Tensor network decompositions for absolutely maximally entangled states

Balázs Pozsgay; Ian M. Wanless

doi:10.22331/q-2024-05-08-1339

Tensor network decompositions for absolutely maximally entangled states

Balázs Pozsgay¹ and Ian M. Wanless²

¹MTA-ELTE ``Momentum'' Integrable Quantum Dynamics Research Group, Department of Theoretical Physics, ELTE Eötvös Loránd University, Hungary
²School of Mathematics, Monash University, Australia

Published:	2024-05-08, volume 8, page 1339
Eprint:	arXiv:2308.07042v3
Doi:	https://doi.org/10.22331/q-2024-05-08-1339
Citation:	Quantum 8, 1339 (2024).

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Abstract

Absolutely maximally entangled (AME) states of $k$ qudits (also known as perfect tensors) are quantum states that have maximal entanglement for all possible bipartitions of the sites/parties. We consider the problem of whether such states can be decomposed into a tensor network with a small number of tensors, such that all physical and all auxiliary spaces have the same dimension $D$. We find that certain AME states with $k=6$ can be decomposed into a network with only three 4-leg tensors; we provide concrete solutions for local dimension $D=5$ and higher. Our result implies that certain AME states with six parties can be created with only three two-site unitaries from a product state of three Bell pairs, or equivalently, with six two-site unitaries acting on a product state on six qudits. We also consider the problem for $k=8$, where we find similar tensor network decompositions with six 4-leg tensors.

Featured image: Tensor network decompositions for a quantum state on six sites.

Popular summary

Entanglement is the distinguishing feature of quantum states, which makes them truly quantum mechanical. Starting from a product state, entanglement can be created by the action of unitary gates. In this work we consider quantum states that have maximal entanglement for all possible bi-partitions. We show, that contrary to general expectations, in certain cases these states can be created by a small number of two-site unitary gates. Equivalently, the states can be decomposed into a tensor network with few elementary tensors. This interpretation is also counter-intuitive, because it is well known that tensor networks describe states with low entanglement. Therefore, our results provide counter-examples to the general expectations, with concrete examples at small system sizes.

► BibTeX data

@article{Pozsgay2024tensornetwork,
  doi = {10.22331/q-2024-05-08-1339},
  url = {https://doi.org/10.22331/q-2024-05-08-1339},
  title = {Tensor network decompositions for absolutely maximally entangled states},
  author = {Pozsgay, Bal{\'{a}}zs and Wanless, Ian M.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1339},
  month = may,
  year = {2024}
}

► References

[1] R. Orús, ``Tensor networks for complex quantum systems,'' Nat. Rev. Phys. 1 (2019) no. 9, 538–550, arXiv:1812.04011 [cond-mat.str-el].
https://doi.org/10.1038/s42254-019-0086-7
arXiv:1812.04011

[2] D. Perez-Garcia, F. Verstraete, M. M. Wolf, and J. I. Cirac, ``Matrix Product State Representations,'' Quantum Info. Comput. 7 (2007) no. 5, 401–430, quant-ph/0608197.
https://doi.org/10.26421/QIC7.5-6-1
arXiv:quant-ph/0608197

[3] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, ``Matrix product states and projected entangled pair states: Concepts, symmetries, theorems,'' Rev. Mod. Phys. 93 (2021) 045003, arXiv:2011.12127 [quant-ph].
https://doi.org/10.1103/RevModPhys.93.045003
arXiv:2011.12127

[4] P. Facchi, G. Florio, G. Parisi, and S. Pascazio, ``Maximally multipartite entangled states,'' Phys. Rev. A 77 (2008) no. 6, 060304, arXiv:0710.2868 [quant-ph].
https://doi.org/10.1103/PhysRevA.77.060304
arXiv:0710.2868

[5] 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) no. 5, 052335, arXiv:1204.2289 [quant-ph].
https://doi.org/10.1103/PhysRevA.86.052335
arXiv:1204.2289

[6] W. Helwig and W. Cui, ``Absolutely Maximally Entangled States: Existence and Applications,'' arXiv e-prints (2013) , arXiv:1306.2536 [quant-ph].
arXiv:1306.2536

[7] D. Goyeneche and K. Życzkowski, ``Genuinely multipartite entangled states and orthogonal arrays,'' Phys. Rev. A 90 (2014) no. 2, 022316, arXiv:1404.3586 [quant-ph].
https://doi.org/10.1103/PhysRevA.90.022316
arXiv:1404.3586

[8] M. Gaeta, A. Klimov, and J. Lawrence, ``Maximally entangled states of four nonbinary particles,'' Phys. Rev. A 91 (2015) no. 1, 012332, arXiv:1411.6178 [quant-ph].
https://doi.org/10.1103/PhysRevA.91.012332
arXiv:1411.6178

