Bounding Entanglement Dimensionality From The Covariance Matrix

Shuheng Liu^1,2,3, Matteo Fadel⁴, Qiongyi He^1,5,6, Marcus Huber^2,3, and Giuseppe Vitagliano^2,3

¹State Key Laboratory for Mesoscopic Physics, School of Physics, Frontiers Science Center for Nano-optoelectronics, & Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
²Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1020 Vienna, Austria
³Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, 1090 Vienna, Austria
⁴Department of Physics, ETH Zürich, 8093 Zürich, Switzerland
⁵Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China
⁶Hefei National Laboratory, Hefei 230088, China

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

Abstract

High-dimensional entanglement has been identified as an important resource in quantum information processing, and also as a main obstacle for simulating quantum systems. Its certification is often difficult, and most widely used methods for experiments are based on fidelity measurements with respect to highly entangled states. Here, instead, we consider covariances of collective observables, as in the well-known Covariance Matrix Criterion (CMC) [1] and present a generalization of the CMC for determining the Schmidt number of a bipartite system. This is potentially particularly advantageous in many-body systems, such as cold atoms, where the set of practical measurements is very limited and only variances of collective operators can typically be estimated. To show the practical relevance of our results, we derive simpler Schmidt-number criteria that require similar information as the fidelity-based witnesses, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin covariances, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems. We conclude by discussing the applicability of our results to a multiparticle ensemble and some open questions for future work.

Featured image: (Above) Entanglement between subsets of levels, signifying the entanglement dimensionality. (Below) Our nonlinear criterion for detecting different entanglement dimensionalities, as an improvement over linear witnesses. $r=1$: separable states, $rgeq 2$: entangled states.

Popular summary

High-dimensional entanglement has been identified as an important resource in quantum information processing, but also as a main obstacle for simulating classically a quantum system. In particular, the resource needed to reproduce the correlations in the quantum state can be quantified by the so-called entanglement dimensionality. Because of this, experiments aim at controlling larger and larger quantum systems and prepare them in high-dimensional entangled states. The question arising is then how to detect such entanglement dimensionality from experimental data, for example through specific entanglement witnesses. Most common methods involve very complex measurements, such as fidelities with respect to highly entangled states, which are often challenging and in some cases, like in ensembles of many atoms, completely inaccessible.

To overcome some of these difficulties, we focus here on quantifying entanglement dimensionality through covariances of global observables, which are typically measured in many-body experiments, such as those involving atomic ensembles in highly entangled spin-squeezed states. Concretely, we generalize well-known entanglement criteria based on covariance matrices of local observables and establish analytical bounds for different entanglement dimensionalities, which, when violated, certify what is the minimal entanglement dimensionality present in the system.

To show the practical relevance of our results, we derive criteria that require similar information as the existing methods in literature, yet can detect a wider set of states. We also consider paradigmatic criteria based on spin operators, similar to spin-squeezing inequalities, which would be very helpful for experimental detection of high-dimensional entanglement in cold atom systems.

As a future outlook, our work also opens interesting research directions and poses further intriguing theoretical questions, such as improving current methods to detect the entanglement dimensionality in multipartite states.

► BibTeX data

@article{Liu2024bounding, doi = {10.22331/q-2024-01-30-1236}, url = {https://doi.org/10.22331/q-2024-01-30-1236}, title = {Bounding entanglement dimensionality from the covariance matrix}, author = {Liu, Shuheng and Fadel, Matteo and He, Qiongyi and Huber, Marcus and Vitagliano, Giuseppe}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {8}, pages = {1236}, month = jan, year = {2024}<br /> }

► References

[1] O. Gühne, P. Hyllus, O. Gittsovich, and J. Eisert. “Covariance Matrices and the Separability Problem”. Phys. Rev. Lett. 99, 130504 (2007).
https://doi.org/10.1103/PhysRevLett.99.130504

[2] E. Schrödinger. “Die gegenwärtige Situation in der Quantenmechanik”. Naturwissenschaften 23, 807–12 (1935).
https://doi.org/10.1007/BF01491891

[3] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. “Quantum entanglement”. Rev. Mod. Phys. 81, 865–942 (2009).
https://doi.org/10.1103/RevModPhys.81.865

