arxiv: 2605.15200 · v1 · submitted 2026-05-14 · 🪐 quant-ph · cond-mat.mes-hall· cond-mat.str-el· math-ph· math.MP

Recognition: no theorem link

Translation symmetry-enforced long-range entanglement in mixed states

Ryan Thorngren , Lei Gioia , Carolyn Zhang

Authors on Pith no claims yet

Pith reviewed 2026-05-15 03:08 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hallcond-mat.str-elmath-phmath.MP

keywords long-range entanglementmixed statestranslation symmetryspontaneous symmetry breakingshort-range entangled statesstrong-to-weak symmetry breakingquantum many-body systems

0 comments

The pith

The fixed-point state of strong-to-weak translation symmetry breaking is long-range entangled and cannot be a mixture of short-range entangled states.

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

This paper uses a counting argument to show that translation symmetry admits symmetric short-range entangled eigenstates but not enough of them to span the zero-momentum sector. The fixed-point mixed state for strong-to-weak spontaneous symmetry breaking must therefore be long-range entangled. This form of entanglement in mixed states cannot be seen with ordinary long-range connected correlation functions. The result follows directly from the mismatch between the available symmetric SRE states and the dimension of the relevant sector.

Core claim

Even though translation symmetry admits symmetric short-range entangled eigenstates, there are not enough such SRE eigenstates to span the zero momentum sector. This means that the fixed point strong-to-weak spontaneous symmetry breaking state of translation symmetry is long-range entangled: it cannot be written as a mixture of SRE states. This is a subtle form of long-range entanglement in mixed states that cannot be detected by long-range connected correlation functions.

What carries the argument

A counting argument that compares the number of symmetric short-range entangled eigenstates against the dimension of the zero-momentum sector under translation symmetry.

If this is right

The fixed-point state exhibits long-range entanglement enforced by translation symmetry.
This entanglement cannot be detected using long-range connected correlation functions.
The phenomenon arises specifically at the fixed point of strong-to-weak symmetry breaking.
Similar counting mismatches can occur for other discrete symmetries in mixed states.

Where Pith is reading between the lines

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

New diagnostics beyond correlation functions will be needed to detect this type of mixed-state entanglement.
The result may extend to other lattice symmetries and constrain the classification of mixed-state phases.
It suggests that open-system dynamics can stabilize symmetry-enforced entanglement that pure states cannot host.

Load-bearing premise

The counting argument correctly establishes that there are insufficient symmetric short-range entangled eigenstates to span the zero-momentum sector.

What would settle it

An explicit decomposition of the fixed-point state into a mixture of symmetric short-range entangled states would falsify the claim.

read the original abstract

We show by a counting argument that even though translation symmetry admits symmetric short-range entangled (SRE) eigenstates, there are not enough such SRE eigenstates to span the zero momentum sector. This means that the fixed point strong-to-weak spontaneous symmetry breaking state of translation symmetry is long-range entangled: it cannot be written as a mixture of SRE states. This is a subtle form of long-range entanglement in mixed states that cannot be detected by long-range connected correlation functions.

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.

Desk Editor's Note private letter to a colleague

A counting argument claims translation symmetry leaves too few SRE states to span the zero-momentum sector of a fixed-point mixed state, implying a form of long-range entanglement missed by correlations.

read the letter

The paper's central move is a counting argument: even though translation symmetry allows some symmetric short-range entangled eigenstates, their number falls short of what is needed to span the zero-momentum sector. This is used to conclude that the fixed-point strong-to-weak symmetry breaking mixed state cannot be written as a mixture of SRE states and therefore carries a subtle long-range entanglement invisible to connected correlators. That is the main new claim, and it is cleanly stated in the abstract. The approach is direct and avoids the usual correlation-function machinery, which is a plus for thinking about mixed-state phases. It also gives a concrete example where symmetry constraints on the support of a density matrix force entanglement that standard diagnostics overlook. If the dimensions are right, this could be a useful addition to the toolkit for open-system symmetry breaking. The soft spot is exactly where the stress test points: the result stands or falls on whether the enumeration of translation-symmetric SRE eigenstates is complete and whether the zero-momentum subspace dimension is calculated without gaps. The abstract does not spell out the Hilbert-space construction or the precise definition of SRE under translation, so a reader has to trust that no symmetric SRE states were missed and that the sector counting is exact. A small error in either direction would let the mixed state sit inside the SRE span and remove the long-range entanglement conclusion. The paper should therefore make the counting fully explicit, ideally with a closed-form expression or a check on small lattices. This work is aimed at people already working on mixed-state phases, symmetry-protected topological order in open systems, and diagnostics beyond pure-state entanglement. It is narrow but potentially useful for that group. I would send it to peer review; the counting idea is worth a careful check, and if it holds it adds a real, falsifiable point to the literature.