[9] Z. Raissi, C. Gogolin, A. Riera, and A. Acín, ``Optimal quantum error correcting codes from absolutely maximally entangled states,'' J. Phys. A 51 (2018) no. 7, 075301, arXiv:1701.03359 [quant-ph].
https://doi.org/10.1088/1751-8121/aaa151
arXiv:1701.03359

[10] D. Goyeneche, D. Alsina, J. I. Latorre, A. Riera, and K. Życzkowski, ``Absolutely maximally entangled states, combinatorial designs, and multiunitary matrices,'' Phys. Rev. A 92 (2015) no. 3, 032316, arXiv:1506.08857 [quant-ph].
https://doi.org/10.1103/PhysRevA.92.032316
arXiv:1506.08857

[11] D. Goyeneche, Z. Raissi, S. Di Martino, and K. Życzkowski, ``Entanglement and quantum combinatorial designs,'' Phys. Rev. A 97 (2018) no. 6, 062326, arXiv:1708.05946 [quant-ph].
https://doi.org/10.1103/PhysRevA.97.062326
arXiv:1708.05946

[12] A. S. Hedayat, N. J. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications. Springer, 1999.

[13] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, ``Chaos in quantum channels,'' J. High Energy Phys. 2016 (2016) 4, arXiv:1511.04021 [hep-th].
https://doi.org/10.1007/JHEP02(2016)004
arXiv:1511.04021

[14] F. Huber and N. Wyderka, ``Table of AME states.'' Online available, 2021. https://www.tp.nt.uni-siegen.de/+fhuber/ame.html.
https://www.tp.nt.uni-siegen.de/+fhuber/ame.html

[15] F. Huber, C. Eltschka, J. Siewert, and O. Gühne, ``Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity,'' J. Phys. A 51 (2018) no. 17, 175301, arXiv:1708.06298 [quant-ph].
https://doi.org/10.1088/1751-8121/aaade5
arXiv:1708.06298

[16] S. A. Rather, A. Burchardt, W. Bruzda, G. Rajchel-Mieldzioć, A. Lakshminarayan, and K. Życzkowski, ``Thirty-six Entangled Officers of Euler: Quantum Solution to a Classically Impossible Problem,'' Phys. Rev. Lett. 128 (2022) no. 8, 080507, arXiv:2104.05122 [quant-ph].
https://doi.org/10.1103/PhysRevLett.128.080507
arXiv:2104.05122

[17] K. Życzkowski, W. Bruzda, G. Rajchel-Mieldzioć, A. Burchardt, S. A. Rather, and A. Lakshminarayan, ``9 $\times$ 4 = 6 $\times$ 6: Understanding the quantum solution to the Euler's problem of 36 officers,'' J. Phys.: Conf. Series 2448 (2023) no. 1, 012003, arXiv:2204.06800 [quant-ph].
https://doi.org/10.1088/1742-6596/2448/1/012003
arXiv:2204.06800

[18] S. A. Rather, N. Ramadas, V. Kodiyalam, and A. Lakshminarayan, ``Absolutely maximally entangled state equivalence and the construction of infinite quantum solutions to the problem of 36 officers of Euler,'' Phys. Rev. A 108 (2023) no. 3, 032412, arXiv:2212.06737 [quant-ph].
https://doi.org/10.1103/PhysRevA.108.032412
arXiv:2212.06737

[19] L. Chen and D. L. Zhou, ``Graph states of prime-power dimension from generalized CNOT quantum circuit,'' Sci. Rep. 6 (2016) 27135, arXiv:1507.05386 [quant-ph].
https://doi.org/10.1038/srep27135
arXiv:1507.05386

[20] A. Cervera-Lierta, J. I. Latorre, and D. Goyeneche, ``Quantum circuits for maximally entangled states,'' Phys. Rev. A 100 (2019) no. 2, 022342, arXiv:1904.07955 [quant-ph].
https://doi.org/10.1103/PhysRevA.100.022342
arXiv:1904.07955

[21] D. N. Page, ``Average entropy of a subsystem,'' Phys. Rev. Lett. 71 (1993) no. 9, 1291–1294, arXiv:gr-qc/9305007 [gr-qc].
https://doi.org/10.1103/PhysRevLett.71.1291
arXiv:gr-qc/9305007

[22] J. Haferkamp, P. Faist, N. B. T. Kothakonda, J. Eisert, and N. Yunger Halpern, ``Linear growth of quantum circuit complexity,'' Nat. Phys. 18 (2022) no. 5, 528–532, arXiv:2106.05305 [quant-ph].
https://doi.org/10.1038/s41567-022-01539-6
arXiv:2106.05305

