doi: 10.3934/amc.2020109

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

Received  February 2020 Revised  May 2020 Published  September 2020

Fund Project: 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)

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 $.

Citation: Yajuan Zang, Guangzhou Chen, Kejun Chen, Zihong Tian. Further results on 2-uniform states arising from irredundant orthogonal arrays. Advances in Mathematics of Communications, doi: 10.3934/amc.2020109
References:
[1]

R. J. R. AbelF. E. Bennett and G. G. Ge, Super-simple holey steiner pentagon systems and related designs, J. Combin. Des., 16 (2008), 301-328.  doi: 10.1002/jcd.20171.  Google Scholar

[2]

C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Phy. Rev. Lett., 68 (1992), 3121-3124.  doi: 10.1103/PhysRevLett.68.3121.  Google Scholar

[3]

C. H. BennettG. BrassardC. CrépeauR. JozsaA. Peres and W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Phys. Rev. Lett., 70 (1993), 1895-1899.  doi: 10.1103/PhysRevLett.70.1895.  Google Scholar

[4]

R. C. Bose, A note on orthogonal arrays, Ann. Math. Stat., 21 (1950), 304-305.   Google Scholar

[5]

K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Stat., 23 (1952), 426-434.  doi: 10.1214/aoms/1177729387.  Google Scholar

[6]

G. Z. ChenK. J. Chen and Y. Zhang, Super-simple (5, 4)-GDDs of group type $g^{u}$, Front. Math. China, 9 (2014), 1001-1018.  doi: 10.1007/s11464-014-0393-3.  Google Scholar

[7]

C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, Chapman and Hall/CRC Press, 2007.  Google Scholar

[8]

Y. H. Chen, Constructions of Optimal Detecting Arrays of Degree 5 and Strength 2, Master Thesis, Soochow University, 2011. Google Scholar

[9]

P. Facchi, Multipartite entanglement in qubit systems, Rend. Lincei Mat. Appl., 20 (2009), 25-67.  doi: 10.4171/RLM/532.  Google Scholar

[10]

P. Facchi, G. Florio, G. Parisi and S. Pascazio, Maximally multipartite entangled states, Phys. Rev. A, 77 (2008), 060304, 1–4. doi: 10.1103/PhysRevA.77.060304.  Google Scholar

[11]

K. Q. FengL. F. JinC. P. Xing and C. Yuan, Multipartite entangled states, symmetric matrices and error-correcting codes, IEEE Trans. Inform. Theory, 63 (2017), 5618-5627.  doi: 10.1109/tit.2017.2700866.  Google Scholar

[12]

D. Goyeneche, Z. Raissi, S. D. Martino and K. $\dot{Z}$yczkowski, Entanglement and quantum combinatorial designs, Physical Review A, 97 (2018), 062326, 1–12. doi: 10.1103/PhysRevA.97.062326.  Google Scholar

[13]

D. Goyeneche and K. $\dot{Z}$yczkowski, Genuinely multipartite entangled states and orthogonal arrays, Phys. Rev. A, 90 (2014), 022316, 1–18. doi: 10.1103/PhysRevA.90.022316.  Google Scholar

[14]

A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Array: Theory and Applications, Springer-Verlag, 1999. doi: 10.1007/978-1-4612-1478-6.  Google Scholar

[15]

S. Hartman, On simple and supersimple transversal designs, J. Comb. Des., 8 (2000), 311-322.  doi: 10.1002/1520-6610(2000)8:5<311::AID-JCD1>3.0.CO;2-1.  Google Scholar

[16]

W. Helwig, Absolutely maximally entangled qudit graph states, preprint, arXiv: 1306.2879. Google Scholar

[17]

W. Helwig, W. Cui, J. I. Latorre, A. Riera and H. K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A, 86 (2012), 052335, 1–5. doi: 10.1103/PhysRevA.86.052335.  Google Scholar

[18]

P. Horodecki, Ł. Rudnicki and K. $\dot{Z}$yczkowski, Five open problems in quantum information, preprint, arXiv: 2002.03233. Google Scholar

[19]

L. J. Ji and J. X. Yin, Construction of new orthogonal arrays and covering arrays of strength three, J. Combin. Theory Ser. A, 117 (2010), 236-247.  doi: 10.1016/j.jcta.2009.06.002.  Google Scholar

[20]