[4] Otfried Gühne and Géza Tóth. “Entanglement detection”. Phys. Rep. 474, 1–75 (2009).
https://doi.org/10.1016/j.physrep.2009.02.004

[5] Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber. “Entanglement certification from theory to experiment”. Nat. Rev. Phys. 1, 72–87 (2019).
https://doi.org/10.1038/s42254-018-0003-5

[6] Irénée Frérot, Matteo Fadel, and Maciej Lewenstein. “Probing quantum correlations in many-body systems: a review of scalable methods”. Reports on Progress in Physics 86, 114001 (2023).
https://doi.org/10.1088/1361-6633/acf8d7

[7] Martin B. Plenio and Shashank Virmani. “An introduction to entanglement measures”. Quant. Inf. Comput. 7, 1–51 (2007).
https://doi.org/10.26421/QIC7.1-2-1

[8] Christian Kokail, Bhuvanesh Sundar, Torsten V. Zache, Andreas Elben, Benoı̂t Vermersch, Marcello Dalmonte, Rick van Bijnen, and Peter Zoller. “Quantum variational learning of the entanglement hamiltonian”. Phys. Rev. Lett. 127, 170501 (2021).
https://doi.org/10.1103/PhysRevLett.127.170501

[9] Christian Kokail, Rick van Bijnen, Andreas Elben, Benoı̂t Vermersch, and Peter Zoller. “Entanglement hamiltonian tomography in quantum simulation”. Nat. Phys. 17, 936–942 (2021).
https://doi.org/10.1038/s41567-021-01260-w

[10] Rajibul Islam, Ruichao Ma, Philipp M. Preiss, M. Eric Tai, Alexander Lukin, Matthew Rispoli, and Markus Greiner. “Measuring entanglement entropy in a quantum many-body system”. Nature 528, 77 (2015).
https://doi.org/10.1038/nature15750

[11] David Gross, Yi-Kai Liu, Steven T. Flammia, Stephen Becker, and Jens Eisert. “Quantum state tomography via compressed sensing”. Phys. Rev. Lett. 105, 150401 (2010).
https://doi.org/10.1103/PhysRevLett.105.150401

[12] Oleg Gittsovich and Otfried Gühne. “Quantifying entanglement with covariance matrices”. Phys. Rev. A 81, 032333 (2010).
https://doi.org/10.1103/PhysRevA.81.032333

[13] Matteo Fadel, Ayaka Usui, Marcus Huber, Nicolai Friis, and Giuseppe Vitagliano. “Entanglement Quantification in Atomic Ensembles”. Phys. Rev. Lett. 127, 010401 (2021).
https://doi.org/10.1103/PhysRevLett.127.010401

[14] Fernando G. S. L. Brandão. “Quantifying entanglement with witness operators”. Phys. Rev. A 72, 022310 (2005).
https://doi.org/10.1103/PhysRevA.72.022310

[15] Marcus Cramer, Martin B. Plenio, and Harald Wunderlich. “Measuring Entanglement in Condensed Matter Systems”. Phys. Rev. Lett. 106, 020401 (2011).
https://doi.org/10.1103/PhysRevLett.106.020401

[16] Oliver Marty, Michael Epping, Hermann Kampermann, Dagmar Bruß, Martin B. Plenio, and M. Cramer. “Quantifying entanglement with scattering experiments”. Phys. Rev. B 89, 125117 (2014).
https://doi.org/10.1103/PhysRevB.89.125117

[17] S. Etcheverry, G. Cañas, E. S. Gómez, W. A. T. Nogueira, C. Saavedra, G. B. Xavier, and G. Lima. “Quantum key distribution session with 16-dimensional photonic states”. Sci. Rep. 3, 2316 (2013).
https://doi.org/10.1038/srep02316

[18] Marcus Huber and Marcin Pawłowski. “Weak randomness in device-independent quantum key distribution and the advantage of using high-dimensional entanglement”. Phys. Rev. A 88, 032309 (2013).
https://doi.org/10.1103/PhysRevA.88.032309

