A pure quantum state of [Formula: see text] subsystems with [Formula: see text] levels is called [Formula: see text]-uniform state if all its reductions to [Formula: see text] qudits are maximally mixed. We construct 3-uniform states for an arbitrary number of [Formula: see text] via orthogonal arrays and the method of adding minus signs mathematically.

The notion of an irredundant orthogonal array (IrOA) was introduced by Goyeneche andŻyczkowski who showed an IrOA λ (t, k, v) corresponds to a t-uniform state of k subsystems with local dimension v (Physical Review A. 90 (2014), 022316). In this paper, we construct some kinds of 2-uniform states by establishing the existence of IrOA λ (2, 5, v) for any integer v ≥ 4, v = 6; IrOA λ (2, 6, v) for any integer v ≥ 2; IrOA λ (2, q, q) and IrOA λ (2, q + 1, q) for any prime power q > 3.

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.

Based on maximally entangled states, we explore the constructions of mutually unbiased bases in bipartite quantum systems. We present a new way to construct mutually unbiased bases by difference matrices in the theory of combinatorial designs. In particular, we establish q mutually unbiased bases with q − 1 maximally entangled bases and one product basis in ℂq ⊗ ℂq for arbitrary prime power q. In addition, we construct maximally entangled bases for dimension of composite numbers of non-prime power, such as five maximally entangled bases in ℂ12⊗ℂ12and ℂ21⊗ℂ21, which improve the known lower bounds for d = 3m, with (3, m) = 1 in ℂd ⊗ ℂd. Furthermore, we construct p + 1 mutually unbiased bases with p maximally entangled bases and one product basis in ℂp ⊗ ℂp2 for arbitrary prime number p.

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