Entanglement characterization using quantum designs

Andreas Ketterer1,2,4, Nikolai Wyderka3,4, and Otfried Gühne4

1Physikalisches Institut, Albert-Ludwigs-Universität Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
2EUCOR Centre for Quantum Science and Quantum Computing, Hermann-Herder-Str. 3, 79104 Freiburg, Germany
3Institut für Theoretische Physik III, Heinrich-Heine-Universität Düsseldorf, Universitätsstr. 1, 40225 Düsseldorf, Germany
4Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany

Find this paper interesting or want to discuss? Scite or leave a comment on SciRate.

Abstract

We present in detail a statistical approach for the reference-frame-independent detection and characterization of multipartite entanglement based on moments of randomly measured correlation functions. We start by discussing how the corresponding moments can be evaluated with designs, linking methods from group and entanglement theory. Then, we illustrate the strengths of the presented framework with a focus on the multipartite scenario. We discuss a condition for characterizing genuine multipartite entanglement for three qubits, and we prove criteria that allow for a discrimination of $W$-type entanglement for an arbitrary number of qubits.

► BibTeX data

► References

[1] O. Gühne and G. Tóth, Phys. Rep. 474, 1 (2009).
https:/​/​doi.org/​10.1016/​j.physrep.2009.02.004

[2] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Rev. Mod. Phys. 79, 555 (2007).
https:/​/​doi.org/​10.1103/​RevModPhys.79.555

[3] R. Ursin, et al., Nat. Phys. 3, 481 (2007).
https:/​/​doi.org/​10.1038/​nphys629

[4] F. Flamini, N. Spagnolo, and F. Sciarrino, Rep. Prog. Phys. 82, 016001 (2018).
https:/​/​doi.org/​10.1088/​1361-6633/​aad5b2

[5] J. G. Rarity, P. R. Tapster, P. M. Gorman, and P. Knight, New J. Phys. 4, 82 (2002).
https:/​/​doi.org/​10.1088/​1367-2630/​4/​1/​382

[6] M. Aspelmeyer, et al., Science 301, 621 (2003).
https:/​/​doi.org/​10.1126/​science.1085593

[7] P. Villoresi, et al., New J. Phys. 10, 033038 (2008).
https:/​/​doi.org/​10.1088/​1367-2630/​10/​3/​033038

[8] C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, New J. Phys. 11, 045017 (2009).
https:/​/​doi.org/​10.1088/​1367-2630/​11/​4/​045017

[9] L. Aolita and S. P. Walborn, Phys. Rev. Lett. 98, 100501 (2007).
https:/​/​doi.org/​10.1103/​PhysRevLett.98.100501

[10] V. D'Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, Nat. Comm. 3, 961 (2012).
https:/​/​doi.org/​10.1038/​ncomms1951

[11] H. Aschauer, J. Calsamiglia, M. Hein, and H. J. Briegel, Quantum Inf. Comput. 4, 383 (2004).
https:/​/​doi.org/​10.26421/​QIC4.5

[12] J. I. de Vicente, Quantum Inf. Comput. 7, 624 (2007).
https:/​/​doi.org/​10.26421/​QIC4.5

[13] J. I. de Vicente, J. Phys. A: Math. Theor. 41, 065309 (2008).
https:/​/​doi.org/​10.1088/​1751-8113/​41/​6/​065309

[14] J. I. de Vicente and M. Huber, Phys. Rev. A 84, 062306 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.062306

[15] P. Badziag, C. Brukner, W. Laskowski, T. Paterek, and M. Żukowski, Phys. Rev. Lett. 100, 140403 (2008).
https:/​/​doi.org/​10.1103/​PhysRevLett.100.140403

[16] W. Laskowski, M. Markiewicz, T. Paterek, and M. Żukowski, Phys. Rev. A 84, 062305 (2011).
https:/​/​doi.org/​10.1103/​PhysRevA.84.062305

[17] T. Lawson, A. Pappa, B. Bourdoncle, I. Kerenidis, D. Markham, and E. Diamanti, Phys. Rev. A 90, 042336 (2014).
https:/​/​doi.org/​10.1103/​PhysRevA.90.042336

[18] C. Klöckl and M. Huber, Phys. Rev. A 91, 042339 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.042339

