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

Yajuan Zang; Guangzhou Chen; Kejun Chen; Zihong Tian

doi:10.3934/amc.2020109

Article Contents

2022, Volume 16, Issue 2: 231-247. Doi: 10.3934/amc.2020109

This issue Previous Article Next Article Two constructions of low-hit-zone frequency-hopping sequence sets

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

1.
School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, 050024, China
2.
Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China
3.
School of Mathematics and Information Science, Nanjing Normal University of Special Education, Nanjing, 210038, China

^* Corresponding author: Zihong Tian
^* Corresponding author: Zihong Tian

Received: February 2020

Revised: May 2020

Early access: September 2020

Published: May 2022

This work was supported by the National Natural Science Foundation of China (Grant No. 11871019) and the Graduate Innovation Project of Hebei Province (Grant No. CXZZBS2019077).

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Abstract

The notion of an irredundant orthogonal array (IrOA) was introduced by Goyeneche and $\dot{Z}$ yczkowski who showed an IrOA $_{\lambda}(t, k, v)$ 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 $_{\lambda}(2, 5, v)$ for any integer $v\geq 4$ , $v\neq 6$ ; IrOA $_{\lambda}(2, 6, v)$ for any integer $v\geq 2$ ; IrOA $_{\lambda}(2, q, q)$ and IrOA $_{\lambda}(2, q+1, q)$ for any prime power $q >3$ .

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Mathematics Subject Classification: Primary: 05B15, 81P99; Secondary: 05B20.

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Table 1. Correspondence between parameters of OAs and quantum states

Parameters	Orthogonal array	Multipartite quantum state $\|\Phi\rangle$
$r$	Runs	Number of linear terms in the state
$k$	Factors	Number of qudits
$v$	Levels	Dimension of the subsystem ( $v=2$ for qubits)
$t$	Strength	Class of entanglement ( $t$ -uniform)

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