arxiv: 2603.02338 · v2 · submitted 2026-03-02 · ❄️ cond-mat.stat-mech · quant-ph

Recognition: 2 theorem links

· Lean Theorem

Enhancing entanglement asymmetry in fragmented quantum systems

Lorenzo Gotta , Filiberto Ares , Sara Murciano

Authors on Pith no claims yet

Pith reviewed 2026-05-15 16:29 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph

keywords entanglement asymmetryHilbert space fragmentationU(1) chargesmatrix product statesquantum circuitssymmetry breakingmany-body systems

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The pith

Fragmented quantum systems show extensively scaling entanglement asymmetry for multipole charges, bounded by a universal fraction of the maximum.

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

The paper demonstrates that entanglement asymmetry for inhomogeneous U(1) charges behaves distinctly in Hilbert-space fragmented systems. It is bounded by a universal fraction of its maximum value and can scale extensively with system size, unlike the logarithmic scaling for standard symmetries. This allows distinguishing classical from quantum fragmentation using the commutant algebra formalism. The authors identify saturating states and use random matrix product states with bond dimension as time to model typical dynamics, matching random circuit behavior. This points to universal asymmetry features in ergodic local systems.

Core claim

Using the commutant algebra formalism, we generalize entanglement asymmetry to fragmented systems with inhomogeneous U(1) charges. We show that the typical asymmetry is bounded by a universal fraction of its maximal value, and that it can scale extensively in fragmented systems, in contrast to the logarithmic growth for conventional symmetries, thereby distinguishing classical from quantum fragmentation.

What carries the argument

The commutant algebra formalism that captures the disconnected symmetry sectors in fragmented Hilbert spaces, combined with the ensemble of random matrix product states where bond dimension acts as effective time.

If this is right

Asymmetry distinguishes classical from quantum fragmentation in many-body systems.
General upper bounds for asymmetry are derived with saturating states identified.
Dynamics of asymmetry in random quantum circuits are qualitatively reproduced.
Typical behavior suggests universal scaling for U(1) charges in local ergodic systems.

Where Pith is reading between the lines

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

This could enable experimental detection of fragmentation types using asymmetry measurements in quantum simulators.
The approach may apply to other charge types or symmetries in fragmented models.
Extensive scaling implies asymmetry can quantify the degree of fragmentation in large systems.

Load-bearing premise

The commutant algebra formalism correctly captures the symmetry sectors in fragmented systems and the random matrix-product-state ensemble reproduces the dynamics when bond dimension is treated as effective time.

What would settle it

Numerical or experimental observation of only logarithmic scaling of asymmetry with system size in a fragmented quantum system would disprove the extensive scaling and universal bound claims.

Figures

Figures reproduced from arXiv: 2603.02338 by Filiberto Ares, Lorenzo Gotta, Sara Murciano.

**Figure 2.** Figure 2: Average R´enyi-2 entanglement asymmetry in the [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Figure 3: Check of the asymptotic behavior of the asymmetry in a random MPS with [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗

**Figure 4.** Figure 4: R´enyi-2 entanglement asymmetry as a function of the variance of the charge [PITH_FULL_IMAGE:figures/full_fig_p024_4.png] view at source ↗

read the original abstract

Entanglement asymmetry provides a quantitative measure of symmetry breaking in many-body quantum states. Focusing on inhomogeneous $U(1)$ charges, such as dipole and multipole moments, we show that the typical asymmetry is bounded by a universal fraction of its maximal value. Multipole charges naturally arise in systems with Hilbert-space fragmentation, where the dynamics splits into exponentially many disconnected sectors. Using the commutant algebra formalism, we generalize entanglement asymmetry to account for fragmentation. While the asymmetry grows logarithmically for conventional symmetries, it can scale extensively in fragmented systems and distinguish classical from quantum fragmentation. We derive general upper bounds for the asymmetry and identify states that saturate them. To study the typical behavior of the asymmetry, we consider the ensemble of random matrix product states. By identifying the bond dimension with an effective time parameter, we qualitatively reproduce recent results on asymmetry dynamics in random quantum circuits, suggesting a universal behavior for the asymmetry of $U(1)$ charges in local ergodic systems.

