A rigidity property of complete systems of mutually unbiased bases

TL;DR

This paper proves a rigidity property of systems of vectors in complex space where vectors are either orthogonal or mutually unbiased, showing such systems with a specific size must form a complete set of mutually unbiased bases.

Contribution

It establishes that systems of vectors with certain orthogonality and unbiasedness conditions are necessarily complete mutually unbiased bases when of a specific size.

Findings

Vectors with the given properties form a complete MUB system when n=d(d+1).

The result confirms a rigidity property of MUB configurations.

The proof links vector properties to the structure of MUBs in quantum information theory.

Abstract

Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$, then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into $d+1$ orthonormal bases, all being mutually unbiased with respect to each other.