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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

k-Uniform States and Quantum Combinatorial Designs

Shanqi Pang, Xiankui PENG, Zhang Xiao, Ruining Zhang, Cuijiao Yin

Published Dec 19, 2021• 4 citations
Publisher

Abstract

Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.

DOI
10.1587/transfun.2021eap1090

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Quantum combinatorial designs are gaining popularity in quantum information theory. Quantum Latin squares can be used to construct mutually unbiased maximally entangled bases and unitary error bases. Here we present a general method for constructing quantum Latin arrangements from irredundant orthogonal arrays. As an application of the method, many new quantum Latin arrangements are obtained. We also find a sufficient condition such that the improved quantum orthogonal arrays [10] are equivalent to quantum Latin arrangements. We further prove that an improved quantum orthogonal array can produce a quantum uniform state.
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We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes, and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way in which quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called kkk-uniform, i.e., multipartite pure states such that every reduction to kkk parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite kkk-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.
We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how a pair of mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construction to an existing construction due to Beth and Wocjan and show that ours is strictly more general.
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.
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Goyeneche et al recently proposed a notion of orthogonality for quantum Latin squares, and showed that orthogonal quantum Latin squares yield quantum codes. We give a simplified characterization of orthogonality for quantum Latin squares, which we show is equivalent to the existing notion. We use this simplified characterization to give an upper bound for the number of mutually orthogonal quantum Latin squares of a given size, and to give the first examples of orthogonal quantum Latin squares that do not arise from ordinary Latin squares. We then discuss quantum Latin isometry squares, generalizations of quantum Latin squares recently introduced by Benoist and Nechita, and define a new orthogonality property for these objects, showing that it also allows the construction of quantum codes. We give a new characterization of unitary error bases using these structures.