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The product structure of MPS-under-permutations

Marta Florido-Llinàs, Álvaro M. Alhambra, Rahul Trivedi, Norbert Schuch, David Pérez-García, J. Ignacio Cirac

TL;DR

This work probes whether matrix product states (MPS) can remain efficient under arbitrary particle permutations and shows that translationally invariant (TI) MPS with the MPS-under-permutations property are fundamentally restricted: they reduce to a GHZ-like superposition of a small number of product states $ig|\psi\rangle = \sum_{i=1}^b \beta_i \big|\phi_i\big>^{\otimes N}$. The authors develop a framework based on normal tensors, the basis of normal tensors (BNT), block-injectivity length, and transfer-matrix purity arguments to establish exact and approximate results for TI and non-TI MPS-up, both with normal and non-normal tensors. They show that, for large system size, TI MPS-up must exhibit either product structure ($b=1$) or a GHZ-like form (finite $b$), while non-TI injective cases likewise collapse to product-like states up to small clusters; these conclusions extend to approximate MPS-up with explicit epsilon bounds. The findings justify using simpler mean-field or product-state Ansätze in many permutation-invariant problems and provide criteria to assess when tensor-network methods are truly advantageous. The work also connects to de Finetti-type intuition and discusses notable examples like the W and Dicke states, illustrating the boundaries of the results and outlining open questions related to border vs tensor rank for broader MPS-under-permutations.

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 (MPS) 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 MPS, as well as further relevant examples of MPS 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.

The product structure of MPS-under-permutations

TL;DR

This work probes whether matrix product states (MPS) can remain efficient under arbitrary particle permutations and shows that translationally invariant (TI) MPS with the MPS-under-permutations property are fundamentally restricted: they reduce to a GHZ-like superposition of a small number of product states . The authors develop a framework based on normal tensors, the basis of normal tensors (BNT), block-injectivity length, and transfer-matrix purity arguments to establish exact and approximate results for TI and non-TI MPS-up, both with normal and non-normal tensors. They show that, for large system size, TI MPS-up must exhibit either product structure () or a GHZ-like form (finite ), while non-TI injective cases likewise collapse to product-like states up to small clusters; these conclusions extend to approximate MPS-up with explicit epsilon bounds. The findings justify using simpler mean-field or product-state Ansätze in many permutation-invariant problems and provide criteria to assess when tensor-network methods are truly advantageous. The work also connects to de Finetti-type intuition and discusses notable examples like the W and Dicke states, illustrating the boundaries of the results and outlining open questions related to border vs tensor rank for broader MPS-under-permutations.

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 (MPS) 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 MPS, as well as further relevant examples of MPS 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.

Paper Structure

This paper contains 20 sections, 15 theorems, 59 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Discussion
  5. Setting and summary of results
  6. Normal and translationally invariant MPS-up
  7. Exact MPS-up with normal tensor
  8. Approximate MPS-up with normal tensor
  9. Non-normal and translationally invariant MPS-up
  10. Exact MPS-up with non-normal tensor
  11. Approximate MPS-up with non-normal tensor
  12. Non-translationally invariant MPS-up
  13. Implications for MPS approximation algorithms
  14. Almost-product MPS-under-permutations
  15. Outlook
  16. ...and 5 more sections

Key Result

Theorem 1

Let $\ket{\psi_N(A)}$ be a TI MPS with the exact MPS-up$_{0,D}$ property on $N$ sites, with $N > pL_{BI}(\log_2 D + 1)$. Then, $\ket{\psi} = \sum_{i=1}^b \beta_i \ket{\phi_i}^{\otimes N}$, where $b$ denotes the number of elements in the BNT of tensor $A$. This means that $\ket{\psi}$ has a GHZ-like

Theorems & Definitions (27)

  • Definition
  • Definition 1
  • Theorem
  • Theorem
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • ...and 17 more