[19] M. C. Tran, B. Dakić, F. Arnault, W. Laskowski, and T. Paterek, Phys. Rev. A 92, 050301(R) (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.92.050301

[20] M. C. Tran, B. Dakić, W. Laskowski, and T. Paterek, Phys. Rev. A 94, 042302 (2016).
https:/​/​doi.org/​10.1103/​PhysRevA.94.042302

[21] A. Ketterer, N. Wyderka, O. Gühne, Phys. Rev. Lett. 122, 120505 (2019).
https:/​/​doi.org/​10.1103/​PhysRevLett.122.120505

[22] M. Krebsbach, Bachelor thesis, Albert-Ludwigs-Universität Freiburg (2019).
https:/​/​doi.org/​10.6094/​UNIFR/​150706

[23] A. Elben, B. Vermersch, M. Dalmonte, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 120, 050406 (2018).
https:/​/​doi.org/​10.1103/​PhysRevLett.120.050406

[24] T. Brydges, A. Elben, P. Jurcevic, B. Vermersch, C. Maier, B. P. Lanyon, P. Zoller, R. Blatt, C. F. Roos, Science 364, 260 (2019).
https:/​/​doi.org/​10.1126/​science.aau4963

[25] A. Elben, B. Vermersch, C. F. Roos, P. Zoller, Phys. Rev. A 99, 052323 (2019).
https:/​/​doi.org/​10.1103/​PhysRevA.99.052323

[26] L. Knips, J. Dziewior, W. Kłobus, W. Laskowski, T. Paterek, P. J. Shadbolt, H. Weinfurter, and J. D. A. Meinecke, npj Quantum Information 6, 51 (2020).
https:/​/​doi.org/​10.1038/​s41534-020-0281-5

[27] A. Elben, B. Vermersch, R. van Bijnen, C. Kokail, T. Brydges, C. Maier, M. K. Joshi, R. Blatt, C. F. Roos, and P. Zoller, Phys. Rev. Lett. 124, 010504 (2020).
https:/​/​doi.org/​10.1103/​PhysRevLett.124.010504

[28] N. Wyderka, F. Huber, and O. Gühne, Phys. Rev. A 97, 060101 (2018).
https:/​/​doi.org/​10.1103/​PhysRevA.97.060101

[29] N. Wyderka and O. Gühne, J. Phys. A: Math. Theor. 53, 345302.
https:/​/​doi.org/​10.1088/​1751-8121/​ab7f0a

[30] C. Eltschka and J. Siewert, Quantum 4, 229 (2020).
https:/​/​doi.org/​10.22331/​q-2020-02-10-229

[31] M. Idel and M. M. Wolf, Lin. Alg. Appl. 471, 76 (2015).
https:/​/​doi.org/​10.1016/​j.laa.2014.12.031

[32] C. Dankert, M.Sc. thesis, University of Waterloo, (2005); also available as e-print quant-ph/​0512217.
arXiv:quant-ph/0512217

[33] P. D. Seymour, T. Zaslavsky, Advances in Mathematics 52, 213 (1984).
https:/​/​doi.org/​10.1016/​0001-8708(84)90022-7

[34] F. G. S. L. Brandão, A. W. Harrow, and M. Horodecki, Phys. Rev. Lett. 116, 170502 (2016).
https:/​/​doi.org/​10.1103/​PhysRevLett.116.170502

[35] Y. Nakata, C. Hirche, M. Koashi, A. Winter, Phys. Rev. X 7, 021006 (2017).
https:/​/​doi.org/​10.1103/​PhysRevX.7.021006

[36] J. Haferkamp, F. Montealegre-Mora, M. Heinrich, J. Eisert, D. Gross, and I. Roth, arXiv:2002.09524.
arXiv:2002.09524

[37] Z. Webb, Quantum Inf. Comput. 16, 1379 (2016).
https:/​/​doi.org/​10.26421/​QIC16.15-16

[38] H. Zhu, R. Kueng, M. Grassl, and D. Gross, arXiv:1609.08172.
arXiv:1609.08172

[39] D. Gross, K. Audenaert, and J. Eisert, J. Math. Phys. 48, 052104 (2007).
https:/​/​doi.org/​10.1063/​1.2716992

[40] R. H. Hardin and N. J. A. Sloane, Discrete & Computational Geometry 15, 429 (1996).
https:/​/​doi.org/​10.1007/​BF02711518

