Quantum combinatorial designs and $k$-uniform states

TL;DR

This paper extends quantum combinatorial designs by introducing incomplete quantum Latin squares and cubes, providing new construction methods, and establishing their connection to generating $k$-uniform entangled states.

Contribution

It introduces incomplete quantum Latin squares and cubes, develops methods for their mutual orthogonality, and links these designs to the creation of $k$-uniform entangled states.

Findings

Introduced incomplete quantum Latin squares and cubes.

Developed construction methods for mutually orthogonal designs.

Connected designs to the generation of $k$-uniform states.

Abstract

Goyeneche et al.\ [Phys.\ Rev.\ A \textbf{97}, 062326 (2018)] introduced several classes of quantum combinatorial designs, namely quantum Latin squares, quantum Latin cubes, and the notion of orthogonality on them. They also showed that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Moreover, they established a relationship between quantum combinatorial designs and a remarkable class of entangled states called $k$-uniform states, i.e., multipartite pure states such that every reduction to $k$ parties is maximally mixed. In this article, we put forward the notions of incomplete quantum Latin squares and orthogonality on them and present construction methods for mutually orthogonal quantum Latin squares and mutually orthogonal quantum Latin cubes. Furthermore, we introduce the notions of generalized mutually orthogonal quantum Latin squares and generalized mutually orthogonal quantum Latin cubes, which are equivalent to quantum orthogonal arrays of size $d^2$ and $d^3$, respectively, and thus naturally provide $2$- and $3$-uniform states.