doi: 10.3934/amc.2020109
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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 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, 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. ZhaI. 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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. 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.  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. ZhaI. 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)
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)
[1]

K. T. Arasu, Manil T. Mohan. Optimization problems with orthogonal matrix constraints. Numerical Algebra, Control & Optimization, 2018, 8 (4) : 413-440. doi: 10.3934/naco.2018026

[2]

Dean Crnković, Bernardo Gabriel Rodrigues, Sanja Rukavina, Loredana Simčić. Self-orthogonal codes from orbit matrices of 2-designs. Advances in Mathematics of Communications, 2013, 7 (2) : 161-174. doi: 10.3934/amc.2013.7.161

[3]

Leonid Berlyand, Giuseppe Cardone, Yuliya Gorb, Gregory Panasenko. Asymptotic analysis of an array of closely spaced absolutely conductive inclusions. Networks & Heterogeneous Media, 2006, 1 (3) : 353-377. doi: 10.3934/nhm.2006.1.353

[4]

Masayuki Sato, Naoki Fujita, A. J. Sievers. Logic operations demonstrated with localized vibrations in a micromechanical cantilever array. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1287-1298. doi: 10.3934/dcdss.2011.4.1287

[5]

Samuel T. Blake, Andrew Z. Tirkel. A multi-dimensional block-circulant perfect array construction. Advances in Mathematics of Communications, 2017, 11 (2) : 367-371. doi: 10.3934/amc.2017030

[6]

Palle E. T. Jorgensen and Steen Pedersen. Orthogonal harmonic analysis of fractal measures. Electronic Research Announcements, 1998, 4: 35-42.

[7]

Artur Avila, Thomas Roblin. Uniform exponential growth for some SL(2, R) matrix products. Journal of Modern Dynamics, 2009, 3 (4) : 549-554. doi: 10.3934/jmd.2009.3.549

[8]

Cónall Kelly, Alexandra Rodkina. Constrained stability and instability of polynomial difference equations with state-dependent noise. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 913-933. doi: 10.3934/dcdsb.2009.11.913

[9]

Hui Ma, Dongxu Qi, Ruixia Song, Tianjun Wang. The complete orthogonal V-system and its applications. Communications on Pure & Applied Analysis, 2007, 6 (3) : 853-871. doi: 10.3934/cpaa.2007.6.853

[10]

Mariusz Lemańczyk, Clemens Müllner. Automatic sequences are orthogonal to aperiodic multiplicative functions. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6877-6918. doi: 10.3934/dcds.2020260

[11]

Xinpeng Wang, Bingo Wing-Kuen Ling, Wei-Chao Kuang, Zhijing Yang. Orthogonal intrinsic mode functions via optimization approach. Journal of Industrial & Management Optimization, 2021, 17 (1) : 51-66. doi: 10.3934/jimo.2019098

[12]

Cuiling Fan, Koji Momihara. Unified combinatorial constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2014, 8 (1) : 53-66. doi: 10.3934/amc.2014.8.53

[13]

T. L. Alderson, K. E. Mellinger. Geometric constructions of optimal optical orthogonal codes. Advances in Mathematics of Communications, 2008, 2 (4) : 451-467. doi: 10.3934/amc.2008.2.451

[14]

Dario Corona. A multiplicity result for orthogonal geodesic chords in Finsler disks. Discrete & Continuous Dynamical Systems, 2021, 41 (11) : 5329-5357. doi: 10.3934/dcds.2021079

[15]

Liqun Qi, Chen Ling, Jinjie Liu, Chen Ouyang. An orthogonal equivalence theorem for third order tensors. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021154

[16]

Kailu Yang, Xiaomiao Wang, Menglong Zhang, Lidong Wang. Some progress on optimal $ 2 $-D $ (n\times m,3,2,1) $-optical orthogonal codes. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021012

[17]

Jintao Li, Jindou Shen, Gang Xu. The global supersonic flow with vacuum state in a 2D convex duct. Electronic Research Archive, 2021, 29 (2) : 2077-2099. doi: 10.3934/era.2020106

[18]

Kalikinkar Mandal, Guang Gong. On ideal $ t $-tuple distribution of orthogonal functions in filtering de bruijn generators. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020125

[19]

Crnković Dean, Vedrana Mikulić Crnković, Bernardo G. Rodrigues. On self-orthogonal designs and codes related to Held's simple group. Advances in Mathematics of Communications, 2018, 12 (3) : 607-628. doi: 10.3934/amc.2018036

[20]

Leetika Kathuria, Madhu Raka. Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless. Advances in Mathematics of Communications, 2012, 6 (4) : 499-503. doi: 10.3934/amc.2012.6.499

2020 Impact Factor: 0.935

Metrics

  • PDF downloads (109)
  • HTML views (389)
  • Cited by (0)

[Back to Top]