[19] Mirdit Doda, Marcus Huber, Gláucia Murta, Matej Pivoluska, Martin Plesch, and Chrysoula Vlachou. “Quantum key distribution overcoming extreme noise: Simultaneous subspace coding using high-dimensional entanglement”. Phys. Rev. Appl. 15, 034003 (2021).
https://doi.org/10.1103/PhysRevApplied.15.034003

[20] Sebastian Ecker, Frédéric Bouchard, Lukas Bulla, Florian Brandt, Oskar Kohout, Fabian Steinlechner, Robert Fickler, Mehul Malik, Yelena Guryanova, Rupert Ursin, and Marcus Huber. “Overcoming noise in entanglement distribution”. Phys. Rev. X 9, 041042 (2019).
https://doi.org/10.1103/PhysRevX.9.041042

[21] Xiao-Min Hu, Chao Zhang, Yu Guo, Fang-Xiang Wang, Wen-Bo Xing, Cen-Xiao Huang, Bi-Heng Liu, Yun-Feng Huang, Chuan-Feng Li, Guang-Can Guo, Xiaoqin Gao, Matej Pivoluska, and Marcus Huber. “Pathways for Entanglement-Based Quantum Communication in the Face of High Noise”. Phys. Rev. Lett. 127, 110505 (2021).
https://doi.org/10.1103/PhysRevLett.127.110505

[22] Benjamin P. Lanyon, Marco Barbieri, Marcelo P. Almeida, Thomas Jennewein, Timothy C. Ralph, Kevin J. Resch, Geoff J. Pryde, Jeremy L. O’Brien, Alexei Gilchrist, and Andrew G. White. “Simplifying quantum logic using higher-dimensional hilbert spaces”. Nat. Phys. 5, 134–140 (2009).
https://doi.org/10.1038/nphys1150

[23] Maarten Van den Nest. “Universal Quantum Computation with Little Entanglement”. Phys. Rev. Lett. 110, 060504 (2013).
https://doi.org/10.1103/PhysRevLett.110.060504

[24] Mario Krenn, Marcus Huber, Robert Fickler, Radek Lapkiewicz, Sven Ramelow, and Anton Zeilinger. “Generation and confirmation of a (100 $times$ 100)-dimensional entangled quantum system”. Proc. Natl. Acad. Sci. U.S.A. 111, 6243–6247 (2014).
https://doi.org/10.1073/pnas.1402365111

[25] Paul Erker, Mario Krenn, and Marcus Huber. “Quantifying high dimensional entanglement with two mutually unbiased bases”. Quantum 1, 22 (2017).
https://doi.org/10.22331/q-2017-07-28-22

[26] Jessica Bavaresco, Natalia Herrera Valencia, Claude Klöckl, Matej Pivoluska, Paul Erker, Nicolai Friis, Mehul Malik, and Marcus Huber. “Measurements in two bases are sufficient for certifying high-dimensional entanglement”. Nat. Phys. 14, 1032–1037 (2018).
https://doi.org/10.1038/s41567-018-0203-z

[27] James Schneeloch, Christopher C. Tison, Michael L. Fanto, Paul M. Alsing, and Gregory A. Howland. “Quantifying entanglement in a 68-billion-dimensional quantum state space”. Nat. Commun. 10, 2785 (2019).
https://doi.org/10.1038/s41467-019-10810-z

[28] Natalia Herrera Valencia, Vatshal Srivastav, Matej Pivoluska, Marcus Huber, Nicolai Friis, Will McCutcheon, and Mehul Malik. “High-Dimensional Pixel Entanglement: Efficient Generation and Certification”. Quantum 4, 376 (2020).
https://doi.org/10.22331/q-2020-12-24-376

[29] Hannes Pichler, Guanyu Zhu, Alireza Seif, Peter Zoller, and Mohammad Hafezi. “Measurement protocol for the entanglement spectrum of cold atoms”. Phys. Rev. X 6, 041033 (2016).
https://doi.org/10.1103/PhysRevX.6.041033

[30] Niklas Euler and Martin Gärttner. “Detecting high-dimensional entanglement in cold-atom quantum simulators” (2023). arXiv:2305.07413.
arXiv:2305.07413

[31] Vittorio Giovannetti, Stefano Mancini, David Vitali, and Paolo Tombesi. “Characterizing the entanglement of bipartite quantum systems”. Phys. Rev. A 67, 022320 (2003).
https://doi.org/10.1103/PhysRevA.67.022320

