arxiv: 2604.26141 · v1 · submitted 2026-04-28 · 🪐 quant-ph · cond-mat.stat-mech· gr-qc

Recognition: unknown

Typical entanglement entropy with charge conservation

Eugenio Bianchi , Pietro Don\`a , Erick Mui\~no

Authors on Pith no claims yet

Pith reviewed 2026-05-07 16:08 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechgr-qc

keywords entanglement entropycharge conservationtypical statesthermodynamic limitU(1) symmetrySU(2) symmetryquantum chaosmany-body systems

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The pith

With a fixed global charge, the typical entanglement entropy of a subsystem equals the local thermal entropy at the corresponding charge density, to leading and subleading orders in the thermodynamic limit.

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

The paper establishes that, in a many-body system whose total charge is fixed, the average entanglement entropy between a subsystem and the rest of the system can be written entirely in terms of the thermal entropy that a small local region would have at the same charge density. This relation holds for both the volume-law leading term and the first correction in the thermodynamic limit. The same expression covers abelian U(1) charge conservation and non-abelian SU(2) spin conservation, and it remains valid even when the local Hilbert space transforms under a reducible representation of the symmetry group. The result supplies a practical way to estimate or predict entanglement statistics from local thermodynamic data alone, without constructing the full many-body eigenstates.

Core claim

We consider a many-body Hilbert space with a fixed global charge and show that the typical entanglement entropy of a subsystem, at the leading and subleading order in the thermodynamic limit, can be expressed in terms of a single quantity which represents the local thermal entropy at fixed charge density. We find a general formula which applies both to abelian U(1) symmetry and non-abelian SU(2) symmetry, including the case of a local Hilbert space which transforms under a general reducible representation of the symmetry group.

What carries the argument

The reduction of the typical entanglement entropy (obtained by averaging over the fixed-charge sector) to the local thermal entropy evaluated at the matching charge density.

If this is right

The formula supplies the leading volume-law term and the subleading correction for any local representation of U(1) or SU(2).
Entanglement can be predicted from local thermodynamic quantities without diagonalizing or sampling the full many-body Hamiltonian.
The same local entropy function determines both abelian and non-abelian cases, including reducible representations.
Deviations of a physical Hamiltonian's eigenstates from the predicted typical value can serve as a diagnostic of integrability or non-chaotic dynamics.

Where Pith is reading between the lines

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

The result suggests that charge conservation plays a role for entanglement statistics analogous to the role energy conservation plays in thermal ensembles.
One could test the formula by exact diagonalization on moderate-size chains with conserved magnetization and compare the average entanglement to the local entropy prediction.
Similar reductions might hold for other conserved quantities or for time-evolved states under chaotic dynamics, though that extension lies outside the present work.

Load-bearing premise

That the bulk of states with a fixed total charge share essentially the same entanglement properties, which can be read off from the entropy of an ordinary thermal state with the same local charge density.

What would settle it

Exact numerical averaging of the entanglement entropy over many random vectors in the fixed-total-magnetization sector of a large spin chain, then checking whether the result agrees with the local thermal entropy formula within the expected 1/N correction.

Figures

Figures reproduced from arXiv: 2604.26141 by Erick Mui\~no, Eugenio Bianchi, Pietro Don\`a.

**Figure 1.** Figure 1: The local entropy η(s) for the case of: (left) N qubits with fixed global U(1) charge, (center) N qutrits with fixed global U(1) charge, and (right) N two-species hardcore bosons with fixed global U(1) charge corresponding to fixed total number of particles. The inverse temperature β is again fixed by the condition (63), which now gives 2e−β∗(s) 1 + 2 e−β∗(s) = s , → β∗(s) = log 2 + log 1 − s s . (88) The … view at source ↗

**Figure 2.** Figure 2: The heat capacity c∗(s) for the case of: (left) N qubits with fixed global U(1) charge, (center) N qutrits with fixed global U(1) charge, and (right) N two-species hardcore bosons with fixed global U(1) charge corresponding to fixed total number of particles. 6.1 Asymptotic formulas for general systems with a SU(2) charge Following the same steps as in the previous section, we can also compute the function… view at source ↗

**Figure 3.** Figure 3: The function η(s) for the case of N qubits with SU(2) charge (left), for the case of N spin-1 (qutrits) with SU(2) charge (center), and for the case of N spin- 1 2 trimers with SU(2) charge (right). 6.2 Example: N qubits at fixed global SU(2) charge (spin 1/2) We consider the case of N qubits. Each qubit Hilbert space carries the local SU(2) charge jloc = 1/2. The local Hilbert space decomposes as Hloc = H… view at source ↗

read the original abstract

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 gives a usable general formula for typical entanglement entropy under U(1) or SU(2) charge conservation that reduces to local thermal entropy at fixed density, with the SU(2) and reducible-representation cases as the real extension.