R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up, Proc. R. Soc. A, 459 (2003), 2011-2032.  doi: 10.1098/rspa.2002.1097.  Google Scholar

[21]

M. S. Li and Y. L. Wang, K-uniform quantum states arising from orthogonal arrays, Phy. Rev. A, 99 (2019), 042332, 1–7.  Google Scholar

[22]

H. K. Lo, M. Curty and B. Qi, Measurement-device-independent quantum key distribution, Phys. Rev. Lett., 108 (2012), 130503, 1–5. doi: 10.1103/PhysRevLett.108.130503.  Google Scholar

[23]

S. Q. PangX. ZhangX. Lin and Q. J. Zhang, Two and three-uniform states from irredundant orthogonal arrays, NPJ Quantum Inf., 5 (2019), 1-10.  doi: 10.1038/s41534-019-0165-8.  Google Scholar

[24]

A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quan-tum evolutions, Phys. Rev. A, 69 (2004), 052330, 1–10. Google Scholar

[25]

E. Seiden and R. Zemach, On orthogonal arrays, Ann. Math. Stat., 37 (1966), 1355-1370.  doi: 10.1214/aoms/1177699280.  Google Scholar

[26]

C. ShiY. Tang and J. X. Yin, The equivalence between optimal detecting arrays and super-simple OAs, Des. Codes Crypogr., 62 (2012), 131-142.  doi: 10.1007/s10623-011-9498-9.  Google Scholar

[27]

Y. J. Zang, H. J. Zuo and Z. H. Tian, 3-uniform states and orthogonal arrays of strength 3, Int. J. Quantum Information, 17 (2019), 1950003, 1–8. doi: 10.1142/S0219749919500035.  Google Scholar

[28]

X. W. ZhaI. Ahmed and Y. P. Zhang, 3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28.   Google Scholar

[29]

X. W. Zha, C. Z. Yuan and Y. P. Zhang, Generalized criterion for a maximally multi-qubit entangled states, Laser Phys. Lett., 10 (2013), 045201, 1–6. doi: 10.1088/1612-2011/10/4/045201.  Google Scholar

show all references

References:
[1]

R. J. R. AbelF. E. Bennett and G. G. Ge, Super-simple holey steiner pentagon systems and related designs, J. Combin. Des., 16 (2008), 301-328.  doi: 10.1002/jcd.20171.  Google Scholar

[2]

C. H. Bennett, Quantum cryptography using any two nonorthogonal states, Phy. Rev. Lett., 68 (1992), 3121-3124.  doi: 10.1103/PhysRevLett.68.3121.  Google Scholar

[3]

C. H. BennettG. BrassardC. CrépeauR. JozsaA. Peres and W. K. Wootters, Teleporting an unknown quantum state via dual classical and einstein-podolsky-rosen channels, Phys. Rev. Lett., 70 (1993), 1895-1899.  doi: 10.1103/PhysRevLett.70.1895.  Google Scholar

[4]

R. C. Bose, A note on orthogonal arrays, Ann. Math. Stat., 21 (1950), 304-305.   Google Scholar

[5]

K. A. Bush, Orthogonal arrays of index unity, Ann. Math. Stat., 23 (1952), 426-434.  doi: 10.1214/aoms/1177729387.  Google Scholar

[6]

G. Z. ChenK. J. Chen and Y. Zhang, Super-simple (5, 4)-GDDs of group type $g^{u}$, Front. Math. China, 9 (2014), 1001-1018.  doi: 10.1007/s11464-014-0393-3.  Google Scholar

[7]

C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, Chapman and Hall/CRC Press, 2007.  Google Scholar

[8]

Y. H. Chen, Constructions of Optimal Detecting Arrays of Degree 5 and Strength 2, Master Thesis, Soochow University, 2011. Google Scholar

[9]

P. Facchi, Multipartite entanglement in qubit systems, Rend. Lincei Mat. Appl., 20 (2009), 25-67.  doi: 10.4171/RLM/532.  Google Scholar

[10]

P. Facchi, G. Florio, G. Parisi and S. Pascazio, Maximally multipartite entangled states, Phys. Rev. A, 77 (2008), 060304, 1–4. doi: 10.1103/PhysRevA.77.060304.  Google Scholar

[11]

K. Q. FengL. F. JinC. P. Xing and C. Yuan, Multipartite entangled states, symmetric matrices and error-correcting codes, IEEE Trans. Inform. Theory, 63 (2017), 5618-5627.  doi: 10.1109/tit.2017.2700866.  Google Scholar

