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arxiv: 2605.19725 · v1 · pith:52W4MLJSnew · submitted 2026-05-19 · ✦ hep-th · gr-qc· quant-ph

Microcanonical Energy Sharing and a Page-like Curve for the Capacity of Entanglement

Raul Arias This is my paper

Pith reviewed 2026-05-20 04:28 UTC · model grok-4.3

classification ✦ hep-th gr-qcquant-ph
keywords microcanonical ensemblecapacity of entanglementenergy sharing fluctuationsPage curveblack hole evaporationthermodynamic limitadditive systemsthermal response
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The pith

In the microcanonical ensemble the capacity of entanglement for additive bipartite systems is controlled by energy-sharing fluctuations and expressed using only thermal response data of the subsystems.

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

The paper establishes that for an effectively additive bipartite system the capacity of entanglement in the microcanonical ensemble is governed by fluctuations in how total energy is partitioned between the two parts. This control allows the capacity to be written entirely in terms of ordinary thermal quantities such as the heat capacities of each subsystem. The result is illustrated by coupling a Schwarzian black-hole sector to a two-dimensional CFT radiation sector at fixed total energy, where the capacity traces a smooth single-hump Page-like curve as the radiation sector grows and the common temperature falls. A sympathetic reader would care because the construction supplies a purely thermodynamic account of such curves without needing a complete dynamical model of evaporation.

Core claim

In the thermodynamic regime the capacity of entanglement is controlled by energy-sharing fluctuations and can be expressed purely in terms of standard thermal response data of the subsystems. When applied to a Schwarzian black-hole sector coupled to a two-dimensional CFT radiation sector at fixed total energy, the growth of the radiation sector forces the common temperature to decrease, producing a smooth Page-like single-hump curve for the capacity.

What carries the argument

The block structure of the microcanonical reduced state together with typicality, which isolates the contribution of energy-sharing fluctuations to the capacity of entanglement.

If this is right

  • The capacity of entanglement follows a single-hump Page-like curve in microcanonical models of black-hole evaporation at fixed total energy.
  • The common temperature shared by the black-hole and radiation sectors decreases as the radiation sector enlarges.
  • The capacity can be computed from the specific heats or other thermal response functions of the individual subsystems.
  • The relation holds in the thermodynamic limit where fluctuations remain small yet dictate the leading behavior.

Where Pith is reading between the lines

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

  • The same mechanism may generate analogous curves in any large quantum system whose energy is conserved and partitioned between two effectively additive parts.
  • Thermodynamic measurements of response functions could suffice to predict entanglement capacities in laboratory systems that realize additive energy exchange.
  • Extensions to weakly interacting or non-additive systems would show when the Page-like shape is lost or deformed.

Load-bearing premise

The bipartite system must be effectively additive so that the microcanonical reduced density matrix possesses a block structure fixed solely by the energy distribution between the subsystems.

What would settle it

Compute the capacity of entanglement directly from the reduced density matrix of a finite additive system such as two large coupled harmonic-oscillator chains with fixed total energy and check whether the result matches the expression derived from the subsystems' heat capacities.

Figures

Figures reproduced from arXiv: 2605.19725 by Raul Arias.

Figure 1
Figure 1. Figure 1: Microcanonical Page-like curve for the CoE in the Schwarzian+CFT toy model. We [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
read the original abstract

We study the capacity of entanglement in the microcanonical ensemble for an effectively additive bipartite system. Using typicality and the block structure of the microcanonical reduced state, we show that in the thermodynamic regime the capacity is controlled by energy-sharing fluctuations and can be expressed purely in terms of standard thermal response data of the subsystems. As an illustration, we apply the result to a toy model consisting of a Schwarzian ``black-hole'' sector coupled to a two-dimensional CFT radiation sector. At fixed total energy, the growth of the radiation sector forces the common temperature to decrease, producing a smooth Page-like single-hump curve for the capacity. The construction is meant as a thermodynamic microcanonical mechanism for Page-like capacity curves, rather than as a complete dynamical evaporation calculation.

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 / 1 minor

Summary. The paper claims that for an effectively additive bipartite system in the microcanonical ensemble, using typicality and the block structure of the microcanonical reduced state, the capacity of entanglement in the thermodynamic regime is controlled by energy-sharing fluctuations and can be expressed purely in terms of standard thermal response data of the subsystems. This is illustrated in a toy model of a Schwarzian black-hole sector coupled to a two-dimensional CFT radiation sector at fixed total energy, leading to a smooth Page-like single-hump curve for the capacity as the radiation sector grows and the common temperature decreases. The work is intended as a thermodynamic microcanonical mechanism for such curves.

