May  2022, 16(2): 231-247. 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  May 2022 Early access  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, 2022, 16 (2) : 231-247. 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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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.

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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[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.

[21]

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

[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.

[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.

[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.

[25]

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

[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.

[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.

[28]

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

[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.

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)
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