[23] T. Farrelly, R. J. Harris, N. A. McMahon, and T. M. Stace, ``Tensor-Network Codes,'' Phys. Rev. Lett. 127 (2021) no. 4, 040507, arXiv:2009.10329 [quant-ph].
https://doi.org/10.1103/PhysRevLett.127.040507
arXiv:2009.10329

[24] C. Cao and B. Lackey, ``Quantum Lego: Building Quantum Error Correction Codes from Tensor Networks,'' PRX Quantum 3 (2022) no. 2, 020332, arXiv:2109.08158 [quant-ph].
https://doi.org/10.1103/PRXQuantum.3.020332
arXiv:2109.08158

[25] T. Farrelly, D. K. Tuckett, and T. M. Stace, ``Local tensor-network codes,'' New J. Phys. 24 (2022) no. 4, 043015, arXiv:2109.11996 [quant-ph].
https://doi.org/10.1088/1367-2630/ac5e87
arXiv:2109.11996

[26] 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. 2015 (2015) 149, arXiv:1503.06237 [hep-th].
https://doi.org/10.1007/JHEP06(2015)149
arXiv:1503.06237

[27] T. Kibe, P. Mandayam, and A. Mukhopadhyay, ``Holographic spacetime, black holes and quantum error correcting codes: A review,'' Eu. Phys. J. C 82 (2022) no. 5, 463, arXiv:2110.14669 [hep-th].
https://doi.org/10.1140/epjc/s10052-022-10382-1
arXiv:2110.14669

[28] A. Jahn and J. Eisert, ``Holographic tensor network models and quantum error correction: a topical review,'' Quantum Sci. Tech. 6 (2021) no. 3, 033002, arXiv:2102.02619 [quant-ph].
https://doi.org/10.1088/2058-9565/ac0293
arXiv:2102.02619

[29] J. Berger and T. J. Osborne, ``Perfect tangles,'' arXiv e-prints (2018) , arXiv:1804.03199 [quant-ph].
arXiv:1804.03199

[30] M. Doroudiani and V. Karimipour, ``Planar maximally entangled states,'' Phys. Rev. A 102 (2020) no. 1, 012427, arXiv:2004.00906 [quant-ph].
https://doi.org/10.1103/PhysRevA.102.012427
arXiv:2004.00906

[31] R. J. Harris, N. A. McMahon, G. K. Brennen, and T. M. Stace, ``Calderbank-Shor-Steane holographic quantum error-correcting codes,'' Phys. Rev. A 98 (2018) 052301, arXiv:1806.06472 [quant-ph].
https://doi.org/10.1103/PhysRevA.98.052301
arXiv:1806.06472

[32] Y.-L. Wang, ``Planar k-uniform states: a generalization of planar maximally entangled states,'' Quant. Inf. Proc. 20 (2021) no. 8, 271, arXiv:2106.12209 [quant-ph].
https://doi.org/10.1007/s11128-021-03204-y
arXiv:2106.12209

[33] B. Bertini, P. Kos, and T. Prosen, ``Exact Correlation Functions for Dual-Unitary Lattice Models in 1+1 Dimensions,'' Phys. Rev. Lett. 123 (2019) no. 21, , arXiv:1904.02140 [cond-mat.stat-mech].
https://doi.org/10.1103/physrevlett.123.210601
arXiv:1904.02140

[34] V. F. R. Jones, ``Planar algebras, I,'' New Zeal. J. Math. 52 (2021) 1–107, arXiv:math/9909027 [math.QA].
https://doi.org/10.53733/172
arXiv:math/9909027

[35] U. Krishnan and V. S. Sunder, ``On Biunitary Permutation Matrices and Some Subfactors of Index 9,'' Trans. Amer. Math. Soc. 348 (1996) no. 12, 4691–4736.
https://doi.org/10.1090/S0002-9947-96-01669-8

[36] G. Evenbly, ``Hyperinvariant Tensor Networks and Holography,'' Phys. Rev. Lett. 119 (2017) no. 14, 141602, arXiv:1704.04229 [quant-ph].
https://doi.org/10.1103/PhysRevLett.119.141602
arXiv:1704.04229

[37] M. Steinberg and J. Prior, ``Conformal Properties of Hyperinvariant Tensor Networks,'' Sci. Rep. 12 (2022) 532, arXiv:2012.09591 [quant-ph].
https://doi.org/10.48550/arXiv.2012.09591
arXiv:2012.09591

[38] D. J. Reutter and J. Vicary, ``Biunitary constructions in quantum information,'' Higher Structures 3 (2019) no. 1, 109–154, arXiv:1609.07775 [quant-ph].
https://doi.org/10.21136/HS.2019.04
arXiv:1609.07775