[41] R. Horodecki and M. Horodecki, Phys. Rev. A 54, 1838 (1996).
https:/​/​doi.org/​10.1103/​PhysRevA.54.1838

[42] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, New York, 2000).
https:/​/​doi.org/​10.1017/​CBO9780511976667

[43] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009).
https:/​/​doi.org/​10.1103/​RevModPhys.81.865

[44] W. Dür, G. Vidal, and J. I. Cirac, Phys. Rev. A 62, 062314 (2000).
https:/​/​doi.org/​10.1103/​PhysRevA.62.062314

[45] A. Acín, D. Bruß, M. Lewenstein, and A. Sanpera, Phys. Rev. Lett. 87, 040401 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.87.040401

[46] R. Raussendorf and H. J. Briegel, Phys. Rev. Lett. 86, 5188 (2001).
https:/​/​doi.org/​10.1103/​PhysRevLett.86.5188

[47] R. Cleve, D. Gottesman, and H.-K. Lo, Phys. Rev. Lett. 83, 648 (1999).
https:/​/​doi.org/​10.1103/​PhysRevLett.83.648

[48] A. Acín, A. Andrianov, L. Costa, E. Jané, J. I. Latorre, and R. Tarrach, Phys. Rev. Lett. 85, 1560 (2000).
https:/​/​doi.org/​10.1103/​PhysRevLett.85.1560

[49] C. Spee, J. I. de Vicente, and B. Kraus, J. Math. Phys. 57, 052201 (2016).
https:/​/​doi.org/​10.1063/​1.4946895

[50] T. Bastin, P. Mathonet, and E. Solano, Phys. Rev. A 91, 022310 (2015).
https:/​/​doi.org/​10.1103/​PhysRevA.91.022310

[51] C. Ritz, C. Spee, and O. Gühne, J. Phys. A: Math. Theor. 52, 335302 (2019).
https:/​/​doi.org/​10.1088/​1751-8121/​ab2f54

[52] H. A. Carteret, A. Higuchi, and A. Sudbery, J. Math. Phys. 41, 7932 (2000).
https:/​/​doi.org/​10.1063/​1.1319516

[53] S. Kıntaş and S. Turgut, J. Math. Phys. 51, 092202 (2010).
https:/​/​doi.org/​10.1063/​1.3481573

[54] J. Maziero, Braz. J. Phys. 45, 575 (2015).
https:/​/​doi.org/​10.1007/​s13538-015-0367-2

[55] D. S. Bernstein, Matrix Mathematics: Theory, Facts, and Formulas - Second Edition, (Princeton University Press, Princeton, 2009).
https:/​/​press.princeton.edu/​books/​paperback/​9780691140391/​matrix-mathematics

Cited by

[1] Zhenhuan Liu, Pei Zeng, You Zhou, and Mile Gu, "Characterizing correlation within multipartite quantum systems via local randomized measurements", Physical Review A 105 2, 022407 (2022).

[2] Andreas Elben, Steven T. Flammia, Hsin-Yuan Huang, Richard Kueng, John Preskill, Benoît Vermersch, and Peter Zoller, "The randomized measurement toolbox", Nature Reviews Physics 5 1, 9 (2022).

[3] Andreas Ketterer, Satoya Imai, Nikolai Wyderka, and Otfried Gühne, "Statistically significant tests of multiparticle quantum correlations based on randomized measurements", Physical Review A 106 1, L010402 (2022).

[4] Cole Miles, Rhine Samajdar, Sepehr Ebadi, Tout T. Wang, Hannes Pichler, Subir Sachdev, Mikhail D. Lukin, Markus Greiner, Kilian Q. Weinberger, and Eun-Ah Kim, "Machine learning discovery of new phases in programmable quantum simulator snapshots", Physical Review Research 5 1, 013026 (2023).

[5] Mazhar Ali, "Partial transpose moments, principal minors and entanglement detection", Quantum Information Processing 22 5, 207 (2023).

[6] Michael E. N. Tschaffon, Johannes Seiler, and Matthias Freyberger, "Average correlation as an indicator for nonclassicality", Physical Review Research 5 2, 023063 (2023).

[7] Aniket Rath, Rick van Bijnen, Andreas Elben, Peter Zoller, and Benoît Vermersch, "Importance Sampling of Randomized Measurements for Probing Entanglement", Physical Review Letters 127 20, 200503 (2021).

