Orthogonality for Quantum Latin Isometry Squares

TL;DR

This paper simplifies the understanding of orthogonality in quantum Latin squares, establishes bounds on their number, introduces quantum Latin isometry squares, and connects these concepts to quantum codes and unitary error bases.

Contribution

It provides a simplified characterization of orthogonality, introduces quantum Latin isometry squares, and links these structures to quantum coding and error bases.

Findings

Derived an upper bound for mutually orthogonal quantum Latin squares.

Constructed the first examples of orthogonal quantum Latin squares not from classical Latin squares.

Established a new characterization of unitary error bases using quantum Latin isometry squares.

Abstract

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.