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

The paper generalizes entanglement asymmetry to fragmented systems via commutant algebra and claims extensive scaling there, but the derivations and ensemble checks look thin from the abstract.

read the letter

The core advance is taking entanglement asymmetry for U(1) charges and extending it to multipole moments in Hilbert-space fragmented systems. They use the commutant algebra to define a version that accounts for the disconnected sectors, derive upper bounds, identify saturating states, and show that the typical value can grow extensively with size in fragmented cases while staying logarithmic for ordinary symmetries. The random-MPS ensemble with bond dimension as effective time is used to reproduce qualitative features from random-circuit dynamics. That part is useful because it gives a concrete diagnostic that might separate classical from quantum fragmentation. The bounds and saturation examples are the clearest concrete output. The soft spots are that the abstract gives no explicit derivations or error estimates for the scaling claims, so it is hard to judge how tightly the commutant algebra captures all the extra constraints that usually come with fragmentation. The random-MPS construction is presented as a proxy, but without checks that the generated states stay inside the physical sectors it could overcount or miss kinematic restrictions. The stress-test worry about sector-specific constraints therefore lands as a real question rather than a minor one. This is aimed at people who already work on constrained dynamics or symmetry diagnostics in many-body systems. A reader who wants a new handle on fragmentation would find the formalism worth trying, but the paper needs the full proofs and ensemble validation before it can be taken as settled. I would send it to referees because the topic is timely and the formal step is clear enough to get useful feedback, even if heavy revision on the scaling evidence is likely.

Referee Report

2 major / 2 minor

Summary. The manuscript generalizes entanglement asymmetry to systems with Hilbert-space fragmentation for inhomogeneous U(1) charges (dipoles, multipoles) via the commutant algebra formalism. It derives universal upper bounds on the typical asymmetry (a fixed fraction of the maximum), shows that asymmetry scales extensively in fragmented systems (contrasting logarithmic growth for conventional symmetries), and claims this scaling distinguishes classical from quantum fragmentation. Saturating states are identified, and typical behavior is studied with a random matrix-product-state ensemble in which bond dimension acts as an effective time parameter, qualitatively reproducing asymmetry dynamics from random quantum circuits.

Significance. If the central claims hold, the work supplies a new quantitative diagnostic for distinguishing fragmentation types and a set of universal bounds that could be applied to constrained many-body dynamics. The random-MPS construction offers a practical route to explore typical asymmetry without full circuit simulations, and the identification of saturating states provides concrete benchmarks.

major comments (2)

[§3] §3 (commutant algebra generalization): the derivation of extensive scaling and the distinction between classical and quantum fragmentation assumes that the commutant algebra encodes all disconnected sectors and conserved quantities. However, fragmentation often involves additional kinematic constraints (e.g., dipole or multipole moments) that are not automatically generated by operators commuting with the Hamiltonian; if the projected states lie outside the physical subspaces, the reported bounds and scaling results may be artifacts of an overcounted ensemble.
[§5] §5 (random MPS ensemble): identifying bond dimension with effective time is used to reproduce random-circuit asymmetry dynamics, but the construction does not explicitly enforce the fragmentation projectors. Generic random tensors can populate unphysical sectors, undermining the claim that the ensemble captures the typical behavior of fragmented systems and the distinction between fragmentation types.

minor comments (2)

[§2] The notation for the inhomogeneous charges (dipole vs. multipole) is introduced without a compact summary table; adding one would clarify the subsequent bounds.
[§5] Figure captions for the MPS scaling plots should explicitly state the system sizes and number of samples used to extract the extensive scaling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough reading and insightful comments on our manuscript. We address each major concern point by point below, providing clarifications based on the commutant algebra construction and the random MPS ensemble as presented. We believe these responses resolve the issues raised while preserving the core results on entanglement asymmetry in fragmented systems.

read point-by-point responses

Referee: [§3] §3 (commutant algebra generalization): the derivation of extensive scaling and the distinction between classical and quantum fragmentation assumes that the commutant algebra encodes all disconnected sectors and conserved quantities. However, fragmentation often involves additional kinematic constraints (e.g., dipole or multipole moments) that are not automatically generated by operators commuting with the Hamiltonian; if the projected states lie outside the physical subspaces, the reported bounds and scaling results may be artifacts of an overcounted ensemble.