[8] Lukas Knips, "A Moment for Random Measurements", Quantum Views 4, 47 (2020).

[9] Michael E. N. Tschaffon and Johannes Seiler, "Average correlation as an indicator for inseparability", Physical Review Research 6 1, 013186 (2024).

[10] Aniket Rath, Cyril Branciard, Anna Minguzzi, and Benoît Vermersch, "Quantum Fisher Information from Randomized Measurements", Physical Review Letters 127 26, 260501 (2021).

[11] Maria Balanzó-Juandó, Michał Studziński, and Felix Huber, "Positive maps from the walled Brauer algebra", Journal of Physics A: Mathematical and Theoretical 57 11, 115202 (2024).

[12] Nikolai Wyderka, Andreas Ketterer, Satoya Imai, Jan Lennart Bönsel, Daniel E. Jones, Brian T. Kirby, Xiao-Dong Yu, and Otfried Gühne, "Complete Characterization of Quantum Correlations by Randomized Measurements", Physical Review Letters 131 9, 090201 (2023).

[13] Satoya Imai, Nikolai Wyderka, Andreas Ketterer, and Otfried Gühne, "Bound Entanglement from Randomized Measurements", Physical Review Letters 126 15, 150501 (2021).

[14] Vaishali Gulati, Arvind, and Kavita Dorai, "Classification and measurement of multipartite entanglement by reconstruction of correlation tensors on an NMR quantum processor", The European Physical Journal D 76 10, 194 (2022).

[15] Benoît Collins, Razvan Gurau, and Luca Lionni, "The Tensor Harish-Chandra–Itzykson–Zuber Integral II: Detecting Entanglement in Large Quantum Systems", Communications in Mathematical Physics 401 1, 669 (2023).

[16] Roy J. Garcia, You Zhou, and Arthur Jaffe, "Quantum scrambling with classical shadows", Physical Review Research 3 3, 033155 (2021).

[17] Donghoon Ha and Jeong San Kim, "Bipartite quantum state discrimination and decomposable entanglement witness", Physical Review A 107 5, 052410 (2023).

[18] Shuheng Liu, Qiongyi He, Marcus Huber, Otfried Gühne, and Giuseppe Vitagliano, "Characterizing Entanglement Dimensionality from Randomized Measurements", PRX Quantum 4 2, 020324 (2023).

[19] Xin Yan, Ye‐Chao Liu, and Jiangwei Shang, "Operational Detection of Entanglement via Quantum Designs", Annalen der Physik 534 5, 2100594 (2022).

[20] Antoine Neven, Jose Carrasco, Vittorio Vitale, Christian Kokail, Andreas Elben, Marcello Dalmonte, Pasquale Calabrese, Peter Zoller, Benoȋt Vermersch, Richard Kueng, and Barbara Kraus, "Symmetry-resolved entanglement detection using partial transpose moments", npj Quantum Information 7 1, 152 (2021).

[21] Satoya Imai, Otfried Gühne, and Stefan Nimmrichter, "Work fluctuations and entanglement in quantum batteries", Physical Review A 107 2, 022215 (2023).

[22] Xiao-Dong Yu, Satoya Imai, and Otfried Gühne, "Optimal Entanglement Certification from Moments of the Partial Transpose", Physical Review Letters 127 6, 060504 (2021).

[23] Sophia Ohnemus, Heinz-Peter Breuer, and Andreas Ketterer, "Quantifying multiparticle entanglement with randomized measurements", Physical Review A 107 4, 042406 (2023).

[24] Lukas Knips, "A Moment for Random Measurements", arXiv:2011.10591, (2020).

[25] Saroch Leedumrongwatthanakun, Luca Innocenti, Alessandro Ferraro, Mauro Paternostro, and Sylvain Gigan, "Classification of quantum states of light using random measurements through a multimode fiber", arXiv:2310.13599, (2023).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-25 05:26:56) and SAO/NASA ADS (last updated successfully 2024-05-25 05:26:57). The list may be incomplete as not all publishers provide suitable and complete citation data.

1 thought on “Entanglement characterization using quantum designs

  1. Pingback: Perspective in Quantum Views by Lukas Knips "A Moment for Random Measurements"