Estimating the entanglement of random multipartite quantum states
1École Polytechnique Fédérale de Lausanne, Switzerland
2Institut Fourier, Université Grenoble Alpes, CNRS, France
3Laboratoire de Physique Théorique, Université de Toulouse, UPS, CNRS, France
| Published: | 2025-10-01, volume 9, page 1870 |
| Editor: | Jens Siewert |
| Eprint: | arXiv:2209.11754v4 |
| Doi: | https://doi.org/10.22331/q-2025-10-01-1870 |
| Citation: | Quantum 9, 1870 (2025). |
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Abstract
Genuine multipartite entanglement of a given multipartite pure quantum state can be quantified through its geometric measure of entanglement, which, up to logarithms, is simply the maximum overlap of the corresponding unit tensor with product unit tensors, a quantity that is also known as the injective norm of the tensor. Our general goal in this work is to estimate this injective norm of randomly sampled tensors. To this end, we study and compare various algorithms, based either on the widely used alternating least squares method or on a novel normalized gradient descent approach, and suited to either symmetrized or non-symmetrized random tensors. We first benchmark their respective performances on the case of symmetrized real Gaussian tensors, whose asymptotic average injective norm is known analytically. Having established that our proposed normalized gradient descent algorithm generally performs best, we then use it to obtain numerical estimates for the average injective norm of complex Gaussian tensors (i.e., up to normalization, uniformly distributed multipartite pure quantum states), with or without permutation-invariance. We also estimate the average injective norm of random matrix product states constructed from Gaussian local tensors, with or without translation-invariance. All these results constitute the first numerical estimates on the amount of genuinely multipartite entanglement typically present in various models of random multipartite pure states. Finally, motivated by our numerical results, we posit two conjectures on the injective norms of random Gaussian tensors (real and complex) and Gaussian MPS in the asymptotic limit of the physical dimension.
Featured image: The genuine multipartite entanglement, a potent quantum resource, present in a tensor $\Psi$ can be measured by finding the closest product state $\hat{\phi}$. Our normalized gradient descent algorithm achieves this through an iterative "update and normalize" design.
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Popular summary
A faithful measure of multipartite entanglement is the injective norm, which quantifies how far a state is from being non-entangled (separable). In this work, we present a novel algorithm, normalized gradient descent (NGD), for estimating the injective norm of arbitrary tensors. We first demonstrate that NGD significantly outperforms existing algorithms and then provide the first numerical estimates for the injective norms of two families of physically relevant quantum states: random multipartite quantum states and random matrix product states.
Finally, supported by our numerical findings, we posit two conjectures. One on the multipartite entanglement of real versus complex tensors, and another on the asymptotic limits of multipartite entanglement in matrix product states. Early follow-up work has already provided additional evidence supporting the former conjecture. We have open-sourced our code and believe it will motivate further advances in these directions.
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► References
[1] Ryszard Horodecki, Paweł Horodecki, Michał Horodecki, and Karol Horodecki. ``Quantum entanglement''. Reviews of Modern Physics 81, 865 (2009).
https://doi.org/10.1103/RevModPhys.81.865
[2] Wolfgang Dür, Guifre Vidal, and J. Ignacio Cirac. ``Three qubits can be entangled in two inequivalent ways''. Physical Review A 62, 062314 (2000).
https://doi.org/10.1103/PhysRevA.62.062314
[3] Valerie Coffman, Joydip Kundu, and William K. Wootters. ``Distributed entanglement''. Physical Review A 61, 052306 (2000).
https://doi.org/10.1103/PhysRevA.61.052306
[4] Howard Barnum and Noah Linden. ``Monotones and invariants for multi-particle quantum states''. Journal of Physics A: Mathematical and General 34, 6787 (2001).
https://doi.org/10.1088/0305-4470/34/35/305
[5] Alexander Wong and Nelson Christensen. ``Potential multiparticle entanglement measure''. Physical Review A 63, 044301 (2001).
https://doi.org/10.1103/PhysRevA.63.044301
[6] Alexandre Grothendieck. ``Résumé de la théorie métrique des produits tensoriels topologiques''. Soc. de Matemática de São Paulo. (1996). url: https://doi.org/10.11606/resimeusp.v2i4.74836.
https://doi.org/10.11606/resimeusp.v2i4.74836
[7] Raymond A. Ryan. ``Introduction to tensor products of banach spaces''. Springer Science & Business Media. (2013).
https://doi.org/10.1007/978-1-4471-3903-4
[8] Guillaume Aubrun and Stanisław J. Szarek. ``Alice and bob meet banach: The interface of asymptotic geometric analysis and quantum information theory''. Volume 223. American Mathematical Soc. (2017).
