On the connection between mutually unbiased bases and orthogonal Latin squares

We offer a piece of evidence that the problems of finding the number of mutually unbiased bases (MUB) and mutually orthogonal Latin squares (MOLS) might not be equivalent. We study a particular procedure which has been shown to relate the two problems and generates complete sets of MUBs in power-of-prime dimensions and three MUBs in dimension six. For these cases, every square from an augmented set of MOLS has a corresponding MUB. We show that this no longer holds for certain composite dimensions.Comment: 6 pages, submitted to Proceedings of CEWQO 200

Searching for incomplete self orthogonal latin squares : a targeted and parallel approach

The primary purpose of this dissertation is in the search for new methods in which to search for Incomplete Self Orthogonal Latin Squares. As such a full understanding of the structures involved must be examined, starting from basic Latin Squares. The structures will be explained and built upon in order to cover Mutually Orthogonal Latin Squares, Frame Latin Squares and Self Orthogonal Latin Squares. In addition the related structure Orthogonal Arrays, will be explained as they relate to Incomplete Self Orthogonal Latin Squares. This paper also dedicates time to explaining basic search methods and optimizations that can be done. The two search methods of focus are the backtracking algorithm and heuristic searches. In our 6nal method the two will work together to achieve an improved result. The methods currently being used to search in parallel are also provided, along with the necessary backup to there structure. The main research of this paper is focused on the search for Incomplete Self Orthogonal Squares. This is done by breaking down the problem into four separate areas of the square. By separating the blocks it enables us to work on a smaller problem while eliminating many incorrect solutions. The solution methodology is broken up into three steps and systematically solving the individual areas of the square. By taking advantage of the properties of squares to constrain our search as much as possible we succeeded in reducing the total search time significantly. Unfortunately, even with our improvement in the overall search time, no open incomplete self orthogonal latin square problems could be solved. Full results and comparisons to existing methods are provided

Orthogonality for Quantum Latin Isometry Squares

Goyeneche et al recently proposed a notion of orthogonality for quantum Latin squares, and showed that orthogonal quantum Latin squares yield quantum codes. We give a simplified characterization of orthogonality for quantum Latin squares, which we show is equivalent to the existing notion. We use this simplified characterization to give an upper bound for the number of mutually orthogonal quantum Latin squares of a given size, and to give the first examples of orthogonal quantum Latin squares that do not arise from ordinary Latin squares. We then discuss quantum Latin isometry squares, generalizations of quantum Latin squares recently introduced by Benoist and Nechita, and define a new orthogonality property for these objects, showing that it also allows the construction of quantum codes. We give a new characterization of unitary error bases using these structures.Comment: In Proceedings QPL 2018, arXiv:1901.0947