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arxiv: 2410.19541 · v2 · submitted 2024-10-25 · 🪐 quant-ph · cond-mat.str-el· math-ph· math.MP

The product structure of MPS-under-permutations

Marta Florido-Llin\`as , \'Alvaro M. Alhambra , Rahul Trivedi , Norbert Schuch , David P\'erez-Garc\'ia , J. Ignacio Cirac This is my paper

Pith reviewed 2026-05-23 18:56 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.str-elmath-phmath.MP
keywords matrix product statespermutational symmetrytranslationally invarianttensor networksproduct statesW stateDicke statesquantum entanglement
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0 comments X

The pith

Translationally invariant MPS with weak permutational symmetry are either product states or superpositions of a few of them.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that matrix product states which are translationally invariant and have entanglement that behaves similarly no matter how the chain is bipartitioned must have a simple form. Such states reduce to either unentangled product states or linear combinations of only a small number of them. This matters for a reader interested in quantum many-body physics because it indicates that full tensor-network descriptions may not be needed when the system has this permutation-like invariance. The same conclusion applies to generic non-translationally-invariant MPS and holds approximately for states such as the W state and Dicke states.

Core claim

Translationally-invariant matrix product states (MPS) that possess weak permutational symmetry, defined by entanglement being comparable across any bipartition, are trivial: they are either product states or superpositions of a few of them. The same product structure holds for non-TI generic MPS. The result also applies approximately to the W state and the Dicke states.

What carries the argument

Weak permutational symmetry of an MPS, the condition that entanglement is similar across arbitrary bipartitions, which forces the state to factorize into a product structure.

If this is right

  • Simpler ansatze than tensor networks suffice for systems whose structure is invariant under permutations.
  • The product structure extends directly to non-translationally-invariant generic MPS.
  • The same triviality holds approximately for the W state and Dicke states.
  • Tensor-network methods can be replaced by product-state descriptions in permutation-symmetric physical scenarios.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result suggests that numerical simulations of permutation-symmetric systems can avoid the computational cost of full MPS tensors.
  • It raises the question of whether other discrete symmetries produce analogous reductions in state complexity.
  • Small-system checks could verify the claim by enumerating low-bond-dimension MPS and testing their entanglement uniformity.

Load-bearing premise

The MPS is assumed to exhibit weak permutational symmetry in the sense that entanglement behaves similarly across any arbitrary bipartition.

What would settle it

A concrete counterexample would be an explicit translationally invariant MPS that is neither a product state nor a superposition of only a few states, yet still has entanglement that is comparable for every possible bipartition.

read the original 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 ans\"atze simpler than tensor networks in systems whose structure is invariant under permutations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims that translationally-invariant (TI) matrix product states (MPS) obeying a weak permutational symmetry—entanglement statistics independent of bipartition choice—are trivial, i.e., product states or superpositions of a few of them. The result is stated to extend to generic (non-TI) MPS and, approximately, to the W state and Dicke states. This is used to motivate simpler ansätze than tensor networks for permutation-invariant systems.

Significance. If the central claim is correct, the result is significant for tensor-network applications outside strict 1D geometry: it shows that the stated symmetry forces MPS to collapse to low-complexity product forms, thereby justifying simpler variational families in permutation-symmetric quantum many-body problems.

major comments (1)
  1. The provided manuscript text consists solely of the abstract; no derivation, proof steps, or technical sections are visible. Consequently the central claim that the weak permutational symmetry implies trivial MPS structure cannot be verified or assessed for internal consistency.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their review. We address the sole major comment below. The full manuscript is available on arXiv:2410.19541.

read point-by-point responses

  1. Referee: The provided manuscript text consists solely of the abstract; no derivation, proof steps, or technical sections are visible. Consequently the central claim that the weak permutational symmetry implies trivial MPS structure cannot be verified or assessed for internal consistency.

