The existence of non-classical orthogonal quantum Latin squares

TL;DR

This paper introduces new types of quantum Latin squares with special properties and provides methods for constructing non-classical examples, expanding the understanding of their existence and applications in quantum information.

Contribution

It proposes novel quantum Latin squares with properties like idempotency and self-orthogonality, along with construction techniques for non-classical variants.

Findings

Existence of non-classical 2-idempotent MOQLS(v) established.

Existence of non-classical 2, 3-MOQLS(v) confirmed.

Existence of non-classical SOQLS(v) demonstrated.

Abstract

Quantum Latin squares are a generalization of classical Latin squares in quantum field and have wide applications in unitary error bases, mutually unbiased bases, $k$-uniform states and quantum error correcting codes. In this paper, we put forward some new quantum Latin squares with special properties, such as idempotent quantum Latin square, self-orthogonal quantum Latin square, holey quantum Latin square, and the notions of orthogonality on them. We present some forceful construction methods including PBD constructions and filling in holes constructions for non-classical quantum Latin squares. As consequences, we establish the existence of non-classical 2-idempotent MOQLS$(v)$, non-classical 2, 3-MOQLS$(v)$ and non-classical SOQLS$(v)$ except possibly for several definite values.