Mutually orthogonal Latin squares from the inner products of vectors in mutually unbiased bases

Abstract

Mutually unbiased bases (MUBs) are important in quantum information theory. While constructions of complete sets of d + 1 MUBs in C d 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 C d 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.

Mutually orthogonal Latin squares and mutually unbiased bases in dimensions of odd prime power

“…The linear combinations also make use of weights. In [6], the weights from the two constructions were found to have very different structures, in the odd prime case. Further investigation of structure and meaning of these weights is required and is currently being undertaken.…”

“…In [6], it is shown that for p an odd prime, the MUBs obtained by Theorems 2.2 and 2.3 can be used to construct complete sets of MOLS. In this paper q − 1 MOLS are constructed from q + 1 MUBs for all odd prime powers q = p n .…”

“…The crux of the construction in [6] uses the way in which the vectors of the standard basis can be expressed as combinations of vectors from the sets B a above. Similarly, start by denoting the standard basis by S = {s 0 , s 1 , .…”

“…These q − 1 Latin striations are obtained by combining the weighted vectors v k ab of the q MUBs in q − 1 uniquely different ways. In [6], the s k , 1 ≤ k ≤ q − 1 are expressed as other combinations of the v ab resulting in the Latin striations. The construction follows on similar lines here.…”

The correspondence between mutually unbiased bases and mutually orthogonal extraordinary supersquares

“…Similarities between MUBs and mutually orthogonal Latin squares have been analyzed during the last years [17,22,23,24,25,26,27,28,29,30]. In Refs.…”