Three Quantum Latin Squares of Order 6 with Cardinalities 13, 15, and 17

TL;DR

This paper presents three explicit quantum Latin squares of order 6 with specific cardinalities, constructed using orthogonal decompositions and Hadamard pairs, advancing the understanding of quantum combinatorial designs.

Contribution

It introduces three new quantum Latin squares of order 6 with unique cardinalities, constructed through novel algebraic and combinatorial methods.

Findings

Constructed quantum Latin squares of orders 6 with cardinalities 13, 15, and 17.

Demonstrated methods based on orthogonal decompositions and Hadamard pairs.

Provided explicit examples expanding the catalog of quantum Latin squares.

Abstract

We give three explicit quantum Latin squares of order $6$, with cardinalities $13$, $15$, and $17$. Throughout, vectors differing only by a global phase are counted as identical. The cardinality-$13$ construction is based on an orthogonal direct-sum decomposition $\C^6=\C^4\oplus\C^2$. The cardinality-$15$ and cardinality-$17$ constructions are based on two-dimensional Hadamard pairs supported on coordinate planes.