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

Abstract

There has been much interest in mutually unbiased bases (MUBs) and their connections with various other discrete structures, such as projective planes, mutually orthogonal Latin squares (MOLS) etc. It has been conjectured by Saniga et al. (J Opt B Quantum Semiclass Opt 6:L19-L20, 2004) that the existence of a complete set of MUBs in C d is linked to the existence of a complete set of MOLS of side length d. Since more is known about MOLS than about MUBs, most research has concentrated on constructing MUBs from MOLS (Roy and Scott, J Math Phys 48:072110, 2007; Paterek et al., Phys Rev A 70:012109, 2009). This paper gives a simple algebraic construction of MOLS from two known constructions of MUBs in the odd prime power case.

Mutually unbiased bases as submodules and subspaces

“…Theorem 11 may be used. This highlights that structures which are not present in sets of vectors, may be present in another way, see also [17]. We use essentially the same proof for the MUBs generated by Theorem 3.…”

“…These structures include planar functions [12], [18], symplectic spreads [11] as well as specific affine planes [8], [17].…”

Constructions and properties of optimally spread subspace packings via symmetric and affine block designs and mutually unbiased bases

“…Mutually Unbiased Bases. Motivated by consistency and, again, to mitigate notational and typographical issues that will arise in the next section, we make a slight modification to the usual definition [6,18,36,42,47,56,67] of mutually unbiased bases -they are usually defined as sets of orthonormal bases with a special property. Equivalently, we reformulate them as families of tight fusion frames comprised of rank one projectors.…”

More Constructions of Approximately Mutually Unbiased Bases

“…Using such a basis to obtain optimal outcomes leads to maximally random results compared to other bases. Therefore, MUBs constitute a basic ingredient in many applications of quantum information processing: quantum tomography, quantum key distribution in cryptography, discrete Wigner function, quantum teleportation, and quantum error correction codes (see [7,9,18,19] and the references therein). MUBs are also closely related to spherical 2-designs [4,10], semifields [4], complex Hadamard matrices [3], orthogonal Latin squares [5], finite geometry [5], frames [3], planar functions [5] and character sums over finite fields [9,17].…”