[39] P. W. Claeys, A. Lamacraft, and J. Vicary, ``From dual-unitary to biunitary: a 2-categorical model for exactly-solvable many-body quantum dynamics,'' arXiv:2302.07280 [quant-ph].
arXiv:2302.07280

[40] C. Jonay, V. Khemani, and M. Ippoliti, ``Triunitary quantum circuits,'' Phys. Rev. Res. 3 (2021) no. 4, 043046, arXiv:2106.07686 [quant-ph].
https://doi.org/10.1103/PhysRevResearch.3.043046
arXiv:2106.07686

[41] R. Milbradt, L. Scheller, C. Aßmus, and C. B. Mendl, ``Ternary unitary quantum lattice models and circuits in $2 + 1$ dimensions,'' Phys. Rev. Lett. 130 (2023) 090601, arXiv:2206.01499 [cond-mat.stat-mech].
https://doi.org/10.1103/PhysRevLett.130.090601
arXiv:2206.01499

[42] Y. Kasim and T. Prosen, ``Dual unitary circuits in random geometries,'' J. Phys. A 56 (2022) no. 2, 025003, arXiv:2206.09665 [cond-mat.stat-mech].
https://doi.org/10.1088/1751-8121/acb1e0
arXiv:2206.09665

[43] G. M. Sommers, D. A. Huse, and M. J. Gullans, ``Crystalline Quantum Circuits,'' PRX Quantum 4 (2023) 030313, arXiv:2210.10808 [quant-ph].
https://doi.org/10.1103/PRXQuantum.4.030313
arXiv:2210.10808

[44] M. Mestyán, B. Pozsgay, and I. M. Wanless, ``Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states,'' SciPost Phys. 16 (2024) no. 1, 010, arXiv:2210.13017 [quant-ph].
https://doi.org/10.21468/SciPostPhys.16.1.010
arXiv:2210.13017

[45] J. I. Latorre and G. Sierra, ``Platonic Entanglement,'' Quantum Inf. Comput. 21 (13 & 14) (2021) 1081–1090, arXiv:2107.04329 [quant-ph].
https://doi.org/10.26421/qic21.13-14-1
arXiv:2107.04329

[46] T. Gombor and B. Pozsgay, ``Superintegrable cellular automata and dual unitary gates from Yang-Baxter maps,'' SciPost Phys. 12 (2022) 102, arXiv:2112.01854 [cond-mat.stat-mech].
https://doi.org/10.21468/SciPostPhys.12.3.102
arXiv:2112.01854

[47] R. Roth and A. Lempel, ``On MDS codes via Cauchy matrices,'' IEEE Trans. Inf. Theory 35 (1989) no. 6, 1314–1319.
https://doi.org/10.1109/18.45291

[48] G. Kéri, ``Types of superregular matrices and the number of $n$-arcs and complete $n$-arcs in PG$(r, q)$,'' J. Comb. Des. 14 (2006) no. 5, 363–390.
https://doi.org/10.1002/jcd.20091

[49] K. A. Bush, ``Orthogonal arrays of index unity,'' Ann. Math. Statistics 23 (1952) 426–434.
https://doi.org/10.1214/aoms/1177729387

[50] G. Gour and N. R. Wallach, ``All maximally entangled four-qubit states,'' J. Math. Phys. 51 (2010) no. 11, 112201–112201, arXiv:1006.0036 [quant-ph].
https://doi.org/10.1063/1.3511477
arXiv:1006.0036

[51] W. Helwig, ``Absolutely Maximally Entangled Qudit Graph States,'' arxiv e-prints (2013) , arXiv:1306.2879 [quant-ph].
arXiv:1306.2879

[52] M. van den Nest, J. Dehaene, and B. de Moor, ``Graphical description of the action of local Clifford transformations on graph states,'' Phys. Rev. A 69 (2004) no. 2, 022316, arXiv:quant-ph/0308151 [quant-ph].
https://doi.org/10.1103/PhysRevA.69.022316
arXiv:quant-ph/0308151

[53] M. Bahramgiri and S. Beigi, ``Graph States Under the Action of Local Clifford Group in Non-Binary Case,'' arXiv e-prints (2006) , arXiv:quant-ph/0610267 [quant-ph].
arXiv:quant-ph/0610267

Cited by

[1] Márton Mestyán, Balázs Pozsgay, and Ian M. Wanless, "Multi-directional unitarity and maximal entanglement in spatially symmetric quantum states", SciPost Physics 16 1, 010 (2024).

[2] Suhail Ahmad Rather, "Construction of perfect tensors using biunimodular vectors", arXiv:2309.01504, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2024-05-23 10:17:08). The list may be incomplete as not all publishers provide suitable and complete citation data.

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