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arXiv:2511.03537

Mutually Unbiased Bases and Orthogonal Latin Squares -- version 3

Stefan Joka

quant-phSubmitted Nov 5, 2025• Announced Nov 6, 2025• 0 citations• Revised Jan 22, 2026 (v3)
v1 · Nov 5, 2025v2 · Jan 7, 2026v3 · Jan 22, 2026
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Abstract

In this paper, we prove that the existence of a complete set of mutually unbiased bases (MUBs) in N-dimensional Hilbert space implies the existence of a complete set of mutually orthogonal Latin squares (MOLSs) of order N. In particular, we prove that a complete set of MUBs does not exist in dimension six (the first dimension which is not a power of prime).

Comments
Version 3 differs from version 2 only in few places (section:proof of main theorem) where, for instance, a word projector is replaced by MUB projector etc

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On the Maximal Number of Real Mutually Unbiased Bases

Paweł Wocjan

In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.

•Feb 3, 2005•1 citation
In this paper, we prove that the existence of a complete set of mutually unbiased bases (MUBs) in N-dimensional Hilbert space implies the existence of a complete set of mutually orthogonal Latin squares (MOLSs) of order N. In particular, we prove that a complete set of MUBs does not exist in dimension six (the first dimension which is not a power of prime).
On first sight, the number of mutually unbiased bases (MUBs) in d dimensions and the number of mutually orthogonal Latin squares (MOLS) of order d seem to be closely related. A careful look however reveals the connections cannot be really strong. In this talk a few similarities and differences between MUBs and MOLS are presented. The first similarity involves their maximal numbers. There are at most d + 1 MUBs and also d + 1 squares in an augmented set of MOLS. This number is attained if d is a power of a prime and we give an algorithm which relates MUBs and MOLS in these dimensions [1]. Another similarity follows from considerations of composite d. One can think about a Hilbert space of a composite dimension as describing several subsystems of prime-power dimensions. Since we can always build MUBs in the global space as tensor products of MUBs for the subspaces, there are at least M + 1 of them for the whole system, where M is the smallest prime-power factor of d. A similar construction exists for MOLS: squares of composite orders can be built as direct products of squares of prime-power orders. The corresponding lower bound on the number of MOLS is called the MacNeish’s bound. In general, both the MacNeish’s bound and the number of MUBs which follows from the tensor product construction are known not to be tight. However, the simplest example today involves dimension d = 676 for MUBs [2], whereas MacNeish’s bound is not tight already for d = 10. We therefore checked whether the connection [1] also works for d = 10, and found it no longer links all MOLS to MUBs [3]. The general results of Ref. [4] reveal in this context that the connection fails whenever MacNeish’s bound is not tight. The last (apparent) similarity to be presented is related to incomplete sets of MOLS and MUBs. There is a Latin square of order 4 which does not have an orthogonal mate (so called bachelor Latin square) and connection [1] links it with a set of MUBs which also cannot be extended. It might be tempting to conjecture that every unextendible augmented set of MOLS has a corresponding unextendible set of MUBs. We already disproved such conjecture for the connection [1] and now we briefly describe that there is no other connection for which the conjecture would hold true. One can see this in at least two ways [5]. The first involves bachelor Latin square of order 5. It was recently shown that bachelor Latin squares exist for all orders d > 3 [6]. If there is a strong relation between MOLS and MUBs, there should correspondingly exist a set of 3 MUBs in all dimensions d > 3. To the contrary, it is known that for d = 5 every set of 3 MUBs can be extended to a complete set of 6 MUBs [7]. The second way is more dramatic. It turns out that for d = 6 there exists a set of two MUBs which cannot be extended any further (and six is the smallest dimension in which this happens). Since an augmented set of MOLS contains at least 3 elements, the strong relation is excluded. For more arguments against a close connection between MOLS and MUBs see Ref. [8].
Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in are known when d is a prime power, it is unknown if such complete sets exist in non-prime power dimensions. It has been conjectured that complete sets of MUBs only exist in if a maximal set of mutually orthogonal Latin squares (MOLS) of side length d also exists. There are several constructions (Roy and Scott 2007 J. Math. Phys. 48 072110; Paterek, Dakić and Brukner 2009 Phys. Rev. A 79 012109) of complete sets of MUBs from specific types of MOLS, which use Galois fields to construct the vectors of the MUBs. In this paper, two known constructions of MUBs (Alltop 1980 IEEE Trans. Inf. Theory 26 350–354; Wootters and Fields 1989 Ann. Phys. 191 363–381), both of which use polynomials over a Galois field, are used to construct complete sets of MOLS in the odd prime case. The MOLS come from the inner products of pairs of vectors in the MUBs.
The mean king's problem with maximal mutually unbiased bases (MUB's) in general dimension ddd is investigated. It is shown that a solution of the problem exists if and only if the maximal number (d+1)(d+1)(d+1) of orthogonal Latin squares exists. This implies that there is no solution in d=6d=6d=6 or d=10d=10d=10 dimensions even if the maximal number of MUB's exists in these dimensions.
In this note I point out that (1) pairs of real mutually unbiased bases (i.e., orthonormal bases of R^d) can only exist in dimensions 2 or d where d is a multiple of 4 and that (2) triples of real mutually unbiased bases can only exist for dimensions d that are also squares. For the case d=2^{2m} the maximal number of real MUBs is given by (d+2)/2 (this follows from known results on extremal euclidean line-sets). In the case d=s^2 where s is an even number that is not a power of 2, we have the lower bound N_MOLS(s)+2, where N_MOLS(s) denotes the maximal number of mutually orthogonal Latin square of order s, provided that there exists a Hadamard matrix of size s. It is not known how good this bound is. Moreover, I observe that the question of deciding if there always two real MUBs is equivalent to the Hadamard conjecture.