Abstract
An \(N \times k\) array A with entries from v-set \({\mathcal {V}}\) is said to be an orthogonal array with v levels, strength t and index \(\lambda \), denoted by OA(N; t, k, v), if every \(N \times t\) sub-array of A contains each t-tuple based on \({\mathcal {V}}\) exactly \(\lambda \) times as a row. An OA(N; t, k, v) is called irredundant, denoted by IrOA(N; t, k, v), if in any \(N\times (k-t )\) sub-array, all of its rows are different. The definition of an IrOA was firstly introduced by Goyeneche and \({\dot{Z}}\)yczkowski (Phys Rev A 90:022316, 2014) who showed an IrOA(N; t, k, v) corresponds to a t-uniform state of k subsystems with local dimension v. In this paper, we construct some kinds of 2-uniform states by establishing the existence of an IrOA\((v^3;2,12,v)\) for any integer \((v\ge 4)\) and \((v\not \equiv 2\pmod 4)\), and an IrOA\((v^3;2,3v,v)\) for any prime or prime power \(v\ge 3\).
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Chen, G., Zhang, X. & Guo, Y. New results for 2-uniform states based on irredundant orthogonal arrays. Quantum Inf Process 20, 43 (2021). https://doi.org/10.1007/s11128-020-02978-x
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DOI: https://doi.org/10.1007/s11128-020-02978-x