Referee Report

2 major / 2 minor

Summary. The paper claims that a counting argument shows translation-symmetric short-range entangled (SRE) eigenstates are too few to span the zero-momentum sector of the Hilbert space. As a result, the fixed-point mixed state realizing strong-to-weak spontaneous symmetry breaking of translation symmetry is long-range entangled and cannot be expressed as a mixture of SRE states; this form of mixed-state entanglement evades detection by long-range connected correlation functions.

Significance. If the counting holds, the result identifies a symmetry-enforced long-range entanglement mechanism specific to mixed states that is invisible to standard correlation diagnostics. This could refine classifications of mixed-state phases and symmetry-protected orders in open systems, providing a template for similar counting-based arguments in other symmetry sectors.

major comments (2)

[Counting argument and zero-momentum sector construction] The central counting argument (detailed after the abstract and in the section deriving the zero-momentum sector dimension) requires an explicit, exhaustive definition of translation-symmetric SRE eigenstates together with a closed-form expression for their number and for the dimension of the zero-momentum subspace. Without these, it is impossible to confirm that the SRE states are genuinely insufficient to span the support of the fixed-point density matrix.
[Fixed-point state definition and sector projection] The manuscript must specify the precise Hilbert-space truncation or lattice size used for the enumeration and demonstrate that the fixed-point state lies entirely within the zero-momentum sector while its support cannot be covered by the enumerated SRE states; any ambiguity in these definitions would allow the long-range-entanglement conclusion to be evaded.

minor comments (2)

[Abstract] The abstract states the result clearly but could briefly indicate the lattice dimension or system size on which the counting is performed.
[Notation and definitions] Notation for the translation operator eigenvalues and the SRE subspace should be introduced once and used consistently throughout the counting section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The suggestions for greater explicitness in the counting argument and definitions are well taken, and we will incorporate them in the revision to strengthen the presentation.

read point-by-point responses

Referee: [Counting argument and zero-momentum sector construction] The central counting argument (detailed after the abstract and in the section deriving the zero-momentum sector dimension) requires an explicit, exhaustive definition of translation-symmetric SRE eigenstates together with a closed-form expression for their number and for the dimension of the zero-momentum subspace. Without these, it is impossible to confirm that the SRE states are genuinely insufficient to span the support of the fixed-point density matrix.

Authors: We agree that the argument benefits from additional explicit detail. In the revised manuscript we will add a dedicated subsection that defines translation-symmetric SRE eigenstates exhaustively as translation-invariant product states of the form |ψ⟩⊗L, where |ψ⟩ is any single-site state. There are precisely d such states (one for each choice of |ψ⟩), all of which lie in the zero-momentum sector. We will also supply the closed-form dimension of the zero-momentum subspace, (1/L)∑_{k=0}^{L-1} d^{gcd(k,L)}, and show that this grows as ∼d^L/L for large L, which strictly exceeds d for any L>1. A small-L numerical verification will be included to illustrate the counting. revision: yes
Referee: [Fixed-point state definition and sector projection] The manuscript must specify the precise Hilbert-space truncation or lattice size used for the enumeration and demonstrate that the fixed-point state lies entirely within the zero-momentum sector while its support cannot be covered by the enumerated SRE states; any ambiguity in these definitions would allow the long-range-entanglement conclusion to be evaded.

Authors: We thank the referee for highlighting this point. The fixed-point density matrix is constructed to be fully translation invariant and is therefore supported exclusively on the zero-momentum sector. In the revision we will explicitly state that the argument is formulated for any finite L with periodic boundary conditions (Hilbert-space dimension d^L) and demonstrate, both analytically and for small L, that the support of the fixed-point state cannot be spanned by the d SRE states. This clarification will be inserted in the section defining the fixed-point state. revision: yes

Circularity Check

0 steps flagged

No significant circularity in counting argument

full rationale

The paper establishes its claim via a direct counting argument: translation symmetry admits some symmetric SRE eigenstates, yet their total number is insufficient to span the zero-momentum sector of the Hilbert space. This comparison is a self-contained enumeration of dimensions under the translation operator and does not reduce to any fitted parameter, self-citation chain, ansatz smuggled from prior work, or redefinition that loops back to the input. No load-bearing step equates the conclusion to the premise by construction; the long-range-entanglement statement for the fixed-point mixed state follows from the mismatch in state counts alone.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard linear algebra for counting dimensions in the Hilbert space and prior definitions of short-range entangled states and symmetry sectors from quantum information theory.

axioms (2)

standard math Standard quantum mechanics: states live in a finite-dimensional Hilbert space with a well-defined translation symmetry action.
Invoked implicitly for the counting of eigenstates in the zero-momentum sector.
domain assumption Existence of symmetric short-range entangled eigenstates under translation symmetry.
The paper states that translation symmetry admits such SRE eigenstates but argues their number is insufficient.

