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An independent certified computation excluding four mutually unbiased bases in dimension six

Manvir Jaswal

Published Oct 13, 2026• 0 citations

Abstract

We show that there are no four mutually unbiased bases in ℂ⁶, so that the maximum number of such bases is three. The proof is computer-assisted and assumes that the machine on which the computation ran carried out binary64 arithmetic correctly, as made precise in Section 3.1. It carries out the discretization proof of a recent preprint by OpenAI, which has not appeared in a refereed journal. Our programs are written from the text of the preprint, and we prove every statement of the preprint that our proof uses. Four mutually unbiased bases could be taken to consist of the standard basis, the columns of a complex Hadamard matrix divided by √6, and two further bases. Finitely many balls in phase coordinates cover all complex Hadamard matrices of order six up to equivalence, and on each small ball a finite search shows that these two further bases do not exist. Every inequality that the programs decide in order to reject a case or to cover a set of matrices is decided in exact integer arithmetic or in interval arithmetic with outward rounding. The programs, with the logs and data of the computation, are deposited.

DOI: 10.5281/zenodo.23255857

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We show that there are no four mutually unbiased bases in ℂ⁶, so that the maximum number of such bases is three. The proof is computer-assisted and assumes that the machine on which the computation ran carried out binary64 arithmetic correctly, as made precise in Section 3.1. It carries out the discretization proof of a recent preprint by OpenAI, which has not appeared in a refereed journal. Our programs are written from the text of the preprint, and we prove every statement of the preprint that our proof uses. Four mutually unbiased bases could be taken to consist of the standard basis, the columns of a complex Hadamard matrix divided by √6, and two further bases. Finitely many balls in phase coordinates cover all complex Hadamard matrices of order six up to equivalence, and on each small ball a finite search shows that these two further bases do not exist. Every inequality that the programs decide in order to reject a case or to cover a set of matrices is decided in exact integer arithmetic or in interval arithmetic with outward rounding. The programs, with the logs and data of the computation, are deposited.

We prove that the maximum number of mutually unbiased bases in complex dimension sixis three. The proof is structural and does not rely on exhaustive classification, numerical search,or probabilistic arguments. The key ingredients are: (1) a Laurent–polynomial obstructionblocking the Haagerup family of complex Hadamard matrices, (2) an analogous obstructionblocking the 2-circulant family, and (3) a structural Bridge Lemma showing that any complexHadamard matrix arising as a transition matrix in a MUB triplet must exhibit an opposite-pairphenomenon in its row-ratio multisets. Combined with the recognition theorem of Matszangoszand Sz¨oll˝osi, this forces all MUB-triplet transition matrices in dimension six into families alreadyobstructed, yielding a contradiction to the existence of a fourth mutually unbiased basis.

The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex $(6\ifmmode\times\else\texttimes\fi{}6)$ Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.

Mutually unbiased bases (MUBs) describe quantum measurements for which certainty in one basis gives uniform outcomes in the others. Their maximum number remains unknown in dimensions that are not powers of a prime. We introduce an ansatz that, in many dimensions, allows us to construct more MUBs than have previously been found. Each basis in the ansatz consists of a diagonal phase matrix applied to a fixed unitary, which is a tensor product of Fourier matrices and, optionally, any real Hadamard matrix. This construction yields five MUBs in dimension 12, six in dimensions 48, 96, and 192, seven in dimensions 36 and 108, and ten in dimension 648. These bases can then be extended to exceed the tensor product bound, asymptotically, in every eighteenth dimension. Moreover, we show that Paley's real Hadamard matrix can be used to construct

q+1

bases in dimension

d = q(q+1)

for every prime power

q\equiv3\pmod4

. This count grows as

\sqrt d

and cannot be exceeded by tensor products of smaller sets.

We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions

d

that are powers of prime numbers, there exists a set of up to

d+1

bases that form an MUB set. However, the maximum number of MUBs in dimensions that are not powers of prime numbers is not known. To address this issue, we introduce three algorithms based on First-Order Logic that can determine the maximum number of bases in an MUB set without numerical approximation. Our algorithms can prove this result in finite time, although the required time is impractical. Additionally, we present a heuristic approach to solve the semi-decision problem of determining if there are

k

MUBs in a given dimension

d

. As a byproduct of our research, we demonstrate that the maximum number of MUBs in any dimension can be achieved with definable complex parameters, computable complex parameters, and other similar fields.