https://doi.org/10.1090/surv/223
[9] Benoit Collins and Ion Nechita. ``Random matrix techniques in quantum information theory''. Journal of Mathematical Physics 57 (2016).
https://doi.org/10.1063/1.4936880
[10] Guillaume Aubrun, Stanislaw J. Szarek, and Deping Ye. ``Entanglement thresholds for random induced states''. Communications on Pure and Applied Mathematics 67, 129–171 (2014).
https://doi.org/10.1002/cpa.21460
[11] Matthew B. Hastings. ``Superadditivity of communication capacity using entangled inputs''. Nature Physics 5, 255–257 (2009).
https://doi.org/10.1038/nphys1224
[12] Karol Życzkowski, Karol A Penson, Ion Nechita, and Benoit Collins. ``Generating random density matrices''. Journal of Mathematical Physics 52, 062201 (2011).
https://doi.org/10.1063/1.3595693
[13] Silvano Garnerone, Thiago R. de Oliveira, and Paolo Zanardi. ``Typicality in random matrix product states''. Physical Review A 81, 032336 (2010).
https://doi.org/10.1103/PhysRevA.81.032336
[14] Benoı̂t Collins, Carlos E. González-Guillén, and David Pérez-García. ``Matrix product states, random matrix theory and the principle of maximum entropy''. Communications in Mathematical Physics 320, 663–677 (2013).
https://doi.org/10.1007/s00220-013-1718-x
[15] Carlos E. González-Guillén, Marius Junge, and Ion Nechita. ``On the spectral gap of random quantum channels'' (2018). url: https://arxiv.org/abs/1811.08847.
arXiv:1811.08847
[16] Cécilia Lancien and David Pérez-García. ``Correlation length in random mps and peps''. Annales Henri Poincaré 23, 141–222 (2022).
https://doi.org/10.1007/s00023-021-01087-4
[17] Jonas Haferkamp, Christian Bertoni, Ingo Roth, and Jens Eisert. ``Emergent statistical mechanics from properties of disordered random matrix product states''. PRX Quantum 2, 040308 (2021).
https://doi.org/10.1103/PRXQuantum.2.040308
[18] Thomas Köhler, Jan Stolpp, and Sebastian Paeckel. ``Efficient and flexible approach to simulate low-dimensional quantum lattice models with large local hilbert spaces''. SciPost Physics 10 (2021).
https://doi.org/10.21468/scipostphys.10.3.058
[19] Nicolas Laflorencie. ``Quantum entanglement in condensed matter systems''. Physics Reports 646, 1–59 (2016).
https://doi.org/10.1016/j.physrep.2016.06.008
[20] Eugenio Bianchi, Lucas Hackl, Mario Kieburg, Marcos Rigol, and Lev Vidmar. ``Volume-law entanglement entropy of typical pure quantum states''. PRX Quantum 3 (2022).
https://doi.org/10.1103/prxquantum.3.030201
[21] Alioscia Hamma, Siddhartha Santra, and Paolo Zanardi. ``Quantum entanglement in random physical states''. Physical Review Letters 109 (2012).
https://doi.org/10.1103/physrevlett.109.040502
[22] Christopher J Hillar and Lek-Heng Lim. ``Most tensor problems are np-hard''. Journal of the ACM (JACM) 60, 45 (2013).
https://doi.org/10.1145/2512329
[23] Yoshio Takane, Forrest W. Young, and Jan de Leeuw. ``Nonmetric individual differences multidimensional scaling: An alternating least squares method with optimal scaling features''. Psychometrika 42, 7–67 (1977).
https://doi.org/10.1007/BF02293745
[24] Antonio Auffinger, Gérard Ben Arous, and Jiří Černỳ. ``Random matrices and complexity of spin glasses''. Communications on Pure and Applied Mathematics 66, 165–201 (2013).
https://doi.org/10.1002/cpa.21422
[25] Amelia Perry, Alexander S. Wein, and Afonso S. Bandeira. ``Statistical limits of spiked tensor models''. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 56, 230 – 264 (2020).
https://doi.org/10.1214/19-AIHP960
[26] Khurshed Fitter, Cecilia Lancien, and Ion Nechita. https://github.com/GlazeDonuts/Tensor-Norms (2022).
https://github.com/GlazeDonuts/Tensor-Norms
[27] Abner Shimony. ``Degree of entanglement''. Annals of the New York Academy of Sciences 755, 675–679 (1995).
https://doi.org/10.1111/j.1749-6632.1995.tb39008.x
[28] Tzu-Chieh Wei and Paul M. Goldbart. ``Geometric measure of entanglement and applications to bipartite and multipartite quantum states''. Physical Review A 68, 042307 (2003).
https://doi.org/10.1103/PhysRevA.68.042307
[29] Huangjun Zhu, Lin Chen, and Masahito Hayashi. ``Additivity and non-additivity of multipartite entanglement measures''. New Journal of Physics 12, 083002 (2010).
https://doi.org/10.1088/1367-2630/13/1/019501
[30] Oleg Evnin. ``Melonic dominance and the largest eigenvalue of a large random tensor''. Letters in Mathematical Physics 111 (2021).
https://doi.org/10.1007/s11005-021-01407-z
[31] Pierre Comon, Xavier Luciani, and André L. F. De Almeida. ``Tensor Decompositions, Alternating Least Squares and other Tales''. Journal of Chemometrics 23, 393–405 (2009).
https://doi.org/10.1002/cem.1236
[32] Ryota Tomioka and Taiji Suzuki. ``Spectral norm of random tensors'' (2014).
