Further results on 2-uniform states arising from irredundant orthogonal arrays

Yajuan Zang; Guangzhou Chen; Kejun Chen; Zihong Tian

doi:10.3934/amc.2020109

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doi: 10.3934/amc.2020109

Further results on 2-uniform states arising from irredundant orthogonal arrays

Yajuan Zang ^1, , Guangzhou Chen ^2, , Kejun Chen ^3, and Zihong Tian ^1,,

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 $.

Keywords: 2-uniform state, multipartite entangled state, orthogonal array, irredundant orthogonal array, difference matrix.

Mathematics Subject Classification: Primary: 05B15, 81P99; Secondary: 05B20.

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. Abel, F. 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. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. 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. Chen, K. 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. Feng, L. F. Jin, C. 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. Pang, X. Zhang, X. 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. Shi, Y. 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. Zha, I. 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. Abel, F. 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. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. 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. Chen, K. 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. Feng, L. F. Jin, C. 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. Pang, X. Zhang, X. 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. Shi, Y. 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. Zha, I. 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)

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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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