Two and three-uniform states from irredundant orthogonal arrays

Abstract

A pure quantum state of N subsystems, each with d levels, is said to be k-uniform if all of its reductions to k qudits are maximally mixed. Only the uniform states obtained from orthogonal arrays (OAs) are considered throughout this work. The Hamming distances of OAs are specially applied to the theory of quantum information. By using difference schemes and orthogonal partitions, we construct a series of infinite classes of irredundant orthogonal arrays (IrOAs), then answer the open questions of whether there exist 3-uniform states of N qubits and 2-uniform states of N qutrits, and whether 3-uniform states of qudits (d > 2) for high values of N can be explicitly constructed. In fact, we obtain 3-uniform states for an arbitrary number of N ≥ 8 qubits and 2uniform states of N qutrits for every N ≥ 4. Additionally, we provide explicit constructions of the 3-uniform states of N ≥ 8 qutrits, N = 6 and N ≥ 8 ququarts and ququints, N ≥ 6 qudits having d levels for any prime power d > 6, and N = 8 and N ≥ 12 qudits having d levels for non-prime-power d ≥ 6. Moreover, we describe an explicit construction scheme for the 2-uniform states of qudits having d ≥ 4 levels. The proofs of existence of the 2-uniform states of N ≥ 6 qubits are simplified by using a class of OAs. Two special 3uniform states are obtained from IrOA(32, 10, 2, 3) and IrOA(32, 11, 2, 3) using the interaction column property of OAs.

Further results on 2-uniform states arising from irredundant orthogonal arrays

“…When t = 3, there are some results for uniform states [11,28,29]. In 2019, Zang, Li and Pang et al presented some 3-uniform states for k subsystems by irredundant orthogonal arrays [21,23,27]. Actually, for some given value k, constructing more kinds of 3-uniform states with v levels of non-prime powers from orthogonal arrays is a very good further research topic.…”

“…Moreover, there exists a perfect match between the parameters of an IrOA(r, k, v, t) and the parameters of a t-uniform state, which is listed in Table 1. [21,23,27]. In this paper, we give some new methods of constructing for irredundant orthogonal arrays, for example, direct construction included difference matrix and computer program, and classical recursive constructions in designs theory, then we obtain some new irredundant orthogonal arrays as well as the parameter situations wherein corresponding 2uniform states exist for these irredundant orthogonal arrays.…”

Constructions of irredundant orthogonal arrays

“…Moreover, they pointed out that a particular orthogonal array, called irredundant orthogonal array, which is corresponding to a t-uniform state. After then, Zang, Li, and Pang et al have obtained the existence of 2, 3-uniform states of k subsystems by irredundant orthogonal arrays [26,30,43]. In 2016, Goyeneche, Bielawski, and Życzkowski [12] firstly introduced irredundant mixed orthogonal arrays to investigate entanglement for heterogeneous systems.…”

“…For more results on irredundant orthogonal arrays, we refer the reader to [26,30,42,43,44] and references therein. In this article, we focus on the constructions of irredundant orthogonal arrays.…”

Quantum Frequency Arrangements, Quantum Mixed Orthogonal Arrays and Entangled States

“…However, the general problem of how to construct genuinely multipartite entangled states remains unresolved. There has been some progress towards a solution [5]- [7], [10], [20], but the task at hand is generally considered a difficult one.…”