We prove that the maximum number of mutually unbiased orthonormal bases in ℂ6 is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
We prove that the maximum number of mutually unbiased orthonormal bases in ℂ6 is three, resolving Zauner's dimension-six MUB conjecture. The upper bound is computer-assisted: under the stated binary64 arithmetic and compiler conditions, a complete execution of the documented verification pipeline excludes four arbitrary complex bases.
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 problem of finding provably maximal sets of mutually unbiased bases in $\CC^d$, for composite dimensions d which are not prime powers, remains completely open. In the first interesting case,~d=6, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in~d=6 by looking at their~`shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers~q≡5mod12, however, we are able to show some curious evidence which --- at least formally --- points in the opposite direction. In $\CC^6$, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.
Evidence for and against Zauner's MUB Conjecture in C6
The problem of finding provably maximal sets of mutually unbiased bases in Cd, for composite dimensions d which are not prime powers, remains completely open. In the first interesting case, d=6, Zauner predicted that there can exist no more than three MUBs. We explore possible algebraic solutions in d=6 by looking at their `shadows' in vector spaces over finite fields. The main result is that if a counter-example to Zauner's conjecture were to exist, then it would leave no such shadow upon reduction modulo several different primes, forcing its algebraic complexity level to be much higher than that of current well-known examples. In the case of prime powers q≡5mod12, however, we are able to show some curious evidence which -- at least formally -- points in the opposite direction. In C6, not even a single vector has ever been found which is mutually unbiased to a set of three MUBs. Yet in these finite fields we find sets of three `generalised MUBs' together with an orthonormal set of four vectors of a putative fourth MUB, all of which lifts naturally to a number field.
We consider the average distance between four bases in six dimensions. The distance between two orthonormal bases vanishes when the bases are the same, and the distance reaches its maximal value of unity when the bases are unbiased. We perform a numerical search for the maximum average distance and find it to be strictly smaller than unity. This is strong evidence that no four mutually unbiased bases exist in six dimensions. We also provide a two-parameter family of three bases which, together with the canonical basis, reach the numerically found maximum of the average distance, and we conduct a detailed study of the structure of the extremal set of bases.
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.