Quantum combinatorial designs and $k$-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.

References in corpus (5)

• Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions
• Maximally multipartite entangled states
• Genuinely multipartite entangled states and orthogonal arrays
• Mutually unbiased bases, orthogonal Latin squares, and hidden-variable models
• Quantum Teamwork for Unconditional Multiparty Communication with Gaussian States

Cited by in corpus (5)

• Multipartite entanglement and quantum error identification in $D$-dimensional cluster states
• Mutually unbiased maximally entangled bases from difference matrices
• Quantum $k$-uniform states from quantum orthogonal arrays
• Absolutely maximally entangled pure states of multipartite quantum systems
• Scalable and fault-tolerant preparation of encoded k-uniform states