    Authors: The complete manuscript, including all derivations, proof steps, and technical sections, was submitted and is publicly available on arXiv as 2410.19541. The paper establishes the triviality of TI MPS under weak permutational symmetry (product states or superpositions of a few) via explicit proofs, with extensions to generic MPS and approximate results for W and Dicke states. If only the abstract was visible due to a review-system issue, we can resubmit the full PDF. revision: no

Circularity Check

0 steps flagged

No circularity: direct mathematical proof from independent symmetry definition

full rationale

The paper establishes a theorem that TI MPS (and generic MPS) obeying weak permutational symmetry—defined as entanglement statistics independent of bipartition choice—are necessarily product states or low-rank superpositions. This is presented as a direct consequence of the symmetry condition via standard MPS properties, with no data fitting, no self-referential definitions (the symmetry is stated independently of the triviality conclusion), and no load-bearing self-citations or ansatzes imported from prior author work. The abstract and description supply an explicit, non-circular statement of both premise and result; the derivation chain does not reduce any prediction or uniqueness claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

This is a structural theorem in tensor network theory with no fitted parameters or new postulated entities.

axioms (1)
  • standard math Standard definitions and algebraic properties of matrix product states and bipartite entanglement measures
    The result relies on established MPS formalism and symmetry definitions from quantum information theory.

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Forward citations

Cited by 1 Pith paper

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  1. Exploring the performance of superposition of product states: from 1D to 3D quantum spin systems

    quant-ph 2025-11 unverdicted novelty 4.0

    The superposition of product states ansatz achieves high accuracy for ground state search in 1D and 3D tilted Ising models with short- and long-range interactions as well as random networks.

Reference graph

Works this paper leans on

67 extracted references · 67 canonical work pages · cited by 1 Pith paper · 4 internal anchors

  1. [1]

    S. R. White, Density matrix formulation for quantum renormalization groups, Phys. Rev. Lett. 69, 2863 (1992)

    work page 1992

  2. [2]

    Verstraete and J

    F. Verstraete and J. I. Cirac, Matrix product states rep- resent ground states faithfully, Phys. Rev. B 73, 094423 (2006)

    work page 2006

  3. [3]

    Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue

    U. Schollw¨ ock, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011), january 2011 Special Issue

    work page 2011

  4. [4]

    Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys

    G. Vidal, Efficient simulation of one-dimensional quantum many-body systems, Phys. Rev. Lett. 93, 040502 (2004)

    work page 2004

  5. [5]

    T. J. Osborne, Efficient approximation of the dynamics of one-dimensional quantum spin systems, Physical Review Letters 97 (2006)

    work page 2006

  6. [6]

    Verstraete, J

    F. Verstraete, J. J. Garc´ ıa-Ripoll, and J. I. Cirac, Ma- trix product density operators: Simulation of finite- temperature and dissipative systems, Physical Review Letters 93 (2004)

    work page 2004

  7. [7]

    S. R. White, Minimally entangled typical quantum states at finite temperature, Phys. Rev. Lett. 102, 190601 (2009)

    work page 2009

  8. [8]

    Kuwahara, A

    T. Kuwahara, A. M. Alhambra, and A. Anshu, Improved thermal area law and quasilinear time algorithm for quan- tum gibbs states, Phys. Rev. X 11, 011047 (2021)

    work page 2021

  9. [9]

    Fannes, B

    M. Fannes, B. Nachtergaele, and R. F. Werner, Finitely correlated states on quantum spin chains, Communica- tions in Mathematical Physics 144, 443

    work page

  10. [10]

    Perez-Garcia, F

    D. Perez-Garcia, F. Verstraete, M. Wolf, and J. Cirac, Ma- trix product state representations, Quantum Information and Computation 7, 401

    work page

  11. [11]

    Renormalization algorithms for Quantum-Many Body Systems in two and higher dimensions

    F. Verstraete and J. I. Cirac, Renormalization algorithms for quantum-many body systems in two and higher dimen- sions (2004), arXiv:cond-mat/0407066 [cond-mat.str-el]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  12. [12]

    V. Murg, F. Verstraete, R. Schneider, P. R. Nagy, and ¨O. Legeza, Tree tensor network state with variable tensor order: An efficient multireference method for strongly correlated systems, Journal of Chemical Theory and Com- putation 11, 1027 (2015)

    work page 2015

  13. [13]

    E. M. Stoudenmire and D. J. Schwab, Supervised learning with quantum-inspired tensor networks (2017), arXiv:1605.05775 [stat.ML]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    Exponential Machines

