Quantum measurements and finite geometry

TL;DR

This paper explores the analogy between certain quantum measurements, like mutually unbiased bases and symmetric informationally complete measurements, and finite affine geometries, investigating their implications for quantum measurement theory.

Contribution

It establishes and analyzes the geometric analogies of quantum measurements, providing insights into their existence and structure in finite-dimensional quantum systems.

Findings

Mutually unbiased bases correspond to affine planes in finite geometry.

Symmetric informationally complete measurements relate to affine planes with roles of points and lines interchanged.

The analogies offer potential criteria for the existence of such measurements.

Abstract

A complete set of mutually unbiased bases for a Hilbert space of dimension N is analogous in some respects to a certain finite geometric structure, namely, an affine plane. Another kind of quantum measurement, known as a symmetric informationally complete positive-operator-valued measure, is, remarkably, also analogous to an affine plane, but with the roles of points and lines interchanged. In this paper I present these analogies and ask whether they shed any light on the existence or non-existence of such symmetric quantum measurements for a general quantum system with a finite-dimensional state space.