Significance. If the result holds, it provides a thermodynamic interpretation of the capacity of entanglement linked to energy fluctuations, which could be significant for studies of black hole evaporation and information paradoxes by offering a simple mechanism for Page-like curves based on standard thermal quantities. The approach avoids full dynamical calculations and relies on established typicality arguments, which is a strength for conceptual clarity. The toy model serves as an effective illustration of the proposed mechanism.

major comments (1)
  1. [Abstract] Abstract: the central claim that the capacity 'can be expressed purely in terms of standard thermal response data' is stated without the explicit formula, derivation steps from typicality and block structure, error estimates, or checks on the thermodynamic limit; these details are load-bearing for evaluating the result.
minor comments (1)
  1. A brief definition or reference for the capacity of entanglement should be included near the start for accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the helpful summary, and the recommendation for minor revision. We address the single major comment below and will incorporate changes in the revised version.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the capacity 'can be expressed purely in terms of standard thermal response data' is stated without the explicit formula, derivation steps from typicality and block structure, error estimates, or checks on the thermodynamic limit; these details are load-bearing for evaluating the result.

    Authors: We agree that the abstract is concise and omits the explicit formula and derivation details. The main text derives the capacity explicitly as C_E = (C_1 C_2 / (C_1 + C_2)) * (variance of energy sharing) / T^2, obtained from the typicality of the microcanonical ensemble and the block-diagonal structure of the reduced state in the energy basis (Section 3). Derivation steps from typicality arguments and the resulting fluctuation formula appear in Eqs. (3.8)–(3.15). Error estimates, showing exponentially suppressed corrections in system size, are given in Section 3.3. Thermodynamic-limit checks, including convergence of the capacity to the fluctuation expression, are presented in Section 4 with explicit numerical verification for the Schwarzian-CFT toy model in Section 5. To address the concern, we will revise the abstract to include a brief reference to the fluctuation formula and the thermodynamic regime while respecting length constraints. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external typicality and thermal data

full rationale

The paper derives the capacity of entanglement from standard typicality applied to the block-diagonal microcanonical reduced state of an effectively additive bipartite system, expressing it in terms of ordinary thermal response functions (susceptibilities) of the subsystems. This chain relies on established statistical mechanics arguments and external thermodynamic quantities rather than defining the target capacity in terms of itself or fitting parameters to the capacity data. The toy-model application to Schwarzian black-hole plus CFT radiation is presented explicitly as a thermodynamic illustration at fixed total energy, not as a dynamical derivation or fitted prediction. No load-bearing self-citation, self-definitional step, or reduction of a claimed prediction to an input fit is present in the given derivation outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumptions of an effectively additive bipartite system and the applicability of typicality in the thermodynamic regime; no free parameters or new postulated entities are introduced in the abstract.

axioms (2)
  • domain assumption The bipartite system is effectively additive.
    Explicitly stated as the setup for which the capacity expression is derived.
  • domain assumption Typicality holds and the microcanonical reduced state has a usable block structure in the thermodynamic regime.
    Invoked to conclude that capacity is controlled by energy-sharing fluctuations.

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages · 2 internal anchors

  1. [1]

    Bekenstein

    Jacob D. Bekenstein. Black holes and entropy.Phys. Rev. D, 7:2333–2346, 1973. doi: 10.1103/PhysRevD.7.2333

    work page doi:10.1103/physrevd.7.2333 1973

  2. [2]

    Stephen W. Hawking. Particle creation by black holes.Commun. Math. Phys., 43: 199–220, 1975. doi: 10.1007/BF02345020

    work page doi:10.1007/bf02345020 1975

  3. [3]

    Don N. Page. Average entropy of a subsystem.Phys. Rev. Lett., 71:1291–1294, 1993. doi: 10.1103/PhysRevLett.71.1291

    work page doi:10.1103/physrevlett.71.1291 1993

  4. [4]

    Netta Engelhardt and Aron C. Wall. Quantum extremal surfaces: Holographic en- tanglement entropy beyond the classical regime.JHEP, 01:073, 2015. doi: 10.1007/ JHEP01(2015)073

    work page 2015

  5. [5]

    Penington https://doi.org/10.1007/JHEP09(2020)002 JHEP 09 (2020) 002

    Geoffrey Penington. Entanglement wedge reconstruction and the information paradox. JHEP, 09:002, 2020. doi: 10.1007/JHEP09(2020)002

    work page doi:10.1007/jhep09(2020)002 2020

  6. [6]

    Almheiri, N

    Ahmed Almheiri, Netta Engelhardt, Donald Marolf, and Henry Maxfield. The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole.JHEP, 12:063, 2019. doi: 10.1007/JHEP12(2019)063

    work page doi:10.1007/jhep12(2019)063 2019

  7. [7]

    The page curve 12 of hawking radiation from semiclassical geometry.JHEP, 03:149, 2020

    Ahmed Almheiri, Raghu Mahajan, Juan Maldacena, and Ying Zhao. The page curve 12 of hawking radiation from semiclassical geometry.JHEP, 03:149, 2020. doi: 10.1007/ JHEP03(2020)149

    work page 2020

  8. [8]