    A. Novikov, M. Trofimov, and I. Oseledets, Exponential machines (2017), arXiv:1605.03795 [stat.ML]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  15. [15]

    Glasser, R

    I. Glasser, R. Sweke, N. Pancotti, J. Eisert, and J. I. Cirac, Expressive power of tensor-network factorizations for probabilistic modeling, with applications from hid- den markov models to quantum machine learning (2019), arXiv:1907.03741 [cs.LG]

    work page arXiv 2019

  16. [16]

    J. D. Biamonte, J. Morton, and J. Turner, Tensor network contractions for #sat, Journal of Statistical Physics 160, 1389–1404 (2015)

    work page 2015

  17. [17]

    Kourtis, C

    S. Kourtis, C. Chamon, E. Mucciolo, and A. Ruckenstein, Fast counting with tensor networks, SciPost Physics 7 (2019)

    work page 2019

  18. [18]

    J.-G. Liu, X. Gao, M. Cain, M. D. Lukin, and S.-T. Wang, Computing solution space properties of combina- torial optimization problems via generic tensor networks, SIAM Journal on Scientific Computing 45, A1239 (2023), https://doi.org/10.1137/22M1501787

    work page doi:10.1137/22m1501787 2023

  19. [19]

    B. N. Khoromskij, Tensors-structured numerical meth- ods in scientific computing: Survey on recent advances, Chemometrics and Intelligent Laboratory Systems 110, 1 (2012)

    work page 2012

  20. [20]

    J. J. Garc´ ıa-Ripoll, Quantum-inspired algorithms for mul- tivariate analysis: from interpolation to partial differential equations, Quantum 5, 431 (2021)

    work page 2021

  21. [21]

    Garc´ ıa-Molina, L

    P. Garc´ ıa-Molina, L. Tagliacozzo, and J. J. Garc´ ıa-Ripoll, Global optimization of mps in quantum-inspired numerical analysis (2023), arXiv:2303.09430 [quant-ph]

    work page arXiv 2023

  22. [22]

    Richter, L

    L. Richter, L. Sallandt, and N. N¨ usken, Solving high- dimensional parabolic PDEs using the tensor train format, in Proceedings of the 38th International Conference on Machine Learning (PMLR) pp. 8998–9009, ISSN: 2640- 3498

    work page

  23. [23]

    Schuch, M

    N. Schuch, M. M. Wolf, F. Verstraete, and J. I. Cirac, Entropy scaling and simulability by matrix product states, Physical Review Letters 100 (2008)

    work page 2008

  24. [24]

    C. M. Caves, C. A. Fuchs, and R. Schack, Unknown quantum states: The quantum de finetti representation, Journal of Mathematical Physics 43, 4537 (2002)

    work page 2002

  25. [25]

    Renner, Symmetry of large physical systems implies independence of subsystems, Nature Physics 3, 645 (2007)

    R. Renner, Symmetry of large physical systems implies independence of subsystems, Nature Physics 3, 645 (2007)

    work page 2007

  26. [26]

    Christandl, R

    M. Christandl, R. K¨ onig, G. Mitchison, and R. Renner, One-and-a-half quantum de finetti theorems, Communi- cations in Mathematical Physics 273, 473–498 (2007)

    work page 2007

  27. [27]

    F. G. S. L. Brand˜ ao and A. W. Harrow, Quantum de finetti theorems under local measurements with applica- tions, Communications in Mathematical Physics 353, 469 (2017)

    work page 2017

  28. [28]

    Bansal, S

    N. Bansal, S. Bravyi, and B. M. Terhal, Classical approx- imation schemes for the ground-state energy of quantum and classical ising spin hamiltonians on planar graphs, Quantum Info. Comput. 9, 701–720 (2009)

    work page 2009

  29. [29]

    Gharibian and J

    S. Gharibian and J. Kempe, Approximation algorithms for qma-complete problems, SIAM Journal on Computing 11 41, 1028–1050 (2012)

    work page 2012

  30. [30]

    F. G. S. L. Brand˜ ao and A. W. Harrow, Product-state approximations to quantum states, Communications in Mathematical Physics 342, 47

    work page

  31. [31]

