Construction of Asymmetric Orthogonal Arrays of Strength &lt;i&gt;t&lt;/i&gt; from Orthogonal Partition of Small Orthogonal Arrays

Pang, Shanqi; Xiao, Lin; Wang, Jing

doi:10.1587/transfun.e101.a.1267

Cited by 18 publications

(12 citation statements)

References 20 publications

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“…Construction of 3-uniform states of N ≥ 8 qutrits (d = 3) By studying the minimal distance of the symmetrical OAs constructed from the orthogonal partition methods in, 32 we can obtain an IrOA(r, N, 3, 3) and 3-uniform states of N qutrits for every N ≥ 12. By constructing OA(81, 10, 3, 3) and OA(243, 11, 3, 3) and computing their minimal distances, we have an IrOA(r, N, 3, 3) for N = 8, 9, 10, 11.…”

Section: Resultsmentioning

confidence: 99%

“…The results presented in this paper will establish a foundation for solving other open problems, such as the construction of k-uniform states of N qudits (d ≥ 2) for k ≥ 4, including the problem stated by Huber et al, 14 and heterogeneous multipartite systems, 17 since the proposed construction methods can be suitable for IrOAs of any strength k ≥ 4 and irredundant mixed orthogonal arrays (IrMOAs). 17,32,33 In the construction process, we often encounter the problem that some uniform states could have fewer terms or qudits. If the Hadamard conjecture is considered, Theorems 2.9 and 2.10 state that the number of terms in many 2-uniform states could be reduced.…”

Section: Discussionmentioning

confidence: 99%

“…[24][25][26][27] Recently, many new methods of constructing OAs of strength k, especially mixed OAs, have been presented, and many new classes of OAs have been obtained. [28][29][30][31][32][33] It is these new developments in OAs that suggest the possibility of constructing infinitely many new k-uniform states from IrOAs.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, as it has been proved that IrOA(r, 7, 2, 3), IrOA(r, 7, 3, 3), IrOA(r, 6, 3, 3), IrOA(r, 4, 6, 2), IrOA(r, 6, 2, 3), IrOA(r, 4, 2, 2), and IrOA(r, 5, 2, 2) do not exist, the corresponding uniform states can not be obtained. However, two special 3-uniform states are obtained from IrOA (32,10,2,3) and IrOA (32,11,2,3) using the interaction column property of OAs.…”

Section: Introductionmentioning

confidence: 99%

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Two and three-uniform states from irredundant orthogonal arrays

et al. 2019

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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.

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Section: Resultsmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Two and three-uniform states from irredundant orthogonal arrays

et al. 2019

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“…As is often the case [15], [17], combinatorics can be useful to quantum information theory, and orthogonal arrays (OAs) are fundamental ingredients in the construction of other useful combinatorial objects [9]. Recently, many new methods of constructing OAs of strength k, especially mixed orthogonal arrays (MOAs), have been presented, and many new classes of OAs have been obtained [3], [16], [18], [19]. It is these new developments in OAs that suggest the possibility of constructing infinitely many new genuinely multipartite entangled states.…”

Section: Introductionmentioning

confidence: 99%

Quantum Frequency Arrangements, Quantum Mixed Orthogonal Arrays and Entangled States

Pang

Zhang

Xiao

2020

IEICE Trans. Fundamentals

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In this work, we introduce notions of quantum frequency arrangements consisting of quantum frequency squares, cubes, hypercubes and a notion of orthogonality between them. We also propose a notion of quantum mixed orthogonal array (QMOA). By using irredundant mixed orthogonal array proposed by Goyeneche et al. we can obtain k-uniform states of heterogeneous systems from quantum frequency arrangements and QMOAs. Furthermore, some examples are presented to illustrate our method.

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On the construction of nested orthogonal arrays with the adjacent numbers of levels

Pang,

Zhu

2024

Stat

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Nested orthogonal arrays (NOAs) provide an option for designing an experimental setup consisting of two experiments, with the expensive higher‐precision experiment nested within a larger and relatively inexpensive lower‐precision experiment. Construction of NOAs with the adjacent numbers of levels is a challenging problem. In this paper, we present several methods for constructing such NOAs and obtain some classes of such new symmetric NOAs in which the larger arrays have minimum run size. These methods are also extended to construction of NOAs with more than two layers. Furthermore, by adding some columns to these symmetric NOAs, we can construct a lot of new asymmetric NOAs. Illustrative examples are given.

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Construction of Asymmetric Orthogonal Arrays of Strength <i>t</i> from Orthogonal Partition of Small Orthogonal Arrays

Cited by 18 publications

References 20 publications

Two and three-uniform states from irredundant orthogonal arrays

Two and three-uniform states from irredundant orthogonal arrays

Quantum Frequency Arrangements, Quantum Mixed Orthogonal Arrays and Entangled States

On the construction of nested orthogonal arrays with the adjacent numbers of levels

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