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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. | |