Table 1.
Correspondence between parameters of OAs and quantum states
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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 |
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,
2022, 16
(2)
: 231-247.
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. |
| [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. 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. |
| [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. |
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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. |
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C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, Chapman
and Hall/CRC Press, 2007. |
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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. 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. |
| [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. 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. |
| [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. 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. |
| [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. Zha, I. 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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. Zha, I. 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. |
| 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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