Table 1.
Correspondence between parameters of IrOAs and quantum states
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Constructions of irredundant orthogonal arrays
| 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 |
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 $-uniform states arise from these irredundant orthogonal arrays.
Mathematics Subject Classification:
Primary: 05B15, 62K15; Secondary: 81P99.
Citation:
Guangzhou Chen, Xiaotong Zhang.
Constructions of irredundant orthogonal arrays. Advances in Mathematics of Communications, doi: 10.3934/amc.2021051
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$k$-uniform quantum states arising from orthogonal arrays, Phy. Rev. A., 99 (2019), 042332.
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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. |
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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. |
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Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, J. High Energy Phys., 6 (2015), 149.
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Nonbinary quantum codes, IEEE Trans. Inform. Theory, 45 (1999), 1827-1832.
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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.
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C. R. Rao,
Factorial experiments derivable from combinational arrangements of arrays, Suppl. J. Roy. Statist. Soc., 9 (1947), 128-139.
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On the construction of asymmetric orthogonal arrays, Statistica Sinica, 11 (2001), 241-260.
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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. |
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3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28.
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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.
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Experimental demonstration of five-photon entanglement and open-destination teleportation, Nature, 430 (2004), 54-58.
doi: 10.1038/nature02643. |
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. |
| [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. |
| [3] |
D. Bouwmeester, J. W. Pan, K. Mattle, M. Eibl, H. Weinfurter and A. Zeilinger,
Experimental quantum teleportation, Nature, 390 (1997), 575-579.
|
| [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. |
| [5] |
C. J. Colbourn and J. H. Dinitz, The CRC Handbook of Combinatorial Designs, CRC Press, Boca Raton, FL, 1996.
doi: 10.1201/9781420049954.
|
| [6] |
A. Dey and R. Mukerjee, Fractional Factorial Plans, John Wiley & Sons, Inc, New York, NY, 1999.
doi: 10.1002/9780470316986. |
| [7] |
A. K. Ekert,
Quantum cryptography based on Bell's theorem, Phys. Rev. Lett., 67 (1991), 661-663.
doi: 10.1103/PhysRevLett.67.661. |
| [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. |
| [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. |
| [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. |
| [11] |
G. Ge,
On (g, 4;1)-difference matrices, Discrete Math., 301 (2005), 164-174.
doi: 10.1016/j.disc.2005.07.004. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [17] |
W. Helwig, Absolutely maximally entangled qudit graph states, preprint, arXiv: 1306.2879v1. |
| [18] |
W. Helwig and W. Cui, Absolutely maximally entangled states: existence and applications, preprint, arXiv: 1306.2536. |
| [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. |
| [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. |
| [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. |
| [22] |
P. Horodecki, Ł. Rudnicki and K. Życzkowski, Five open problems in quantum information, prepint, arXiv: 2002.03233v1. |
| [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. |
| [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. |
| [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. |
| [26] |
M. S. Li and Y. L. Wang,
$k$-uniform quantum states arising from orthogonal arrays, Phy. Rev. A., 99 (2019), 042332.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [32] |
E. M. Rains,
Nonbinary quantum codes, IEEE Trans. Inform. Theory, 45 (1999), 1827-1832.
doi: 10.1109/18.782103. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [39] |
F. Shi, Y. Shen, L. Chen and X. Zhang, Constructions of $k$-uniform states from mixed orthogonal arrays, preprint, arXiv: 2006.04086v1. |
| [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. |
| [41] |
C. Suen, A. Das and A. Dey,
On the construction of asymmetric orthogonal arrays, Statistica Sinica, 11 (2001), 241-260.
|
| [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. |
| [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. |
| [44] |
X. W. Zha, I. Ahmed and Y. P. Zhang,
3-uniform states and orthognal arrays, Results Phys., 6 (2016), 26-28.
|
| [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. |
| [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. |
| 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) |
| 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 |
| $(\mathbb{C}^s)^{\otimes k}$ | 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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