[32] Bernd Lücke, Jan Peise, Giuseppe Vitagliano, Jan Arlt, Luis Santos, Géza Tóth, and Carsten Klempt. “Detecting Multiparticle Entanglement of Dicke States”. Phys. Rev. Lett. 112, 155304 (2014).
https://doi.org/10.1103/PhysRevLett.112.155304

[33] Giuseppe Vitagliano, Giorgio Colangelo, Ferran Martin Ciurana, Morgan W. Mitchell, Robert J. Sewell, and Géza Tóth. “Entanglement and extreme planar spin squeezing”. Phys. Rev. A 97, 020301(R) (2018).
https://doi.org/10.1103/PhysRevA.97.020301

[34] Luca Pezzè, Augusto Smerzi, Markus K. Oberthaler, Roman Schmied, and Philipp Treutlein. “Quantum metrology with nonclassical states of atomic ensembles”. Rev. Mod. Phys. 90, 035005 (2018).
https://doi.org/10.1103/RevModPhys.90.035005

[35] Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke, Carsten Klempt, and Géza Tóth. “Entanglement and extreme spin squeezing of unpolarized states”. New J. Phys. 19, 013027 (2017).
https://doi.org/10.1088/1367-2630/19/1/013027

[36] Flavio Baccari, Jordi Tura, Matteo Fadel, Albert Aloy, Jean.-Daniel Bancal, Nicolas Sangouard, Maciej Lewenstein, Antonio Acín, and Remigiusz Augusiak. “Bell correlation depth in many-body systems”. Phys. Rev. A 100, 022121 (2019).
https://doi.org/10.1103/PhysRevA.100.022121

[37] Matteo Fadel and Manuel Gessner. “Relating spin squeezing to multipartite entanglement criteria for particles and modes”. Phys. Rev. A 102, 012412 (2020).
https://doi.org/10.1103/PhysRevA.102.012412

[38] Brian Julsgaard, Alexander Kozhekin, and Eugene S. Polzik. “Experimental long-lived entanglement of two macroscopic objects”. Nature 413, 400–403 (2001).
https://doi.org/10.1038/35096524

[39] Matteo Fadel, Tilman Zibold, Boris Décamps, and Philipp Treutlein. “Spatial entanglement patterns and Einstein-Podolsky-Rosen steering in Bose-Einstein condensates”. Science 360, 409–413 (2018).
https://doi.org/10.1126/science.aao1850

[40] Philipp Kunkel, Maximilian Prüfer, Helmut Strobel, Daniel Linnemann, Anika Frölian, Thomas Gasenzer, Martin Gärttner, and Markus K. Oberthaler. “Spatially distributed multipartite entanglement enables EPR steering of atomic clouds”. Science 360, 413–416 (2018).
https://doi.org/10.1126/science.aao2254

[41] Karsten Lange, Jan Peise, Bernd Lücke, Ilka Kruse, Giuseppe Vitagliano, Iagoba Apellaniz, Matthias Kleinmann, Géza Tóth, and Carsten Klempt. “Entanglement between two spatially separated atomic modes”. Science 360, 416–418 (2018).
https://doi.org/10.1126/science.aao2035

[42] Giuseppe Vitagliano, Matteo Fadel, Iagoba Apellaniz, Matthias Kleinmann, Bernd Lücke, Carsten Klempt, and Géza Tóth. “Number-phase uncertainty relations and bipartite entanglement detection in spin ensembles”. Quantum 7, 914 (2023).
https://doi.org/10.22331/q-2023-02-09-914

[43] M. Cramer, A. Bernard, N. Fabbri, L. Fallani, C. Fort, S. Rosi, F. Caruso, M. Inguscio, and M.B. Plenio. “Spatial entanglement of bosons in optical lattices”. Nat. Commun. 4, 2161 (2013).
https://doi.org/10.1038/ncomms3161

[44] Bjarne Bergh and Martin Gärttner. “Experimentally accessible bounds on distillable entanglement from entropic uncertainty relations”. Phys. Rev. Lett. 126, 190503 (2021).
https://doi.org/10.1103/PhysRevLett.126.190503

