Estimating the entanglement of random multipartite quantum states

• URL: http://arxiv.org/abs/2209.11754v4
• Date: Thu, 25 Sep 2025 20:16:43 GMT
• Title: Estimating the entanglement of random multipartite quantum states
• Authors: Khurshed P. Fitter, Cecilia Lancien, Ion Nechita,
• Abstract summary: We study and compare various algorithms to estimate the injective norm of randomly sampled tensors.<n>First numerical estimates on the amount of genuinely multipartite entanglement typically present in various models of random multipartite pure states.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• A moment-based approach to the injective norm of random tensors [0.0]
  We present a method to establish upper bounds on the expected injective norm of real and complex random tensors.<n>Our results yield rigorous estimates on the ground-state energy of real and complex, possibly non-Gaussian, spin glass models.<n>They also establish bounds on the geometric entanglement of random bosonic states and of random states with bounded multipartite Schmidt rank.
  arXiv  Detail & Related papers  (2026-03-02T00:50:33Z)
• On the injective norm of random fermionic states and skew-symmetric tensors [0.0]
  We study the injective norm of random skew-symmetric tensors and the associated fermionic quantum states.<n> Extending recent advances on random quantum states, we analyze both real and complex skew-symmetric ensembles.
  arXiv  Detail & Related papers  (2025-10-29T12:52:07Z)
• Tensor cumulants for statistical inference on invariant distributions [49.80012009682584]
  We show that PCA becomes computationally hard at a critical value of the signal's magnitude. We define a new set of objects, which provide an explicit, near-orthogonal basis for invariants of a given degree. It also lets us analyze a new problem of distinguishing between different ensembles.
  arXiv  Detail & Related papers  (2024-04-29T14:33:24Z)
• Quantization of Large Language Models with an Overdetermined Basis [73.79368761182998]
  We introduce an algorithm for data quantization based on the principles of Kashin representation. Our findings demonstrate that Kashin Quantization achieves competitive or superior quality in model performance.
  arXiv  Detail & Related papers  (2024-04-15T12:38:46Z)
• Injective norm of real and complex random tensors I: From spin glasses to geometric entanglement [0.0]
  In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states. In this paper, we give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors.
  arXiv  Detail & Related papers  (2024-04-04T17:49:23Z)
• Quantum tomography of helicity states for general scattering processes [55.2480439325792]
  Quantum tomography has become an indispensable tool in order to compute the density matrix $rho$ of quantum systems in Physics. We present the theoretical framework for reconstructing the helicity quantum initial state of a general scattering process.
  arXiv  Detail & Related papers  (2023-10-16T21:23:42Z)
• Page curves and typical entanglement in linear optics [0.0]
  We study entanglement within a set of squeezed modes that have been evolved by a random linear optical unitary. We prove various results on the typicality of entanglement as measured by the R'enyi-2 entropy. Our main make use of a symmetry property obeyed by the average and the variance of the entropy that dramatically simplifies the averaging over unitaries.
  arXiv  Detail & Related papers  (2022-09-14T18:00:03Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• When Random Tensors meet Random Matrices [50.568841545067144]
  This paper studies asymmetric order-$d$ spiked tensor models with Gaussian noise. We show that the analysis of the considered model boils down to the analysis of an equivalent spiked symmetric textitblock-wise random matrix.
  arXiv  Detail & Related papers  (2021-12-23T04:05:01Z)
• Near optimal sample complexity for matrix and tensor normal models via geodesic convexity [5.191641077435773]
  We show nonasymptotic bounds for the error achieved by the maximum likelihood estimator (MLE) in several natural metrics. In the same regimes as our sample complexity bounds, we show that an iterative procedure to compute the MLE known as the flip-flop algorithm converges linearly with high probability.
  arXiv  Detail & Related papers  (2021-10-14T17:47:00Z)
• A Random Matrix Perspective on Random Tensors [40.89521598604993]
  We study the spectra of random matrices arising from contractions of a given random tensor. Our technique yields a hitherto unknown characterization of the local maximum of the ML problem. Our approach is versatile and can be extended to other models, such as asymmetric, non-Gaussian and higher-order ones.
  arXiv  Detail & Related papers  (2021-08-02T10:42:22Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)
• On Linear Stochastic Approximation: Fine-grained Polyak-Ruppert and Non-Asymptotic Concentration [115.1954841020189]
  We study the inequality and non-asymptotic properties of approximation procedures with Polyak-Ruppert averaging. We prove a central limit theorem (CLT) for the averaged iterates with fixed step size and number of iterations going to infinity.
  arXiv  Detail & Related papers  (2020-04-09T17:54:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.