Constructions of irredundant orthogonal arrays

Guangzhou Chen; Xiaotong Zhang

doi:10.3934/amc.2021051

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Constructions of irredundant orthogonal arrays

Guangzhou Chen ^1,, and Xiaotong Zhang ^2,

1.	Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China
2.	School of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China

^* Corresponding author: chenguangzhou0808@163.com

Received April 2021 Revised August 2021 Early access November 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (Grant Nos. 11871417 and 11501181)

An $ N \times k $ array $ A $ with entries from $ v $-set $ \mathcal{V} $ is said to be an orthogonal array with $ v $ levels, strength $ t $ and index $ \lambda $, denoted by OA$ (N,k,v,t) $, if every $ N\times t $ sub-array of $ A $ contains each $ t $-tuple based on $ \mathcal{V} $ exactly $ \lambda $ times as a row. An OA$ (N,k,v,t) $ is called irredundant, denoted by IrOA$ (N,k,v,t) $, if in any $ N\times (k-t ) $ sub-array, all of its rows are different. Goyeneche and $ \dot{Z} $yczkowski firstly introduced the definition of an IrOA and showed that an IrOA$ (N,k,v,t) $ corresponds to a $ t $-uniform state of $ k $ subsystems with local dimension $ v $ (Physical Review A. 90 (2014), 022316). In this paper, we present some new constructions of irredundant orthogonal arrays by using difference matrices and some special matrices over finite fields, respectively, as a consequence, many infinite families of irredundant orthogonal arrays are obtained. Furthermore, several infinite classes of $ t $-uniform states arise from these irredundant orthogonal arrays.

Keywords: Orthogonal array, t-uniform states, irredundant orthogonal array, difference matrix.

Mathematics Subject Classification: Primary: 05B15, 62K15; Secondary: 81P99.

Citation: Guangzhou Chen, Xiaotong Zhang. Constructions of irredundant orthogonal arrays. Advances in Mathematics of Communications, doi: 10.3934/amc.2021051

References:

[1]	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
[2]	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
[3]	D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, Experimental quantum teleportation, Nature, 390 (1997), 575-579. Google Scholar
[4]	G. Chen, X. Zhang and Y. Guo, New results for 2-uniform states based on irredundant orthogonal arrays, Quantum Inf. Process., 20 (2021), 43. doi: 10.1007/s11128-020-02978-x. Google Scholar
[5]	C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996. doi: 10.1201/9781420049954. Google Scholar
[6]	A. Dey and R. Mukerjee, Fractional Factorial Plans, John Wiley & Sons, Inc, New York, NY, 1999. doi: 10.1002/9780470316986. Google Scholar
[7]	A. K. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett., 67 (1991), 661-663. doi: 10.1103/PhysRevLett.67.661. Google Scholar
[8]	P. Facchi, Multipartite entanglement in qubit systems, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 20 (2009), 25-67. doi: 10.4171/RLM/532. Google Scholar
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[10]	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
[11]	G. Ge, On (g, 4;1)-difference matrices, Discrete Math., 301 (2005), 164-174. doi: 10.1016/j.disc.2005.07.004. Google Scholar
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[13]	D. Goyeneche, Z. Raissi, S. D. Martino and K. Życzkowski, Entanglement and quantum combinatorial designs, Phys. Rev. A., 97 (2018), 062326. doi: 10.1103/PhysRevA.97.062326. Google Scholar
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[15]	M. Grassl and M. Rötteler, Quantum MDS codes over small fields,, IEEE International Symposium on Information Theory (ISIT), (2015), 1104–1108. doi: 10.1109/ISIT.2015.7282626. Google Scholar
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[23]	F. Huber, O. Gühne and J. Siewert, Absolutely maximally entangled states of seven qubits do not exist, Phys. Rev. Lett., 118 (2017), 200502. doi: 10.1103/PhysRevLett.118.200502. Google Scholar
[24]	L. Ji and J. Yin, Constructions of new orthogonal arrays and covering arrays of strength three, J. Combi. Theory, Ser. A, 117 (2010), 236-247. doi: 10.1016/j.jcta.2009.06.002. Google Scholar
[25]	R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2011-2032. doi: 10.1098/rspa.2002.1097. Google Scholar
[26]	M. S. Li and Y. L. Wang, $k$-uniform quantum states arising from orthogonal arrays, Phy. Rev. A., 99 (2019), 042332. Google Scholar
[27]	H. K. Lo, M. Curty and B. Qi, Measurement-device-independent quantum key distribution, Phys. Rev. Lett., 108 (2012), 130503. doi: 10.1103/PhysRevLett.108.130503. Google Scholar
[28]	S. Q. Pang, X. Zhang, J. Du and T. Wang, Multipartite entanglement states of higher uniformity, J. Phys. A: Math. Theor., 54 (2021), 015305. doi: 10.1088/1751-8121/abc9a4. Google Scholar
[29]	S. Q. Pang, X. Zhang, S. M. Fei and Z. J. Zheng, Quantum $k$-uniform states for heterogeneous systems from irredundant mixed orthogonal arrays, Quantum Inf. Process., 20 (2021), 156. doi: 10.1007/s11128-021-03040-0. Google Scholar
[30]	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
[31]	F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, J. High Energy Phys., 6 (2015), 149. doi: 10.1007/JHEP06(2015)149. Google Scholar
[32]	E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, 45 (1999), 1827-1832. doi: 10.1109/18.782103. Google Scholar
[33]	Z. Raissi, A. Teixido, C. Gogolin and A. Acín, Constructions of $k$-uniform and absolutely maximally entangled states beyond maximum distance codes, Physical Review Research, 2 (2020), 033411. doi: 10.1103/PhysRevResearch.2.033411. Google Scholar
[34]	C. R. Rao, Factorial experiments derivable from combinational arrangements of arrays, Suppl. J. Roy. Statist. Soc., 9 (1947), 128-139. doi: 10.2307/2983576. Google Scholar
[35]	S. A. Rather, A. Burchardt, W. Bruzda, G. R.-Mieldzioć, A. Lakshminarayan and K. Życzkowski, Thirty-six entangled officers of Euler, preprint, arXiv: 2104.05122v1. Google Scholar
[36]	M. Riebe, H. Haffner, F. C. Roos and et al, Deterministic quantum teleportation with atoms, Nature, 429 (2004), 734-737. doi: 10.1038/nature02570. Google Scholar
[37]	C. F. Roos, M. Riebe, H. Haffner and et al, Control and measurement of three-qubit entangled states, Science, 304 (2004), 1478-1480. doi: 10.1126/science.1097522. Google Scholar
[38]	A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions, Phys. Rev. A., 69 (2004), 052330. doi: 10.1103/PhysRevA.69.052330. Google Scholar
[39]	F. Shi, Y. Shen, L. Chen and X. Zhang, Constructions of $k$-uniform states from mixed orthogonal arrays, preprint, arXiv: 2006.04086v1. Google Scholar
[40]	D. R. Stinson, Ideal ramp schemes and related combinatorial objects, Discrete Math., 341 (2018), 299-307. doi: 10.1016/j.disc.2017.08.041. Google Scholar
[41]	C. Suen, A. Das and A. Dey, On the construction of asymmetric orthogonal arrays, Statistica Sinica, 11 (2001), 241-260. Google Scholar
[42]	Y. J. Zang, G. Z. Chen, K. J. Chen and Z. H. Tian, Further results on $2$-uniform states arising from irredundant orthogonal arrays, Advances in Mathematics of Communications, 2020. doi: 10.3934/amc.2020109. Google Scholar
[43]	Y. J. Zang, H. J. Zuo and Z. H. Tian, 3-uniform states and orthogonal arrays of strength 3, Int. J. Quantum Inf., 17 (2019), 1950003. doi: 10.1142/S0219749919500035. Google Scholar
[44]	X. W. Zha, I. Ahmed and Y. P. Zhang, 3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28. Google Scholar
[45]	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. doi: 10.1088/1612-2011/10/4/045201. Google Scholar
[46]	Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel and J. Pan, Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature, 430 (2004), 54-58. doi: 10.1038/nature02643. Google Scholar

show all references

References:

[1]	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
[2]	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
[3]	D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger, Experimental quantum teleportation, Nature, 390 (1997), 575-579. Google Scholar
[4]	G. Chen, X. Zhang and Y. Guo, New results for 2-uniform states based on irredundant orthogonal arrays, Quantum Inf. Process., 20 (2021), 43. doi: 10.1007/s11128-020-02978-x. Google Scholar
[5]	C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996. doi: 10.1201/9781420049954. Google Scholar
[6]	A. Dey and R. Mukerjee, Fractional Factorial Plans, John Wiley & Sons, Inc, New York, NY, 1999. doi: 10.1002/9780470316986. Google Scholar
[7]	A. K. Ekert, Quantum cryptography based on Bell's theorem, Phys. Rev. Lett., 67 (1991), 661-663. doi: 10.1103/PhysRevLett.67.661. Google Scholar
[8]	P. Facchi, Multipartite entanglement in qubit systems, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., 20 (2009), 25-67. doi: 10.4171/RLM/532. Google Scholar
[9]	P. Facchi, G. Florio, G. Parisi and S. Pascazio, Maximally multipartite entangled states, Phys. Rev. A., 77 (2008), 060304. doi: 10.1103/PhysRevA.77.060304. Google Scholar
[10]	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
[11]	G. Ge, On (g, 4;1)-difference matrices, Discrete Math., 301 (2005), 164-174. doi: 10.1016/j.disc.2005.07.004. Google Scholar
[12]	D. Goyeneche, J. Bielawski and K. Życzkowski, Multipartite entanglement in heterogeneous systems, Phys. Rev. A., 94 (2016), 012346. doi: 10.1103/PhysRevA.94.012346. Google Scholar
[13]	D. Goyeneche, Z. Raissi, S. D. Martino and K. Życzkowski, Entanglement and quantum combinatorial designs, Phys. Rev. A., 97 (2018), 062326. doi: 10.1103/PhysRevA.97.062326. Google Scholar
[14]	D. Goyeneche and K. Życzkowski, Genuinely multipartite entangled states and orthogonal arrays, Phys. Rev. A., 90 (2014), 022316. doi: 10.1103/PhysRevA.90.022316. Google Scholar
[15]	M. Grassl and M. Rötteler, Quantum MDS codes over small fields,, IEEE International Symposium on Information Theory (ISIT), (2015), 1104–1108. doi: 10.1109/ISIT.2015.7282626. Google Scholar
[16]	A. S. Hedayat, N. J. A. Sloane and J. Stufken, Orthogonal Arrays: Theory and Applications, Springer Series in Statistics. Springer-Verlag, New York, 1999. doi: 10.1007/978-1-4612-1478-6. Google Scholar
[17]	W. Helwig, Absolutely maximally entangled qudit graph states, preprint, arXiv: 1306.2879v1. Google Scholar
[18]	W. Helwig and W. Cui, Absolutely maximally entangled states: existence and applications, preprint, arXiv: 1306.2536. Google Scholar
[19]	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. doi: 10.1103/PhysRevA.86.052335. Google Scholar
[20]	A. Higuchi and A. Sudbery, How entangled can two couples get?, Phys. Lett. A, 273 (2000), 213-217. doi: 10.1016/S0375-9601(00)00480-1. Google Scholar
[21]	R. Horodecki, P. Horodecki, M. Horodecki and K. Horodecki, Quantum entanglement, Rev. Modern Phys, 81 (2009), 865-942. doi: 10.1103/RevModPhys.81.865. Google Scholar
[22]	P. Horodecki, Ł. Rudnicki and K. Życzkowski, Five open problems in quantum information, prepint, arXiv: 2002.03233v1. Google Scholar
[23]	F. Huber, O. Gühne and J. Siewert, Absolutely maximally entangled states of seven qubits do not exist, Phys. Rev. Lett., 118 (2017), 200502. doi: 10.1103/PhysRevLett.118.200502. Google Scholar
[24]	L. Ji and J. Yin, Constructions of new orthogonal arrays and covering arrays of strength three, J. Combi. Theory, Ser. A, 117 (2010), 236-247. doi: 10.1016/j.jcta.2009.06.002. Google Scholar
[25]	R. Jozsa and N. Linden, On the role of entanglement in quantum computational speed-up, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 459 (2003), 2011-2032. doi: 10.1098/rspa.2002.1097. Google Scholar
[26]	M. S. Li and Y. L. Wang, $k$-uniform quantum states arising from orthogonal arrays, Phy. Rev. A., 99 (2019), 042332. Google Scholar
[27]	H. K. Lo, M. Curty and B. Qi, Measurement-device-independent quantum key distribution, Phys. Rev. Lett., 108 (2012), 130503. doi: 10.1103/PhysRevLett.108.130503. Google Scholar
[28]	S. Q. Pang, X. Zhang, J. Du and T. Wang, Multipartite entanglement states of higher uniformity, J. Phys. A: Math. Theor., 54 (2021), 015305. doi: 10.1088/1751-8121/abc9a4. Google Scholar
[29]	S. Q. Pang, X. Zhang, S. M. Fei and Z. J. Zheng, Quantum $k$-uniform states for heterogeneous systems from irredundant mixed orthogonal arrays, Quantum Inf. Process., 20 (2021), 156. doi: 10.1007/s11128-021-03040-0. Google Scholar
[30]	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
[31]	F. Pastawski, B. Yoshida, D. Harlow and J. Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, J. High Energy Phys., 6 (2015), 149. doi: 10.1007/JHEP06(2015)149. Google Scholar
[32]	E. M. Rains, Nonbinary quantum codes, IEEE Trans. Inform. Theory, 45 (1999), 1827-1832. doi: 10.1109/18.782103. Google Scholar
[33]	Z. Raissi, A. Teixido, C. Gogolin and A. Acín, Constructions of $k$-uniform and absolutely maximally entangled states beyond maximum distance codes, Physical Review Research, 2 (2020), 033411. doi: 10.1103/PhysRevResearch.2.033411. Google Scholar
[34]	C. R. Rao, Factorial experiments derivable from combinational arrangements of arrays, Suppl. J. Roy. Statist. Soc., 9 (1947), 128-139. doi: 10.2307/2983576. Google Scholar
[35]	S. A. Rather, A. Burchardt, W. Bruzda, G. R.-Mieldzioć, A. Lakshminarayan and K. Życzkowski, Thirty-six entangled officers of Euler, preprint, arXiv: 2104.05122v1. Google Scholar
[36]	M. Riebe, H. Haffner, F. C. Roos and et al, Deterministic quantum teleportation with atoms, Nature, 429 (2004), 734-737. doi: 10.1038/nature02570. Google Scholar
[37]	C. F. Roos, M. Riebe, H. Haffner and et al, Control and measurement of three-qubit entangled states, Science, 304 (2004), 1478-1480. doi: 10.1126/science.1097522. Google Scholar
[38]	A. J. Scott, Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions, Phys. Rev. A., 69 (2004), 052330. doi: 10.1103/PhysRevA.69.052330. Google Scholar
[39]	F. Shi, Y. Shen, L. Chen and X. Zhang, Constructions of $k$-uniform states from mixed orthogonal arrays, preprint, arXiv: 2006.04086v1. Google Scholar
[40]	D. R. Stinson, Ideal ramp schemes and related combinatorial objects, Discrete Math., 341 (2018), 299-307. doi: 10.1016/j.disc.2017.08.041. Google Scholar
[41]	C. Suen, A. Das and A. Dey, On the construction of asymmetric orthogonal arrays, Statistica Sinica, 11 (2001), 241-260. Google Scholar
[42]	Y. J. Zang, G. Z. Chen, K. J. Chen and Z. H. Tian, Further results on $2$-uniform states arising from irredundant orthogonal arrays, Advances in Mathematics of Communications, 2020. doi: 10.3934/amc.2020109. Google Scholar
[43]	Y. J. Zang, H. J. Zuo and Z. H. Tian, 3-uniform states and orthogonal arrays of strength 3, Int. J. Quantum Inf., 17 (2019), 1950003. doi: 10.1142/S0219749919500035. Google Scholar
[44]	X. W. Zha, I. Ahmed and Y. P. Zhang, 3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28. Google Scholar
[45]	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. doi: 10.1088/1612-2011/10/4/045201. Google Scholar
[46]	Z. Zhao, Y. A. Chen, A. N. Zhang, T. Yang, H. J. Briegel and J. Pan, Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature, 430 (2004), 54-58. doi: 10.1038/nature02643. Google Scholar