[12]

D. Goyeneche, Z. Raissi, S. D. Martino and K. $\dot{Z}$yczkowski, Entanglement and quantum combinatorial designs, Physical Review A, 97 (2018), 062326, 1–12. doi: 10.1103/PhysRevA.97.062326.  Google Scholar

[13]

D. Goyeneche and K. $\dot{Z}$yczkowski, Genuinely multipartite entangled states and orthogonal arrays, Phys. Rev. A, 90 (2014), 022316, 1–18. doi: 10.1103/PhysRevA.90.022316.  Google Scholar

[14]

A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Array: Theory and Applications, Springer-Verlag, 1999. doi: 10.1007/978-1-4612-1478-6.  Google Scholar

[15]

S. Hartman, On simple and supersimple transversal designs, J. Comb. Des., 8 (2000), 311-322.  doi: 10.1002/1520-6610(2000)8:5<311::AID-JCD1>3.0.CO;2-1.  Google Scholar

[16]

W. Helwig, Absolutely maximally entangled qudit graph states, preprint, arXiv: 1306.2879. Google Scholar

[17]

W. Helwig, W. Cui, J. I. Latorre, A. Riera and H. K. Lo, Absolute maximal entanglement and quantum secret sharing, Phys. Rev. A, 86 (2012), 052335, 1–5. doi: 10.1103/PhysRevA.86.052335.  Google Scholar

[18]

P. Horodecki, Ł. Rudnicki and K. $\dot{Z}$yczkowski, Five open problems in quantum information, preprint, arXiv: 2002.03233. Google Scholar

[19]

L. J. Ji and J. X. Yin, Construction of new orthogonal arrays and covering arrays of strength three, J. Combin. Theory Ser. A, 117 (2010), 236-247.  doi: 10.1016/j.jcta.2009.06.002.  Google Scholar

[20]

R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up, Proc. R. Soc. A, 459 (2003), 2011-2032.  doi: 10.1098/rspa.2002.1097.  Google Scholar

[21]

M. S. Li and Y. L. Wang, K-uniform quantum states arising from orthogonal arrays, Phy. Rev. A, 99 (2019), 042332, 1–7.  Google Scholar

[22]

H. K. Lo, M. Curty and B. Qi, Measurement-device-independent quantum key distribution, Phys. Rev. Lett., 108 (2012), 130503, 1–5. doi: 10.1103/PhysRevLett.108.130503.  Google Scholar

[23]

S. Q. PangX. ZhangX. Lin and Q. J. Zhang, Two and three-uniform states from irredundant orthogonal arrays, NPJ Quantum Inf., 5 (2019), 1-10.  doi: 10.1038/s41534-019-0165-8.  Google Scholar

[24]

A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quan-tum evolutions, Phys. Rev. A, 69 (2004), 052330, 1–10. Google Scholar

[25]

E. Seiden and R. Zemach, On orthogonal arrays, Ann. Math. Stat., 37 (1966), 1355-1370.  doi: 10.1214/aoms/1177699280.  Google Scholar

[26]

C. ShiY. Tang and J. X. Yin, The equivalence between optimal detecting arrays and super-simple OAs, Des. Codes Crypogr., 62 (2012), 131-142.  doi: 10.1007/s10623-011-9498-9.  Google Scholar

[27]

Y. J. Zang, H. J. Zuo and Z. H. Tian, 3-uniform states and orthogonal arrays of strength 3, Int. J. Quantum Information, 17 (2019), 1950003, 1–8. doi: 10.1142/S0219749919500035.  Google Scholar

[28]

X. W. ZhaI. Ahmed and Y. P. Zhang, 3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28.   Google Scholar

[29]

X. W. Zha, C. Z. Yuan and Y. P. Zhang, Generalized criterion for a maximally multi-qubit entangled states, Laser Phys. Lett., 10 (2013), 045201, 1–6. doi: 10.1088/1612-2011/10/4/045201.  Google Scholar

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)
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)
[1]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[2]

Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079

[3]

Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

[4]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012

[5]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[6]

Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031

[7]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[8]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[9]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[10]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378

[11]

S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435

[12]

Yuri Fedorov, Božidar Jovanović. Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi–Mumford systems. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020375

[13]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[14]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

2019 Impact Factor: 0.734

Article outline

Figures and Tables

[Back to Top]