[33] Cecilia Lancien and Ion Nechita. ``Estimating the injective norm of high-dimensional random tensors''. work in progress.
[34] Matthew B. Hastings. ``Solving gapped hamiltonians locally''. Phys. Rev. B 73, 085115 (2006).
https://doi.org/10.1103/PhysRevB.73.085115
[35] Zeph Landau, Umesh Vazirani, and Thomas Vidick. ``A polynomial time algorithm for the ground state of one-dimensional gapped local Hamiltonians''. Nature Physics 11, 566–569 (2015).
https://doi.org/10.1038/nphys3345
[36] S. Dartois and B. McKenna. ``Injective norm of real and complex random tensors i: From spin glasses to geometric entanglement''. https://arxiv.org/abs/2404.03627 (2024).
arXiv:2404.03627
[37] Naoki Sasakura. ``Signed eigenvalue/vector distribution of complex order-three random tensor''. Progress of Theoretical and Experimental Physics 2024, 053A04 (2024).
https://doi.org/10.1093/ptep/ptae062
[38] N. Kiesel, C. Schmid, G. Tóth, E. Solano, and H. Weinfurter. ``Experimental observation of four-photon entangled dicke state with high fidelity''. Phys. Rev. Lett. 98, 063604 (2007).
https://doi.org/10.1103/PhysRevLett.98.063604
[39] R. Prevedel, G. Cronenberg, M. S. Tame, M. Paternostro, P. Walther, M. S. Kim, and A. Zeilinger. ``Experimental realization of dicke states of up to six qubits for multiparty quantum networking''. Phys. Rev. Lett. 103, 020503 (2009).
https://doi.org/10.1103/PhysRevLett.103.020503
[40] D. B. Hume, C. W. Chou, T. Rosenband, and D. J. Wineland. ``Preparation of dicke states in an ion chain''. Physical Review A 80 (2009).
https://doi.org/10.1103/physreva.80.052302
[41] I. Jex, G. Alber, S.M. Barnett, and A. Delgado. ``Antisymmetric multi-partite quantum states and their applications''. Fortschritte der Physik 51, 172–178 (2003).
https://doi.org/10.1002/prop.200310021
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[1] Aaditya Rudra and Maria Anastasia Jivulescu, "Calculating the Projective Norm of Higher-Order Tensors Using a Gradient Descent Algorithm", Mathematics 14 1, 105 (2025).
[2] Lisa T. Weinbrenner and Otfried Gühne, "Quantifying entanglement from the geometric perspective", EPL (Europhysics Letters) 151 6, 68001 (2025).
[3] Swastik Majumder and Naoki Sasakura, "Three Cases of Complex Eigenvalue/Vector Distributions of Symmetric Order-Three Random Tensors", Progress of Theoretical and Experimental Physics 2024 9, 093A01 (2024).
[4] Jonathan Steinberg and Otfried Gühne, "Finding maximal quantum resources", Physical Review A 110 6, 062428 (2024).
[5] Max Regalado Kloos and Naoki Sasakura, "Usefulness of signed eigenvalue/vector distributions of random tensors", Letters in Mathematical Physics 114 3, 80 (2024).
[6] Stephane Dartois and Gilles Zémor, "The injective norm of CSS quantum error-correcting codes", arXiv:2510.23736, (2025).
[7] Naoki Sasakura, "Signed Eigenvalue/vector Distribution of Complex Order-Three Random Tensor", Progress of Theoretical and Experimental Physics 2024 5, 053A04 (2024).
[8] Stephane Dartois and Parham Radpay, "On the injective norm of random fermionic states and skew-symmetric tensors", arXiv:2510.25474, (2025).
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[10] Anwesha Chakraborty, Lucas Hackl, and Mario Kieburg, "Random matrix prediction of average entanglement entropy in non-Abelian symmetry sectors", arXiv:2512.22942, (2025).
[11] Piotr Masajada, Aby Philip, and Alexander Streltsov, "ENTCALC: Toolkit for calculating geometric entanglement in multipartite quantum systems", arXiv:2512.10884, (2025).
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