Table 1. Correspondence between parameters of IrOAs and quantum states

Parameters	Irredundant Orthogonal array	Multipartite quantum state $\|\psi\rangle$
N	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)

Table Options

Download as excel

Parameters	Irredundant Orthogonal array	Multipartite quantum state $\|\psi\rangle$
N	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)

Table 2. Existence of t-uniform states of k qudits (s ≥ 2)

$(\mathbb{C}^s)^{\otimes k}$	Existence	Nonexistence	Unknown
1-uniform	$s\geq 2, k\geq 2$	no	no
2-uniform	$s\geq 2, k\geq 4$ except (s,k) = (2,4)	s = 2 and k = 4	no
3-uniform	$s\geq 2, k\geq 6$ except s ≡ 2 (mod 4) and k = 7	s = 2 and k = 7	s ≥ 6, s ≡ 2 (mod 4) and k = 7

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Download as excel

$(\mathbb{C}^s)^{\otimes k}$	Existence	Nonexistence	Unknown
1-uniform	$s\geq 2, k\geq 2$	no	no
2-uniform	$s\geq 2, k\geq 4$ except (s,k) = (2,4)	s = 2 and k = 4	no
3-uniform	$s\geq 2, k\geq 6$ except s ≡ 2 (mod 4) and k = 7	s = 2 and k = 7	s ≥ 6, s ≡ 2 (mod 4) and k = 7

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2020 Impact Factor: 0.935

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2020 Impact Factor: 0.935

American Institute of Mathematical Sciences

Constructions of irredundant orthogonal arrays

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American Institute of Mathematical Sciences

Constructions of irredundant orthogonal arrays

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