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  • Open Access

The Product Structure of Matrix Product States under Permutations

Marta Florido-Llinàs1,2,*, Álvaro M. Alhambra3, Rahul Trivedi1,2, Norbert Schuch4,5, David Pérez-García6, and J. Ignacio Cirac1,2

  • *Contact author: marta.florido.llinas@mpq.mpg.de
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PRX Quantum 6, 040338 – Published 17 November, 2025

DOI: https://doi.org/10.1103/8sbs-t24w

Abstract

Tensor network methods have proved to be highly effective in addressing a wide variety of physical scenarios, including those lacking an intrinsic one-dimensional geometry. In such contexts, it is possible for the problem to exhibit a weak form of permutational symmetry, in the sense that entanglement behaves similarly across any arbitrary bipartition. In this paper, we show that translationally-invariant (TI) matrix product states (MPSs) with this property are trivial, meaning that they are either product states or superpositions of a few of them. The results also apply to non-TI generic MPSs, as well as further relevant examples of MPSs including the W state and the Dicke states in an approximate sense. Our findings motivate the usage of Ansätze simpler than tensor networks in systems whose structure is invariant under permutations.

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The study of complex systems is at the heart of many diverse fields, ranging from quantum many-body physics to high-dimensional data analysis. They all share a common challenge: they are often too intricate and require more straightforward and manageable representations. Designing optimal representations (or Ansätze) to make these problems tractable is a key question. Our work addresses this issue in scenarios where entanglement across any arbitrary partition is limited. We find that such systems can be well approximated by a remarkably simple Ansatz: a superposition of just a few product states.

To arrive at this result, we define a class of quantum states called “MPS-under-permutations,” which captures the above property and can naturally arise in contexts such as quantum chemistry, machine learning, and solving high-dimensional differential equations. These states are rooted in the widely used matrix product state (MPS) framework, a powerful tool for analyzing quantum many-body systems. By leveraging the well-developed analytical toolbox of MPS, we demonstrate in a mathematically rigorous way that MPS-under-permutations exhibit a very simple product structure.

Therefore, our results advance the fundamental understanding of the entanglement properties of physically relevant many-body states and pave the way for future generalizations to broader classes of states beyond the MPS considered here. Furthermore, they represent a step toward the overarching goal of fully characterizing the structure of systems under specific physical constraints, which is an essential task for tackling complex challenges in quantum physics and beyond.

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PRX Quantum 6, 040338– Published 17 November, 2025
  • Received 13 December 2024
  • Revised 8 June 2025
  • Accepted 9 October 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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