[45] Bjarne Bergh and Martin Gärttner. “Entanglement detection in quantum many-body systems using entropic uncertainty relations”. Phys. Rev. A 103, 052412 (2021).
https://doi.org/10.1103/PhysRevA.103.052412

[46] Barbara M. Terhal and Paweł Horodecki. “Schmidt number for density matrices”. Phys. Rev. A 61, 040301(R) (2000).
https://doi.org/10.1103/PhysRevA.61.040301

[47] Anna Sanpera, Dagmar Bruß, and Maciej Lewenstein. “Schmidt-number witnesses and bound entanglement”. Phys. Rev. A 63, 050301(R) (2001).
https://doi.org/10.1103/PhysRevA.63.050301

[48] Steven T. Flammia and Yi-Kai Liu. “Direct Fidelity Estimation from Few Pauli Measurements”. Phys. Rev. Lett. 106, 230501 (2011).
https://doi.org/10.1103/PhysRevLett.106.230501

[49] M. Weilenmann, B. Dive, D. Trillo, E. A. Aguilar, and M. Navascués. “Entanglement Detection beyond Measuring Fidelities”. Phys. Rev. Lett. 124, 200502 (2020).
https://doi.org/10.1103/PhysRevLett.124.200502

[50] Asher Peres. “Separability criterion for density matrices”. Phys. Rev. Lett. 77, 1413–1415 (1996).
https://doi.org/10.1103/PhysRevLett.77.1413

[51] Michał Horodecki and Paweł Horodecki. “Reduction criterion of separability and limits for a class of distillation protocols”. Phys. Rev. A 59, 4206–4216 (1999).
https://doi.org/10.1103/PhysRevA.59.4206

[52] N. J. Cerf, C. Adami, and R. M. Gingrich. “Reduction criterion for separability”. Phys. Rev. A 60, 898–909 (1999).
https://doi.org/10.1103/PhysRevA.60.898

[53] Kai Chen, Sergio Albeverio, and Shao-Ming Fei. “Concurrence of arbitrary dimensional bipartite quantum states”. Phys. Rev. Lett. 95, 040504 (2005).
https://doi.org/10.1103/PhysRevLett.95.040504

[54] Julio I. de Vicente. “Lower bounds on concurrence and separability conditions”. Phys. Rev. A 75, 052320 (2007).
https://doi.org/10.1103/PhysRevA.75.052320

[55] Claude Klöckl and Marcus Huber. “Characterizing multipartite entanglement without shared reference frames”. Phys. Rev. A 91, 042339 (2015).
https://doi.org/10.1103/PhysRevA.91.042339

[56] Nathaniel Johnston and David W. Kribs. “Duality of entanglement norms”. Houston J. Math. 41, 831 – 847 (2015).
https://doi.org/10.48550/arXiv.1304.2328

[57] O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. “Unifying several separability conditions using the covariance matrix criterion”. Phys. Rev. A 78, 052319 (2008).
https://doi.org/10.1103/PhysRevA.78.052319

[58] Holger F. Hofmann and Shigeki Takeuchi. “Violation of local uncertainty relations as a signature of entanglement”. Phys. Rev. A 68, 032103 (2003).
https://doi.org/10.1103/PhysRevA.68.032103

[59] Roger A. Horn and Charles R. Johnson. “Topics in matrix analysis”. Page 209 theorem 3.5.15. Cambridge University Press. (1991).
https://doi.org/10.1017/CBO9780511840371

[60] Shuheng Liu, Qiongyi He, Marcus Huber, Otfried Gühne, and Giuseppe Vitagliano. “Characterizing entanglement dimensionality from randomized measurements”. PRX Quantum 4, 020324 (2023).
https://doi.org/10.1103/PRXQuantum.4.020324

[61] Nikolai Wyderka and Andreas Ketterer. “Probing the geometry of correlation matrices with randomized measurements”. PRX Quantum 4, 020325 (2023).
https://doi.org/10.1103/PRXQuantum.4.020325

[62] Satoya Imai, Otfried Gühne, and Stefan Nimmrichter. “Work fluctuations and entanglement in quantum batteries”. Phys. Rev. A 107, 022215 (2023).
https://doi.org/10.1103/PhysRevA.107.022215