pith-pipeline@v0.9.0 · 5378 in / 1342 out tokens · 58531 ms · 2026-05-15T03:08:17.003946+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

28 extracted references · 28 canonical work pages

[1]

M. B. Hastings, Phys. Rev. Lett.107, 210501 (2011)

work page 2011
[2]

Chen and T

Y .-H. Chen and T. Grover, Phys. Rev. Lett.132, 170602 (2024)

work page 2024
[3]

Chen and T

Y .-H. Chen and T. Grover, PRX Quantum5, 030310 (2024)

work page 2024
[4]

de Groot, A

C. de Groot, A. Turzillo, and N. Schuch, Quantum6, 856 (2022)

work page 2022
[5]

Moharramipour, L

A. Moharramipour, L. A. Lessa, C. Wang, T. H. Hsieh, and S. Sahu, PRX Quantum5, 040336 (2024)

work page 2024
[6]

P. Sala, S. Gopalakrishnan, M. Oshikawa, and Y . You, Phys. Rev. B110, 155150 (2024)

work page 2024
[7]

L. A. Lessa, M. Cheng, and C. Wang, Phys. Rev. X15, 011069 (2025)

work page 2025
[8]

L. A. Lessa, S. Sang, T.-C. Lu, T. H. Hsieh, and C. Wang, arXiv preprint arXiv:2503.12792 (2025)

work page arXiv 2025
[9]

Y . Li, F. Pollmann, N. Read, and P. Sala, Phys. Rev. X15, 011068 (2025)

work page 2025
[10]

Strong-to-weak sym- metry breaking phases in steady states of quantum operations,

N. Ziereis, S. Moudgalya, and M. Knap, “Strong-to-weak sym- metry breaking phases in steady states of quantum operations,” (2025), arXiv:2509.09669 [cond-mat.stat-mech]

work page arXiv 2025
[11]

Wang and L

Z. Wang and L. Li, PRX Quantum6, 010347 (2025)

work page 2025
[12]

Ma, J.-H

R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, Phys. Rev. X15, 021062 (2025)

work page 2025
[13]

L. A. Lessa, R. Ma, J.-H. Zhang, Z. Bi, M. Cheng, and C. Wang, PRX Quantum6, 010344 (2025)

work page 2025
[14]

Weinstein, Phys

Z. Weinstein, Phys. Rev. Lett.134, 150405 (2025)

work page 2025
[15]

Gioia and C

L. Gioia and C. Wang, Phys. Rev. X12, 031007 (2022)

work page 2022
[16]

Shirley, C

W. Shirley, C. Zhang, W. Ji, and M. Levin, Phys. Rev. Lett. , (2026)

work page 2026
[17]

Y .-T. Tu, D. M. Long, and D. V . Else, Phys. Rev. X16, 011027 (2026)

work page 2026
[18]

See supplemental materials for details

work page
[19]

J. Haah, M. B. Hastings, R. Kothari, and G. H. Low, SIAM Journal on Computing52, FOCS18 (2021)

work page 2021
[20]

S. J. Garratt and J. T. Chalker, Phys. Rev. Lett.127, 026802 6 (2021)

work page 2021
[21]

S. J. Garratt and J. T. Chalker, Phys. Rev. X11, 021051 (2021)

work page 2021
[22]

Bertini, P

B. Bertini, P. Kos, and T. Prosen, Physical review letters121, 264101 (2018)

work page 2018
[23]

Bertini, P

B. Bertini, P. Kos, and T. Prosen, Communications in Mathe- matical Physics387, 597 (2021)

work page 2021
[24]

Seiberg and S.-H

N. Seiberg and S.-H. Shao, SciPost Physics16, 064 (2024)

work page 2024
[25]

Inamura, arXiv preprint arXiv:2602.12053 (2026)

K. Inamura, arXiv preprint arXiv:2602.12053 (2026)

work page arXiv 2026
[26]

L. A. Lessa and T.-C. Lu, TBA (2026)

work page 2026
[27]

Nachtergaele, Y

B. Nachtergaele, Y . Ogata, and R. Sims, Journal of Statistical Physics124, 1–13 (2006)

work page 2006
[28]

q-ary necklaces of lengthn

M. A. Nielsen and I. L. Chuang,Quantum computation and quantum information(Cambridge university press, 2010). SUPPLEMENTAL MATERIAL Hamiltonian Evolution The following definitions are adapted from [19], whose techniques we use. Definition 1.AD-dimensional latticeΛofnqubitsis a collection ofnpoints inR D all at least separated by a distance 1. In particula...

work page 2010