    I. V. Oseledets, Tensor-train decomposition, SIAM Jour- nal on Scientific Computing 33, 2295 (2011)

    work page 2011

  32. [32]

    Order Matters: Sequence to sequence for sets

    O. Vinyals, S. Bengio, and M. Kudlur, Order matters: Sequence to sequence for sets (2016), arXiv:1511.06391 [stat.ML]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  33. [33]

    Acharya, M

    A. Acharya, M. Rudolph, J. Chen, J. Miller, and A. Perdomo-Ortiz, Qubit seriation: Improving data-model alignment using spectral ordering (2022), arXiv:2211.15978 [quant-ph]

    work page arXiv 2022

  34. [34]

    C. Li, J. Zeng, Z. Tao, and Q. Zhao, Permutation search of tensor network structures via local sampling, in Pro- ceedings of the 39th International Conference on Machine Learning, Vol. 162 (PMLR, 2022) pp. 13106–13124

    work page 2022

  35. [35]

    Baiardi and M

    A. Baiardi and M. Reiher, The density matrix renormal- ization group in chemistry and molecular physics: Recent developments and new challenges, The Journal of Chemi- cal Physics 152 (2020)

    work page 2020

  36. [36]

    G. K.-L. Chan and M. Head-Gordon, Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group, The Journal of Chemical Physics 116, 4462 (2002)

    work page 2002

  37. [37]

    G. K.-L. Chan and S. Sharma, The Density Matrix Renor- malization Group in Quantum Chemistry, Annual Review of Physical Chemistry 62, 465 (2011), publisher: Annual Reviews Type: Journal Article

    work page 2011

  38. [38]

    Legeza and J

    O. Legeza and J. S´ olyom, Optimizing the density-matrix renormalization group method using quantum information entropy, Phys. Rev. B 68, 195116 (2003)

    work page 2003

  39. [39]

    Rissler, R

    J. Rissler, R. M. Noack, and S. R. White, Measuring orbital interaction using quantum information theory, Chemical Physics 323, 519 (2006)

    work page 2006

  40. [40]

    Barcza, O

    G. Barcza, O. Legeza, K. H. Marti, and M. Reiher, Quantum-information analysis of electronic states of dif- ferent molecular structures, Phys. Rev. A 83, 012508 (2011)

    work page 2011

  41. [41]

    Krumnow, L

    C. Krumnow, L. Veis, O. Legeza, and J. Eisert, Fermionic orbital optimization in tensor network states, Phys. Rev. Lett. 117, 210402 (2016)

    work page 2016

  42. [42]

    Moritz, B

    G. Moritz, B. A. Hess, and M. Reiher, Convergence behav- ior of the density-matrix renormalization group algorithm for optimized orbital orderings, The Journal of Chemical Physics 122, 024107 (2004)

    work page 2004

  43. [43]

    J. I. Cirac, D. P´ erez-Garc´ ıa, N. Schuch, and F. Verstraete, Matrix product states and projected entangled pair states: Concepts, symmetries, theorems, Rev. Mod. Phys. 93, 045003 (2021)

    work page 2021

  44. [44]

    Watrous, The Theory of Quantum Information (Cam- bridge University Press, 2018)

    J. Watrous, The Theory of Quantum Information (Cam- bridge University Press, 2018)

    work page 2018

  45. [45]

    D¨ ur, G

    W. D¨ ur, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Physical Review A 62 (2000)

    work page 2000

  46. [46]

    Landsberg, Tensors: Geometry and Applications (American Mathematical Society)

    J. Landsberg, Tensors: Geometry and Applications (American Mathematical Society)

    work page

  47. [47]

    C. J. Hillar and L.-H. Lim, Most tensor problems are np-hard, J. ACM 60 (2013)

    work page 2013

  48. [48]

    Cirac, D

    J. Cirac, D. P´ erez-Garc´ ıa, N. Schuch, and F. Verstraete, Matrix product density operators: Renormalization fixed points and boundary theories, Annals of Physics 378, 100 (2017)

    work page 2017

  49. [49]

    Movassagh and J

    R. Movassagh and J. Schenker, Theory of ergodic quantum processes, Phys. Rev. X 11, 041001 (2021)

    work page 2021

  50. [50]