[63] Fabian Steinlechner, Sebastian Ecker, Matthias Fink, Bo Liu, Jessica Bavaresco, Marcus Huber, Thomas Scheidl, and Rupert Ursin. “Distribution of high-dimensional entanglement via an intra-city free-space link”. Nat. Commun. 8, 15971 (2017).
https://doi.org/10.1038/ncomms15971

[64] Mehul Malik, Manuel Erhard, Marcus Huber, Mario Krenn, Robert Fickler, and Anton Zeilinger. “Multi-photon entanglement in high dimensions”. Nat. Photonics 10, 248–252 (2016).
https://doi.org/10.1038/nphoton.2016.12

[65] Lukas Bulla, Matej Pivoluska, Kristian Hjorth, Oskar Kohout, Jan Lang, Sebastian Ecker, Sebastian P. Neumann, Julius Bittermann, Robert Kindler, Marcus Huber, Martin Bohmann, and Rupert Ursin. “Nonlocal temporal interferometry for highly resilient free-space quantum communication”. Phys. Rev. X 13, 021001 (2023).
https://doi.org/10.1103/PhysRevX.13.021001

[66] Otfried Gühne and Norbert Lütkenhaus. “Nonlinear entanglement witnesses”. Phys. Rev. Lett. 96, 170502 (2006).
https://doi.org/10.1103/PhysRevLett.96.170502

[67] Otfried Gühne, Mátyás Mechler, Géza Tóth, and Peter Adam. “Entanglement criteria based on local uncertainty relations are strictly stronger than the computable cross norm criterion”. Phys. Rev. A 74, 010301(R) (2006).
https://doi.org/10.1103/PhysRevA.74.010301

[68] Cheng-Jie Zhang, Yong-Sheng Zhang, Shun Zhang, and Guang-Can Guo. “Optimal entanglement witnesses based on local orthogonal observables”. Phys. Rev. A 76, 012334 (2007).
https://doi.org/10.1103/PhysRevA.76.012334

[69] K. G. H. Vollbrecht and R. F. Werner. “Entanglement measures under symmetry”. Phys. Rev. A 64, 062307 (2001).
https://doi.org/10.1103/PhysRevA.64.062307

[70] Marcus Huber, Ludovico Lami, Cécilia Lancien, and Alexander Müller-Hermes. “High-dimensional entanglement in states with positive partial transposition”. Phys. Rev. Lett. 121, 200503 (2018).
https://doi.org/10.1103/PhysRevLett.121.200503

[71] Satoshi Ishizaka. “Bound entanglement provides convertibility of pure entangled states”. Phys. Rev. Lett. 93, 190501 (2004).
https://doi.org/10.1103/PhysRevLett.93.190501

[72] Marco Piani and Caterina E. Mora. “Class of positive-partial-transpose bound entangled states associated with almost any set of pure entangled states”. Phys. Rev. A 75, 012305 (2007).
https://doi.org/10.1103/PhysRevA.75.012305

[73] Ludovico Lami and Marcus Huber. “Bipartite depolarizing maps”. J. Math. Phys. 57, 092201 (2016).
https://doi.org/10.1063/1.4962339

[74] Géza Tóth, Christian Knapp, Otfried Gühne, and Hans J. Briegel. “Spin squeezing and entanglement”. Phys. Rev. A 79, 042334 (2009).
https://doi.org/10.1103/PhysRevA.79.042334

[75] Satoya Imai, Nikolai Wyderka, Andreas Ketterer, and Otfried Gühne. “Bound entanglement from randomized measurements”. Phys. Rev. Lett. 126, 150501 (2021).
https://doi.org/10.1103/PhysRevLett.126.150501

[76] Beatrix C Hiesmayr. “Free versus bound entanglement, a np-hard problem tackled by machine learning”. Sci. Rep. 11, 19739 (2021).
https://doi.org/10.1038/s41598-021-98523-6

[77] Marcin Wieśniak. “Two-qutrit entanglement: 56-years old algorithm challenges machine learning” (2022). arXiv:2211.03213.
arXiv:2211.03213

