Mutually Unbiased Bases and Finite Projective Planes

TL;DR

This paper explores the deep connection between the existence of finite projective planes of non-prime power order and the existence of mutually unbiased bases in non-prime power dimensions, highlighting a longstanding open problem.

Contribution

It investigates the conjectured relationship between finite projective planes and mutually unbiased bases, proposing a link that could impact quantum information theory and finite geometry.

Findings

No definitive proof of the existence of such projective planes or MUBs in non-prime power dimensions.

Highlights the equivalence conjecture linking projective planes and MUBs.

Provides a new perspective on longstanding open problems in finite geometry and quantum information.

Abstract

It is conjectured that the question of the existence of projective planes whose order is not a power of prime is intimately linked with the problem whether there exists a set of d+1 mutually unbiased bases in a d-dimensional Hilbert space if d differs from a power of prime.