An Overview of Quantum Latin Squares in Quantum Information Theory

Quantum Latin Squares have emerged as mathematical structures of interest in quantum information theory. This paper provides a comprehensive overview of quantum Latin squares. We aim to synthesize the current knowledge for the benefit of researchers in quantum information theory. We begin by establishing the theoretical foundation linking classical Latin squares with their quantum counterparts. This provides insight into their potential applications, particularly in the realm of quantum error correction. Through a detailed examination, we outline known constructions and elucidate the properties of quantum Latin squares, highlighting their role within quantum computing frameworks. We then explore the relationships of quantum Latin squares with mutually unbiased bases and unitary error bases. This underscores their importance in the construction of quantum error correcting codes using mutually orthogonal quantum Latin squares. We give significant focus to a comparative analysis between classical and quantum Latin squares, particularly in their applications to error correcting codes. The applications are analogous to the application of classical Latin squares to classical error correcting codes. This overview aims to not only compile existing research but also to inspire further exploration into Quantum Latin squares and their applications in quantum information theory.