Maximum number of mutually unbiased bases

Problem

For each integer $d\geq2$, determine the maximum number $\mu(d)$ of pairwise mutually unbiased orthonormal bases of $\mathbb C^d$. Two orthonormal bases $\mathcal B_r=\{|e_i^{(r)}\rangle\}_{i=1}^{d}$ and $\mathcal B_s=\{|e_j^{(s)}\rangle\}_{j=1}^{d}$ are mutually unbiased when

\begin{equation} \bigl|\langle e_i^{(r)}|e_j^{(s)}\rangle\bigr|^2=\frac1d \qquad\text{for every }i,j\in\{1,\ldots,d\}. \tag{1} \end{equation}

Thus the extremal quantity defined by Eq. (1) is

\begin{equation} \mu(d):=\max\left\{m:\text{there exist $m$ orthonormal bases of $\mathbb C^d$ that are pairwise mutually unbiased}\right\}. \tag{2} \end{equation}

Determine Eq. (2) in the non-prime-power regime, in particular for $d\in\{6,10,12,14,15\}$.

Source

Krüger and Werner explicitly pose determination of the maximum number of mutually unbiased bases in arbitrary dimension; McNulty and Weigert retain the composite-dimensional cases as open [KW05], [MW26].

Progress

The universal dimension bound is $\mu(d)\leq d+1$, and finite-field constructions attain it whenever $d$ is a prime power. Hence $\mu(d)=d+1$ throughout the prime-power regime [WF89], [KR04].
If $d=\prod_r p_r^{a_r}$ is the prime-power factorization, tensor products give $\mu(d)\geq1+\min_r p_r^{a_r}$. Moreover, any family of $d$ mutually unbiased bases extends to $d+1$ bases. Consequently $\mu(6)\in\{3,4,5,7\}$; in particular, six bases cannot be maximal [KR04], [Wei13].
No fourth mutually unbiased basis in $\mathbb C^6$ is known. Numerical searches support $\mu(6)=3$, while symmetry-reduced semidefinite hierarchies provide convergent certificate frameworks but have not resolved the unrestricted case [BH07], [GP24], [MW26].

Comment

Determining the exact value in Eq. (2) subsumes the dimension-six milestone of constructing four bases or excluding seven: these give only $\mu(6)\geq4$ or $\mu(6)\leq5$, respectively. The milestone is therefore recorded here rather than maintained separately [HRZ22].

References

[WF89]: W. K. Wootters and B. D. Fields, “Optimal State-Determination by Mutually Unbiased Measurements,” Annals of Physics 191, 363–381 (1989).DOI
[KR04]: A. Klappenecker and M. Rötteler, “Constructions of Mutually Unbiased Bases,” in Finite Fields and Applications, LNCS 2948, 137–144 (Springer, 2004).DOI arXiv
[Wei13]: M. Weiner, “A Gap for the Maximum Number of Mutually Unbiased Bases,” Proceedings of the American Mathematical Society 141, 1963–1969 (2013).DOI arXiv
[BH07]: P. Butterley and W. Hall, “Numerical Evidence for the Maximum Number of Mutually Unbiased Bases in Dimension Six,” Physics Letters A 369, 5–8 (2007).DOI arXiv
[GP24]: S. Gribling and S. Polak, “Mutually Unbiased Bases: Polynomial Optimization and Symmetry,” Quantum 8, 1318 (2024).DOI arXiv
[MW26]: D. McNulty and S. Weigert, “Mutually Unbiased Bases in Composite Dimensions—A Review,” Quantum 10, 2051 (2026).DOI arXiv
[KW05]: O. Krüger and R. F. Werner (eds.), “Some Open Problems in Quantum Information Theory,” arXiv:quant-ph/0504166 (2005).DOI arXiv
[HRZ22]: P. Horodecki, Ł. Rudnicki, and K. Życzkowski, “Five Open Problems in Quantum Information Theory,” PRX Quantum 3, 010101 (2022).DOI arXiv

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