河北师范大学学报—

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刊名：河北师范大学学报（自然科学版）Journal of Hebei Normal University (Natural Science)
主办：河北师范大学
ISSN： 1000-5854
CN： 13-1061/N
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基于非冗余正交阵列的2-均匀态

臧亚娟¹

陈思琦²

田子红³

（1.河北师范大学初等教育系,河北石家庄 050024; 2.唐山市第四十九中学，河北唐山 063021； 3. 河北师范大学数学科学学院，河北石家庄 050024）
DOI： 10.13763/j.cnki.jhebnu.nse.202401005

Two-uniform States Based on Irredundant Orthogonal Arrays

ZANG Yajuan¹,CHEN Siqi²,TIAN Zihong³

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摘要/Abstract

摘要：

一个强度为 t,约束数为 k,指标为 λ 的正交阵列,记为 OA(N,k,v,t),是一个元素取自 v 元集 V上的 N×k 阵列,要求在它的任意 N×t 子阵列中,V 上的任意有序 t 元组作为行在该子阵列中恰好出现 λ 次,其中 λ=N/v^t.一个正交阵列OA(N,k,v,t)称为是非冗余的,记为 IrOA(N,k,v,t),要求其任意 N×(k－t) 子阵列,每行互不相同.在 2014年,Goyeneche 等给出非冗余正交阵列概念的同时,也指出非冗余正交阵列与 t均匀态有着密切的联系.本文利用差矩阵给出带有部分强度 t>2子阵列的IrOA(N,k,v,2) 的构造方法,进而给出一些 2均匀态的存在性结果.

Abstract：

An N×k array A with entries from a v-set V is said to be an orthogonal array with k constraints,strength t and index λ,denoted by OA(N,k,v,t),if every N×t sub-array of A contains each t-tuple based on V exactly λ times as a row,where λ=N/v^t.An OA(N,k,v,t) is called irredundant,denoted by IrOA(N,k,v,t),if in any N×(k－t) sub-array,all of its rows are different.In 2014,Goyeneche et al. introduced the definition of irredundant orthogonal array and pointed out the close relationship between an irredundant orthogonal array and a t-uniform state.In this paper,we present some new constructions of IrOA(N,k,v,2) with subarrays of strength t>2 by using difference matrices.Furthermore,some classes of 2-uniform states arise from these irredundant orthogonal arrays.

关键词

关键词： 正交阵列 ; t均匀态 ; 非冗余正交阵列 ; 差矩阵

Key words： orthogonal array ; t-uniform state ; irredundant orthogonal array ; difference matrix

参考文献 22

[1] BENNETT C H.Quantum Cryptography Using Any Two Nonorthogonal States[J].Physical Review Letters,1992,68(21):3121-3124.doi:10.1103/PhysRevLett.68.3121
[2] BENNETT C H,BRASSARD G,CRPEAU C,et al.Teleporting an Unknown Quantum State via Dual Classical and Einstein-podolsky-rosen Channels[J].Physical Review Letters,1993,70(13):1895-1899.doi:10.1103/PhysRevLett.70.1895
[3] JOZSA R,LINDEN N.On the Role of Entanglement in Quantum Computational Speed-up[J].Proceedings of Royal Society A,2003,459(2036):2011-2032.doi:10.1098/rspa.2002.1097
[4] LO H K,CURTY M,QI B.Measurement-device-independent Quantum Key Distribution[J].Physical Review Letters,2012,108(13):130503.doi:10.1103/PhysRevLett.108.130503
[5] GOYENECHE D,YCZKOWSKI K.Genuinely Multipartite Entangled States and Orthogonal Arrays[J].Physical Review A,2014,90(2):022316.doi:10.1103/PhysRevA.90.022316
[6] GOYENECHE D,ALSINA D,LATORRE J I,et al.Absolutely Maximally Entangled States,Combinatorial Designs and Multiunitary Matrices[J].Physical Review A,2015,92(3):032316.doi:10.1103/PhysRevA.92.032316
[7] HELWIG W,CUI W,LATORRE J I,et al.Absolute Maximal Entanglement and Quantum Secret Sharing[J].Physical Review A,2012,86(3):052335.doi:10.1103/PhysRevA.86.052335
[8] RAISSI Z,GOGOLIN C,RIERA A,et al.Constructing Optimal Quantum Error Correcting Codes from Absolute Maximally Entangled States[J].Journal of Physics A:Mathematical and Theoretical,2018,51:075301.doi:10.1088/1751-8121/aaa151
[9] LI M S,WANG Y L.k-Uniform Quantum States Arising from Orthogonal Arrays[J].Physical Review A,2019,99(4):042332.doi:10.1103/PhysRevA.99.042332
[10] PANG S Q,ZHANG X,LIN X,et al.Two and Three-uniform States from Irredundant Orthogonal Arrays[J].npj Quantum Information,2019,5:52.doi:10.1038/s41534-019-0165-8
[11] ZANG Y J,ZUO H J,TIAN Z H.3-uniform States and Orthogonal Arrays of Strength 3[J].International Journal of Quantum Information,2019,17(1):1950003.doi:10.1142/S0219749919500035
[12] ZANG Y J,CHEN G Z,CHEN K J,et al.Further Results on 2-uniform States Arising from Irredundant Orthogonal Arrays[J].Advances in Mathematics of Communications,2022,16(2):231-247.doi:10.3934/amc.2020109
[13] ZANG Y J,FACCHI P,TIAN Z H.Quantum Combinatorial Designs and k-uniform States[J].Journal of Physics A:Mathematical and Theoretical,2021,54:505204.doi:10.1088/17518121/ac3705
[14] CHEN G Z,ZHANG X T,GUO Y.New Results for 2-uniform States Based on Irredundant Orthogonal Arrays[J].Quantum Information Processing,2021,20:43.doi:10.1007/s11128-020-02978-x
[15] CHEN G Z,ZHANG X T.Constructions of Irredundant Orthogonal Arrays[J].Advances in Mathematics of Communications,2023,17(6)：13141337.doi:10.3934/amc.2021051
[16] BURATTI M.Recursive Constructions for Difference Matrices and Relative Difference Families[J].Journal of Combinatorial Designs,1998,6(3):165182.
[17] DRAKE D A.Partial λ-geometries and Generalized Hadamard Matrices over Groups[J].Canadian Journal of Mathematics,1979,31(3):617-627.doi:10.4153/CJM-1979-062-1
[18] EVANS A B.On Orthogonal Orthomorphisms of Cyclic and Non-abelian Groups.Ⅱ[J].Journal of Combinatorial Designs,2007,15(3):195-209.doi:10.1002/jcd.20138
[19] GE G.On (g,4;1)-difference Matrices[J].Discrete Mathematics,2005,301(2-3):164-174.doi:10.1016/j.disc.2005.07.004
[20] JI L J,YIN J X.Constructions of New Orthogonal Arrays and Covering Arrays of Strength Three[J].Journal of Combinatorial Theory,Series A,2010,117(3):236-247.doi:10.1016/j.jcta.2009.06.002
[21] HEDAYAT A S,SLOANE N J A,STUFKEN J.Orthogonal Arrays:Theory and Applications[M].New York:Springer Series in Statistics.Springer-verlag,1999,XXIII,417.doi:10.1007/978-1-4612-1478-6
[22] 陈思琦.变强度混合覆盖阵列的构造[D].石家庄:河北师范大学,2020. CHEN S Q.The Constructions of Variable Strength Mixed Covering Arrays[D].Shijiazhuang:Hebei Normal Univer-sity,2020.