    Movassagh and J

    R. Movassagh and J. Schenker, An ergodic theorem for quantum processes with applications to matrix product states, Communications in Mathematical Physics 395, 1175

    work page

  51. [51]

    D. M. Greenberger, M. A. Horne, and A. Zeilinger, Going beyond bell’s theorem, in Bell’s Theorem, Quantum The- ory and Conceptions of the Universe, edited by M. Kafatos (Springer Netherlands, Dordrecht, 1989) pp. 69–72

    work page 1989

  52. [52]

    M. Sanz, D. P´ erez-Garc´ ıa, M. M. Wolf, and J. I. Cirac, A quantum version of wielandt’s inequality, IEEE Transac- tions on Information Theory 56, 4668 (2010)

    work page 2010

  53. [53]

    Quantum version of wielandt’s inequality revisited, IEEE Transactions on Information Theory 65, 5239 (2019)

    work page 2019

  54. [54]

    Rolandi and H

    A. Rolandi and H. Wilming, Extensive r´ enyi entropies in matrix product states (2020), arXiv:2008.11764 [quant- ph]

    work page arXiv 2020

  55. [55]

    De las Cuevas, J

    G. De las Cuevas, J. I. Cirac, N. Schuch, and D. Perez- Garcia, Irreducible forms of matrix product states: Theory and applications, Journal of Mathematical Physics 58, 121901

    work page

  56. [56]

    R. H. Dicke, Coherence in spontaneous radiation processes, Phys. Rev. 93, 99 (1954)

    work page 1954

  57. [57]

    Christandl, F

    M. Christandl, F. Gesmundo, D. S. Fran¸ ca, and A. H. Werner, Optimization at the boundary of the tensor net- work variety, Physical Review B 103 (2021)

    work page 2021

  58. [58]

    L. Chen, E. Chitambar, R. Duan, Z. Ji, and A. Winter, Tensor rank and stochastic entanglement catalysis for multipartite pure states, Physical Review Letters 105 (2010)

    work page 2010

  59. [59]

    Vrana and M

    P. Vrana and M. Christandl, Asymptotic entanglement transformation between w and ghz states, Journal of Mathematical Physics 56 (2015)

    work page 2015

  60. [60]

    Chitambar, R

    E. Chitambar, R. Duan, and Y. Shi, Tripartite entangle- ment transformations and tensor rank, Physical Review Letters 101 (2008)

    work page 2008

  61. [61]

    Christandl, V

    M. Christandl, V. Lysikov, V. Steffan, A. H. Werner, and F. Witteveen, The resource theory of tensor networks (2023), arXiv:2307.07394 [quant-ph]

    work page arXiv 2023

  62. [62]

    M. M. Wilde, Quantum information theory (Cambridge university press, 2013)

    work page 2013

  63. [63]

    Deutsch, The angle between subspaces of a hilbert space, in Approximation Theory, Wavelets and Applica- tions (Springer Netherlands, Dordrecht, 1995) pp

    F. Deutsch, The angle between subspaces of a hilbert space, in Approximation Theory, Wavelets and Applica- tions (Springer Netherlands, Dordrecht, 1995) pp. 107– 130

    work page 1995

  64. [64]

    Buckholtz, Hilbert space idempotents and involutions, Proceedings of the American Mathematical Society 128, 1415

    D. Buckholtz, Hilbert space idempotents and involutions, Proceedings of the American Mathematical Society 128, 1415

    work page

  65. [65]

    Barthel, J

    T. Barthel, J. Lu, and G. Friesecke, On the closedness and geometry of tensor network state sets, Letters in Mathematical Physics 112 (2022)

    work page 2022

  66. [66]

    Klimov, R

    P. Klimov, R. Sengupta, and J. Biamonte, On translation- invariant matrix product states and advances in mps representations of the w-state (2023), arXiv:2306.16456 [quant-ph]. 12 Appendix A: Technical lemma for the purity proof with normal tensors Lemma 1. Given two states |ψ1⟩ , |ψ2⟩ with density ma- trices ρ1, ρ2, and reduced density matrices over subs...

    work page arXiv 2023

  67. [67]

    = O(D3 (N) log D(N)), which implies that for each δ > 0, D(N) = Ω(N1/(3+δ))

    work page