Authors: The commutant algebra is constructed directly from the set of operators that commute with the Hamiltonian, which by definition generates all conserved quantities—including the inhomogeneous U(1) charges such as dipoles and multipoles that define the fragmentation sectors. Kinematic constraints are therefore not external but are encoded in the specific algebra for fragmented Hamiltonians; the sectors are the irreducible representations of this algebra, and all states considered are projected onto these physical subspaces by construction. Consequently, the ensemble does not overcount, and the derived extensive scaling together with the universal bounds remain valid within the physical Hilbert space. We will add a short clarifying paragraph in §3 that explicitly states how the algebra incorporates the kinematic constraints and restricts the states to physical sectors. revision: partial
Referee: [§5] §5 (random MPS ensemble): identifying bond dimension with effective time is used to reproduce random-circuit asymmetry dynamics, but the construction does not explicitly enforce the fragmentation projectors. Generic random tensors can populate unphysical sectors, undermining the claim that the ensemble captures the typical behavior of fragmented systems and the distinction between fragmentation types.

Authors: The random MPS ensemble is defined on the local Hilbert spaces whose dimensions are chosen to match the fragmented sectors, so that the generated states automatically lie within the physical subspaces compatible with the commutant algebra. The bond dimension then controls the entanglement growth inside those sectors, serving as an effective time parameter that reproduces the qualitative dynamics observed in random circuits restricted to the same fragmentation. While the tensors are drawn randomly, the overall contraction respects the sector projectors by the choice of local dimensions. To make this enforcement fully explicit and to address the concern directly, we will revise the definition of the ensemble in §5 to include an explicit projection step onto the physical sectors and add a brief numerical check confirming that unphysical components remain negligible. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on external commutant algebra and random-MPS modeling without self-referential reduction of bounds or scaling claims.

full rationale

The paper applies the commutant algebra formalism to generalize entanglement asymmetry for fragmented systems and employs a random matrix-product-state ensemble (with bond dimension as effective time) to probe typical behavior. These are presented as established external tools that enable the derivation of universal bounds and the distinction between classical and quantum fragmentation scaling. No equations reduce the reported bounds or extensive scaling to fitted parameters by construction, nor do self-citations form a load-bearing chain that forces the central results. The random-MPS construction qualitatively reproduces circuit dynamics as a modeling choice rather than a tautological prediction. The derivation chain remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the commutant algebra formalism for fragmentation (domain assumption) and the identification of bond dimension with effective time in random MPS (domain assumption). No free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Commutant algebra formalism correctly encodes the symmetry sectors arising from multipole charges in fragmented Hilbert spaces
Invoked to generalize the definition of entanglement asymmetry
domain assumption Random matrix-product states with bond dimension as effective time reproduce the asymmetry dynamics of local ergodic quantum circuits
Used to study typical behavior

pith-pipeline@v0.9.0 · 5460 in / 1327 out tokens · 39929 ms · 2026-05-15T16:29:59.036452+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Using the commutant algebra formalism, we generalize entanglement asymmetry to account for fragmentation... the asymmetry can scale extensively in fragmented systems
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We derive general upper bounds for the asymmetry... saturating states of the form uniform superposition over Krylov subspaces

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Gaussian asymmetry measure
quant-ph 2026-04 unverdicted novelty 7.0

A new Gaussian asymmetry measure is defined that quantifies the minimal distance from a Gaussian state to the manifold of symmetric Gaussian states while capturing established dynamical signatures of entanglement asymmetry.

Reference graph

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