A rigidity property of complete systems of mutually unbiased bases

Abstract

Suppose that for some unit vectors $b_1,\dots b_n$ in ${\mathbb{C}}^d$ we have that for any $j\ne k$ $b_j$ is either orthogonal to $b_k$ or $|⟨b_j,b_k⟩{|}^2=1/d$ (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.

References in corpus (5)

• On mutually unbiased bases
• Mubs and Hadamards of Order Six
• Maximal Sets of Mutually Unbiased Quantum States in Dimension Six
• A generalized Pauli problem and an infinite family of MUB-triplets in dimension 6
• A Survey of Finite Algebraic Geometrical Structures Underlying Mutually Unbiased Quantum Measurements

Cited by in corpus (1)

• Projective toric designs, quantum state designs, and mutually unbiased bases