MUBs, polytopes, and finite geometries

Abstract

A complete set of N+1 mutually unbiased bases (MUBs) exists in Hilbert spaces of dimension N = p^k, where p is a prime number. They mesh naturally with finite affine planes of order N, that exist when N = p^k. The existence of MUBs for other values of N is an open question, and the same is true for finite affine planes. I explore the question whether the existence of complete sets of MUBs is directly related to the existence of finite affine planes. Both questions can be shown to be geometrical questions about a convex polytope, but not in any obvious way the same question.

15 pages; talk at the Vaxjo conference on probability and physics

Cited by in corpus (9)

• Quantification and manipulation of magic states
• MUBs inequivalence and affine planes
• Discrete phase-space structure of $n$-qubit mutually unbiased bases
• Mutually unbiased triplets from non-affine families of complex Hadamard matrices in dimension six
• Maximal Entanglement, Collective Coordinates and Tracking the King
• Detecting three-qubit bound MUB diagonal entangled states via Nonlinear optimal entanglement witnesses
• Radon Transform in Finite Dimensional Hilbert Space
• Partially Unbiased Entangled Bases
• Classical and Quantum Mechanical State Reconstruction