Mutually unbiased bases in composite dimensions

Find the most mutually unbiased bases possible in dimension d when d is not a prime power; for d = 6 it is 3, 4 or 5, and a claimed proof of 3 awaits checking.

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	1	+# Mutually unbiased bases in composite dimensions
	2	+
	3	+Two bases are mutually unbiased when a measurement in one reveals nothing about the outcome of a measurement in the other. A $d$-dimensional quantum system has at most $d+1$ such bases, and that many exist whenever $d$ is a prime power. In every other dimension the maximum is unknown, starting with $d=6$.
	4	+
	5	+## Problem
	6	+
	7	+Two orthonormal bases $\{\lvert e_i\rangle\}$ and $\{\lvert f_j\rangle\}$ of $\mathbb{C}^d$ are mutually unbiased if
	8	+
	9	+$$\lvert\langle e_i\vert f_j\rangle\rvert^2=\frac1d\qquad\text{for all } i,j .$$
	10	+
	11	+Let $\mu(d)$ be the largest number of orthonormal bases of $\mathbb{C}^d$ that are pairwise mutually unbiased. It is known that $\mu(d)\le d+1$.
	12	+
	13	+Question. Determine $\mu(d)$ for every $d$ that is not a prime power. The first cases are $d=6,10,12,14,15$. In particular, is $\mu(6)=3$?
	14	+
	15	+## Background
	16	+
	17	+Mutually unbiased bases give the most efficient set of projective measurements for reconstructing an unknown quantum state; Wootters and Fields studied them in this role in 1989 [1]. They also appear in quantum key distribution and in the "mean king" retrodiction puzzle, and they are closely tied to complex Hadamard matrices: a family that contains the standard basis amounts to a list of such matrices. Zauner conjectured that $\mu(6)=3$; the conjecture goes back to his 1991 diploma thesis and appears in his 1999 doctoral thesis [5]. The problem is No. 13 in the Open Quantum Problems collection of IQOQI Vienna, which singles out dimension six [6].
	18	+
	19	+## What is known
	20	+
	21	+- If $d$ is a prime power, $\mu(d)=d+1$, by constructions over finite fields [1, 2].
	22	+- Tensor products of these constructions give $\mu(d)\ge\min_i\,(p_i^{a_i}+1)$ for $d=\prod_i p_i^{a_i}$ [2]. Hence $\mu(6)\ge3$ and $\mu(10)\ge3$.
	23	+- Weiner proved that any $d$ mutually unbiased bases can be completed to $d+1$, so $\mu(d)\ne d$ [3]. Before 2026 this left $\mu(6)\in\{3,4,5,7\}$.
	24	+- Numerical searches find no fourth basis in $\mathbb{C}^6$ ([Butterley and Hall, 2007](https://doi.org/10.1016/j.physleta.2007.04.059)). Symmetry-reduced semidefinite hierarchies give sum-of-squares proofs that $d+2$ bases do not exist for $d\le8$, and their numerics suggest that such a proof excluding four bases in $d=6$ needs polynomials of degree at least 12 [4].
	25	+- In September 2026 OpenAI released two preprints on $d=6$ [5]. One, on exact Fourier certificates for complex Hadamard matrices of order six, yields $\mu(6)\le5$, and this bound has a Lean proof. The other claims $\mu(6)=3$, but its exclusion of four bases is a computer-assisted verification in binary64 floating-point arithmetic, valid under stated compiler conditions; it is unrefereed and not formalized, and the Lean proof does not cover it. The formal-conjectures project therefore still lists $\mu(6)$ as open, with $3\le\mu(6)\le5$.
	26	+
	27	+## What would count as progress
	28	+
	29	+- An independent, rigorous proof that four mutually unbiased bases do not exist in $\mathbb{C}^6$, for example an exact or interval-arithmetic recheck of the floating-point exclusion in [5], or a sum-of-squares certificate. This would settle $\mu(6)=3$.
	30	+- Four explicit bases of $\mathbb{C}^6$ that are pairwise unbiased, checkable to any precision. This would overturn the conjecture.
	31	+- Any new bound for $d=10$, such as four explicit mutually unbiased bases of $\mathbb{C}^{10}$, or a proof that a complete set of eleven does not exist.
	32	+- Lean statements for $d=6,10,12,14,15$ and for general $d$ are in [OpenQuantumProblems/13.lean](https://github.com/google-deepmind/formal-conjectures/blob/main/FormalConjectures/OpenQuantumProblems/13.lean) of formal-conjectures; a proof of `mutuallyUnbiasedBases_dim6` there would close the main case formally.
	33	+
	34	+## References
	35	+
	36	+1. W. K. Wootters and B. D. Fields, "Optimal state-determination by mutually unbiased measurements", Annals of Physics 191, 363–381 (1989). [doi:10.1016/0003-4916(89)90322-9](https://doi.org/10.1016/0003-4916(89)90322-9)
	37	+2. A. Klappenecker and M. Rötteler, "Constructions of mutually unbiased bases", in Finite Fields and Applications, LNCS 2948, 137–144 (Springer, 2004). [arXiv:quant-ph/0309120](https://arxiv.org/abs/quant-ph/0309120)
	38	+3. M. Weiner, "A gap for the maximum number of mutually unbiased bases", Proceedings of the American Mathematical Society 141, 1963–1969 (2013). [doi:10.1090/S0002-9939-2013-11487-5](https://doi.org/10.1090/S0002-9939-2013-11487-5)
	39	+4. S. Gribling and S. Polak, "Mutually unbiased bases: polynomial optimization and symmetry", Quantum 8, 1318 (2024). [doi:10.22331/q-2024-04-30-1318](https://doi.org/10.22331/q-2024-04-30-1318)
	40	+5. OpenAI, "The maximum number of mutually unbiased bases in dimension six" and "Exact Fourier certificates for complex Hadamard matrices of order six", preprints (September 2026). [First PDF](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026/The-maximum-number-of-mutually-unbiased-bases-in-dimension-six-September-24-2026.pdf); [second PDF](https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026/Exact-Fourier-certificates-for-complex-Hadamard-matrices-of-order-six-September-24-2026.pdf); [scope of the Lean proof](https://github.com/openai/math/blob/main/lean/docs/266.md)
	41	+6. B.-G. Englert (contact), "Mutually unbiased bases", Open Quantum Problems, IQOQI Vienna, problem 13 (posted 2003). [Archived page](https://web.archive.org/web/20231029013422/https://oqp.iqoqi.oeaw.ac.at/mutually-unbiased-bases)
	42	+
	43	+## Source
	44	+
	45	+Imported from [QIQCOP Zoo op_c9c62042b15fcb06](https://qiqc-op.com/problem/op_c9c62042b15fcb06/), curated by Bikun Li, Qicheng Tang, Changhao Li, Chengkai Zhu, Minbo Gao, Zhong-Xia Shang, Bin Cheng and Naixu Guo. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.