Local equivalence of quantum orthogonal arrays and orthogonal arrays

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Local equivalence of quantum orthogonal arrays and orthogonal arrays Jiao Du1,2 · Cuijiao Yin1,2 · Shanqi Pang1,2

· Tianyin Wang3

Received: 1 April 2020 / Accepted: 3 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020

Abstract Two orthogonal arrays (OAs) are locally equivalent if they lead to locally equivalent quantum states. By studying permutations of the rows or levels of each factor, we present the local equivalence between two OAs. Using the tensor products of unitary matrices, we find that two infinite classes of OAs, OA(d n , n + 1, d, n) and OA(d, n + 1, d, 1), are locally equivalent. Therefore, we provide a positive answer to the open problem of which OAs are locally equivalent, i.e., OA(r , N , d, k) ∼loc OA(r  , N , d, k  ), in a sense that they lead to locally equivalent quantum states. In addition, an improved quantum orthogonal array (IQOA) is defined. The equivalence and local equivalence of IQOAs are investigated. Keywords Local equivalence · Improved quantum orthogonal array · Quantum state · Unitary matrix

1 Introduction Unitary operation is one of the most basic components of quantum mechanics. Research on various properties of unitary operation is the core of quantum information processing. Entanglement is invariant under the choice of the local basis [1], which opens the possibility of applying tensor network techniques to describing the ground states of relevant physical systems [2], as well as phenomena such as quantum phase transitions [3]. There is no doubt that the construction of multipartite entangled states is a key problem in quantum information theory [4], as these states have numerous

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Shanqi Pang [email protected]

1

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China

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Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, Henan Normal University, Xinxiang 453007, China

3

School of Mathematical Science, Luoyang Normal University, Luoyang 471934, China 0123456789().: V,-vol

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applications to quantum error-correcting codes (QECCs) [5,6], key distribution [7], dense coding [8], quantum cryptography [9,10], and quantum teleportation [11–14]. Orthogonal arrays (OAs) are well known to play important roles in coding theory, cryptography, computer science, and statistics, especially in the design of experiments. We refer to the book [15] for more applications of orthogonal arrays (OAs). Recently, OAs have been applied to quantum information theory such as [16] and decoupling schemes [17]. A pure quantum state of N subsystems, each with d levels, is said to be kuniform if all of its reductions to k qudits are maximally mixed [4,18,19]. Interestingly, ((N , 1, k + 1))d QECCs have a one-to-one connection with the k-uniform states of N qudits [4,20,21]. An OA(r , N , d1n 1 d2n 2 · · · dlnl , k) is an r × N matrix, having n i l factors with di levels, where i = 1, 2, . . . , l, l is an integer, N = i=1 n i , and di = d j for i = j, with