Mutually Unbiased Bases in Dimension Six

Let $\mathcal H_6$ be a six-dimensional complex Hilbert space. For an integer $K$, let $\mathcal B_m=\{\psi_i^m:1\leq i\leq 6\}$, $1\leq m\leq K$, be orthonormal bases of $\mathcal H_6$. The bases are mutually unbiased when every pair of vectors from distinct bases satisfies

$$|\langle\psi_i^m,\psi_j^n\rangle|^2=\frac{1}{6}\qquad (m\ne n, ext{ and all }i,j\in\{1,\ldots,6\}).$$

The question is to establish either that such bases exist for some integer $K$ with $4\leq K\leq 7$, or that no such family exists for $K=7$. The upper limit $7$ is the general bound of dimension plus one. Horodecki, Rudnicki, and Życzkowski, Five open problems in theory of quantum information

Background

Mutually unbiased bases represent maximally complementary measurements: the outcome probabilities for a basis vector measured in any other basis are all equal. They are also useful in quantum-state tomography; a complete family of $d+1$ bases is optimal in the sense discussed in the six-dimensional study. The general upper bound is $d+1$, and complete families are known when $d$ is a prime power. For composite $d$, the cited open-problem account gives a general construction of at least $p_1^{k_1}+1$ bases when $d=\prod_j p_j^{k_j}$ and $p_1^{k_1}$ is the smallest prime-power factor. Thus dimension six has three known bases as a baseline, while seven would be a complete family. Horodecki, Rudnicki, and Życzkowski, Five open problems in theory of quantum information Mutually unbiased bases in dimension six: The four most distant bases

Dimension six is the smallest dimension that is not a prime power. Zauner’s conjecture, reported in the six-dimensional literature, asserts that no more than three mutually unbiased bases exist there. The search for additional bases has consequently produced both structural and computational approaches, but numerical failure to find a configuration is not a proof of nonexistence. Prat Colomer et al., Three numerical approaches to find mutually unbiased bases using Bell inequalities Mutually unbiased bases in dimension six: The four most distant bases

Known results

The general upper bound rules out more than seven bases. Moreover, if six mutually unbiased bases exist in dimension six, the seventh also exists; hence the possible maximum cannot be six. The 2020 account reports constructions of three bases in dimension six, but does not report a construction of four or a proof excluding seven. Earlier numerical searches for small mutually unbiased constellations found no configurations beyond the three-basis case and described this as evidence against four bases, not as a nonexistence theorem. Horodecki, Rudnicki, and Życzkowski, Five open problems in theory of quantum information Maximal Sets of Mutually Unbiased Quantum States in Dimension Six

A numerical optimization of the average squared distance among four bases in dimension six reported a value of $0.9983$, strictly below the value $1$ attained exactly by pairwise unbiased bases. The authors also gave a two-parameter family attaining that value, but the numerical search does not establish that this is a global maximum. In 2022, three numerical methods based on Bell inequalities likewise failed to find four bases and supported Zauner’s conjecture. Separately, a 2024 polynomial-optimization study gave sum-of-squares proofs of the general nonexistence of $d+2$ bases for $d=2,\ldots,8$; its reported numerical evidence suggests that, in its hierarchy, refuting the existence of more than three bases in dimension six requires polynomials of total degree at least $12$. These computational results do not settle either alternative in the question. Mutually unbiased bases in dimension six: The four most distant bases Prat Colomer et al., Three numerical approaches to find mutually unbiased bases using Bell inequalities Gribling and Polak, Mutually unbiased bases: polynomial optimization and symmetry