces a framework linking quantum state geometry to probability distributions over symmetric POVM outcomes, including properties of these POVMs.

Findings

Quantum states can be characterized by distributions over symmetric POVMs.

Properties of symmetric POVMs are detailed and analyzed.

This approach offers new insights into quantum state geometry.

Abstract

The geometry of the Quantum State Space, described by Bloch vectors, is a very intricate one. A deeper understanding of this geometry could lead to the solution of some difficult problems in Quantum Foundations and Quantum Information such as the existence of SIC-POVMs and the cardinality of the maximal set of MUBs. In this paper we show that the geometry of quantum states can be described by the probability distributions that quantum states induce over the outcomes of symmetric POVMs not necessarily of arbitrary rank or informationally complete. We also describe the properties of these symmetric POVMs.