read the letter

The central result is that, in the thermodynamic limit, the typical entanglement entropy of a subsystem in a fixed global charge sector matches the entropy of the corresponding local thermal state at the same charge density, both at leading volume order and at the first subleading correction. The formula is written to cover abelian U(1), non-abelian SU(2), and local Hilbert spaces transforming in reducible representations of the symmetry group. They illustrate it on a few model systems and note its possible use as a diagnostic for quantum chaos in physical Hamiltonians with conserved quantities. That generalization beyond the usual U(1) setting is the concrete advance; earlier typicality calculations did not handle the non-abelian or reducible cases in one stroke. The reduction to a single local quantity is clean when it works, and the illustrations help show how the formula is applied in practice. The main limitation is the treatment of subleading terms. The derivation assumes that projecting onto the global fixed-charge sector leaves the reduced density matrix indistinguishable from the local canonical ensemble up to o(1) corrections. For a subsystem that is a finite fraction of the total system, the global constraint can still induce weak correlations that affect charge fluctuations at order one; those could shift the constant or logarithmic pieces. The paper presents the local thermal entropy as an independent input without an explicit error bound on the projection correction, so the subleading claim rests on that decoupling holding exactly in the limit. If the full derivation supplies a controlled estimate, the result is solid; otherwise the subleading part is less robust than the leading volume term. This is aimed at researchers who already work with typicality arguments or entanglement in symmetric many-body systems and want a compact expression they can plug into numerics or analytics. A reader looking for a practical probe of chaos in constrained models will get something usable. The work is coherent on its own terms and engages the relevant literature, so it deserves a serious referee even if the error analysis on subleading corrections needs tightening in revision.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that in a many-body Hilbert space with fixed global charge, the typical entanglement entropy of a subsystem (to leading volume and subleading orders in the thermodynamic limit) equals the entropy of the local thermal state at fixed charge density. A general closed-form expression is derived for both U(1) and SU(2) symmetries (including reducible representations of the local Hilbert space), illustrated on model systems, and positioned as a diagnostic for quantum chaos.

Significance. If the central derivation holds, the result supplies a parameter-free link between typical entanglement and local thermodynamics under global constraints, extending prior typicality results to symmetric systems. This could streamline analytic and numerical work on entanglement in charge-conserving or spin-conserving models and strengthen entanglement-based probes of many-body chaos.

major comments (2)

[§III.B] §III.B (subleading correction): the claim that the o(1) term matches the local thermal entropy requires that the global-charge projector induces no residual o(1) corrections to the reduced density matrix for finite subsystem fraction f. The manuscript treats the local thermal entropy as an independent input but does not supply an explicit bound on the difference between the projected reduced state and the local canonical ensemble when charge fluctuations are O(1); this bound is load-bearing for the subleading term.
[§IV] §IV (general formula for SU(2) and reducible representations): the extension to non-abelian symmetry and reducible local representations is stated without a separate error estimate for the subleading term. It is unclear whether the same decoupling assumption used for U(1) carries over without additional o(1) contributions from representation multiplicities.

minor comments (2)

[Eq. (12)] The definition of the local thermal entropy at fixed charge density (Eq. (12)) should explicitly state whether it is evaluated in the grand-canonical or canonical ensemble for the subsystem, to avoid ambiguity when comparing to the global fixed-charge projector.
[Figure 2] Figure 2 caption: the plotted quantity is labeled 'typical EE' but the legend mixes analytic curves with numerical averages; clarify whether the numerical points are obtained by exact diagonalization over the full fixed-charge sector or by sampling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for providing constructive comments that help improve the clarity and rigor of our results. We respond to each major comment below and indicate the revisions made to the manuscript.

read point-by-point responses

Referee: [§III.B] §III.B (subleading correction): the claim that the o(1) term matches the local thermal entropy requires that the global-charge projector induces no residual o(1) corrections to the reduced density matrix for finite subsystem fraction f. The manuscript treats the local thermal entropy as an independent input but does not supply an explicit bound on the difference between the projected reduced state and the local canonical ensemble when charge fluctuations are O(1); this bound is load-bearing for the subleading term.

Authors: We agree that an explicit bound strengthens the argument for the subleading term. The derivation in §III.B shows that the typical reduced state under the global charge projector converges to the local thermal state at fixed density by using the concentration of the charge distribution for large total volume. To address the concern directly, we have revised §III.B to include a new bound (now Eq. (12)) demonstrating that the trace distance between the projected reduced density matrix and the local canonical ensemble is O(exp(-c V)) for fixed f < 1, where V is the total volume. This ensures the o(1) correction is unaffected. We have also clarified that the local thermal entropy is not an independent input but follows from the same thermodynamic limit applied locally. revision: yes
Referee: [§IV] §IV (general formula for SU(2) and reducible representations): the extension to non-abelian symmetry and reducible local representations is stated without a separate error estimate for the subleading term. It is unclear whether the same decoupling assumption used for U(1) carries over without additional o(1) contributions from representation multiplicities.

Authors: The general formula in §IV is obtained by replacing the U(1) sector dimensions with the appropriate SU(2) representation dimensions and multiplicities while preserving the structure of the projector. The decoupling of the global constraint from the subsystem holds for the same reason as in the U(1) case: the total volume is taken to infinity with fixed subsystem fraction f. We have revised §IV to add an explicit error estimate showing that representation multiplicities contribute only to the leading extensive term (via the effective dimension) and produce no additional o(1) corrections to the entropy beyond those already present in the local thermal entropy. This is justified by the same large-deviation arguments used for U(1), now applied to the multiplicity-weighted dimensions. We have also included a brief remark on reducible representations to make the extension explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: typical EE derived from independent local thermal entropy input

full rationale

The paper's central result states that typical entanglement entropy (leading and subleading orders) in the fixed global charge sector equals the local thermal entropy at fixed charge density. This local quantity is defined independently from the local Hilbert space, symmetry representation (U(1) or SU(2)), and thermodynamic limit, without being fitted from the global entanglement data or defined in terms of the target EE. The derivation uses standard typicality arguments and does not reduce any prediction to its own inputs by construction, self-definition, or self-citation chains. Model illustrations serve as checks rather than circular fits. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard domain assumptions of quantum statistical mechanics; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

domain assumption Typical states dominate entanglement properties in the fixed-charge sector of the many-body Hilbert space in the thermodynamic limit.
Required to obtain the leading and subleading orders expressed solely in terms of local thermal entropy.

pith-pipeline@v0.9.0 · 5403 in / 1234 out tokens · 79156 ms · 2026-05-07T16:08:21.357718+00:00 · methodology

discussion (0)

Reference graph

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