[78] Marcel Seelbach Benkner, Jens Siewert, Otfried Gühne, and Gael Sentís. “Characterizing generalized axisymmetric quantum states in $d times d$ systems”. Phys. Rev. A 106, 022415 (2022).
https://doi.org/10.1103/PhysRevA.106.022415

[79] Marcus Huber and Julio I. de Vicente. “Structure of multidimensional entanglement in multipartite systems”. Phys. Rev. Lett. 110, 030501 (2013).
https://doi.org/10.1103/PhysRevLett.110.030501

[80] Oleg Gittsovich, Philipp Hyllus, and Otfried Gühne. “Multiparticle covariance matrices and the impossibility of detecting graph-state entanglement with two-particle correlations”. Phys. Rev. A 82, 032306 (2010).
https://doi.org/10.1103/PhysRevA.82.032306

[81] Natalia Herrera Valencia, Vatshal Srivastav, Matej Pivoluska, Marcus Huber, Nicolai Friis, Will McCutcheon, and Mehul Malik. “High-Dimensional Pixel Entanglement: Efficient Generation and Certification”. Quantum 4, 376 (2020).
https://doi.org/10.22331/q-2020-12-24-376

[82] Frank Verstraete, Jeroen Dehaene, and Bart De Moor. “Normal forms and entanglement measures for multipartite quantum states”. Phys. Rev. A 68, 012103 (2003).
https://doi.org/10.1103/PhysRevA.68.012103

[83] John Schliemann. “Entanglement in su(2)-invariant quantum spin systems”. Phys. Rev. A 68, 012309 (2003).
https://doi.org/10.1103/PhysRevA.68.012309

[84] John Schliemann. “Entanglement in su(2)-invariant quantum systems: The positive partial transpose criterion and others”. Phys. Rev. A 72, 012307 (2005).
https://doi.org/10.1103/PhysRevA.72.012307

[85] Kiran K. Manne and Carlton M. Caves. “Entanglement of formation of rotationally symmetric states”. Quantum Info. Comput. 8, 295–310 (2008).

Cited by

[1] Irénée Frérot, Matteo Fadel, and Maciej Lewenstein, “Probing quantum correlations in many-body systems: a review of scalable methods”, Reports on Progress in Physics 86 11, 114001 (2023).

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

[3] Nikolai Wyderka and Andreas Ketterer, “Probing the Geometry of Correlation Matrices with Randomized Measurements”, PRX Quantum 4 2, 020325 (2023).

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

The above citations are from SAO/NASA ADS (last updated successfully 2024-01-30 11:09:58). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2024-01-30 11:09:56: Could not fetch cited-by data for 10.22331/q-2024-01-30-1236 from Crossref. This is normal if the DOI was registered recently.

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

SEO Powered Content & PR Distribution. Get Amplified Today.
PlatoData.Network Vertical Generative Ai. Empower Yourself. Access Here.
PlatoAiStream. Web3 Intelligence. Knowledge Amplified. Access Here.
PlatoESG. Carbon, CleanTech, Energy, Environment, Solar, Waste Management. Access Here.
PlatoHealth. Biotech and Clinical Trials Intelligence. Access Here.
Source: https://quantum-journal.org/papers/q-2024-01-30-1236/

Generative Data Intelligence

Bounding entanglement dimensionality from the covariance matrix

Abstract

Popular summary

► BibTeX data

► References

Cited by

DOJ Disputes Roman Storm’s Characterization of Tornado Cash Operations in New Filing

Top 5 Crypto Presales: BDAG Leads the Pack with 30,000x ROI Potential

Latest Intelligence

Forbes Unveils 20 Crypto ‘Zombies,’ Declares Ripple And XRP Among The Undead

Consob, Italy’s Watchdog, Bans Additional Forex And Cryptocurrency Trading Websites – CryptoInfoNet

Fed’s Inflation Gauge Heats Up, Rate Cuts on Hold as Bitcoin and Stocks Slide

Explore Top 6 Cryptos for 2024: BlockDAG Leads with Unprecedented ROI Potential

Franklin Templeton’s Ethereum spot ETF listed on DTCC

Bitcoin Bloodbath: Crypto Analyst Spots ‘Death Cross’ After 8% Price Drop

Chat with us