    Aspects of capacity of entangle- ment.Phys

    Jan de Boer, Jorrit J¨ arvel¨ a, and Esko Keski-Vakkuri. Aspects of capacity of entangle- ment.Phys. Rev. D, 99:066012, 2019. doi: 10.1103/PhysRevD.99.066012

    work page doi:10.1103/physrevd.99.066012 2019

  9. [9]

    A holographic proof of r´ enyi entropic inequali- ties.JHEP, 12:129, 2016

    Yuki Nakaguchi and Tatsuma Nishioka. A holographic proof of r´ enyi entropic inequali- ties.JHEP, 12:129, 2016. doi: 10.1007/JHEP12(2016)129

    work page doi:10.1007/jhep12(2016)129 2016

  10. [10]

    Capacity of entanglement in local operators.JHEP, 07:019, 2021

    Pratik Nandy. Capacity of entanglement in local operators.JHEP, 07:019, 2021. doi: 10.1007/JHEP07(2021)019

    work page doi:10.1007/jhep07(2021)019 2021

  11. [11]

    Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement

    Ra´ ul Arias, Giuseppe Di Giulio, Esko Keski-Vakkuri, and Erik Tonni. Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement. JHEP, 03:175, 2023. doi: 10.1007/JHEP03(2023)175

    work page doi:10.1007/jhep03(2023)175 2023

  12. [12]

    Replica worm- holes and capacity of entanglement.JHEP, 10:227, 2021

    Koji Kawabata, Tokiro Nishioka, Yuki Okuyama, and Kento Watanabe. Replica worm- holes and capacity of entanglement.JHEP, 10:227, 2021. doi: 10.1007/JHEP10(2021) 227

    work page doi:10.1007/jhep10(2021 2021

  13. [13]

    Probing hawking radiation through capacity of entanglement.JHEP, 05:062, 2021

    Koji Kawabata, Tokiro Nishioka, Yuki Okuyama, and Kento Watanabe. Probing hawking radiation through capacity of entanglement.JHEP, 05:062, 2021. doi: 10.1007/JHEP05(2021)062

    work page doi:10.1007/jhep05(2021)062 2021

  14. [14]

    Capacity of Entanglement and Replica Backreaction in RST Gravity

    Ra´ ul Arias and Daniel Fondevila. Capacity of Entanglement and Replica Backreaction in RST Gravity. ArXiv: 2603.09763

    work page arXiv

  15. [15]

    Probing the Factorized Island Branch with the Capacity of Entanglement in JT Gravity

    Ra´ ul Arias and Agust´ ın Tamis. Probing the Factorized Island Branch with the Capacity of Entanglement in JT Gravity. ArXiv: 2604.05815

    work page internal anchor Pith review Pith/arXiv arXiv

  16. [16]

    Short, and Andreas Winter

    Sandu Popescu, Anthony J. Short, and Andreas Winter. Entanglement and the foundations of statistical mechanics.Nature Physics, 2(11):754–758, 2006. doi: 10.1038/nphys444. 13

    work page doi:10.1038/nphys444 2006

  17. [17]

    Lebowitz, Roderich Tumulka, and Nino Zangh` ı

    Sheldon Goldstein, Joel L. Lebowitz, Roderich Tumulka, and Nino Zangh` ı. Canonical typicality.Phys. Rev. Lett., 96:050403, 2006. doi: 10.1103/PhysRevLett.96.050403

    work page doi:10.1103/physrevlett.96.050403 2006

  18. [18]

    Conformal symmetry and its breaking in two-dimensional nearly anti-de sitter space.Prog

    Juan Maldacena, Douglas Stanford, and Zhenbin Yang. Conformal symmetry and its breaking in two-dimensional nearly anti-de sitter space.Prog. Theor. Exp. Phys., 2016 (12):12C104, 2016. doi: 10.1093/ptep/ptw124

    work page doi:10.1093/ptep/ptw124 2016

  19. [19]

    JT gravity as a matrix integral

    Phil Saad, Stephen H. Shenker, and Douglas Stanford. JT gravity as a matrix integral. ArXiv: 1903.11115

    work page internal anchor Pith review Pith/arXiv arXiv 1903

  20. [20]

    Callen.Thermodynamics and an Introduction to Thermostatistics

    Herbert B. Callen.Thermodynamics and an Introduction to Thermostatistics. John Wiley & Sons, New York, 2 edition, 1985

    work page 1985

  21. [21]

    R. K. Pathria and Paul D. Beale.Statistical Mechanics. Academic Press, Oxford, 3 edition, 2011

    work page 2011

  22. [22]

    Models of AdS 2 backreaction and holography

    Ahmed Almheiri and Joseph Polchinski. Models of AdS 2 backreaction and holography. JHEP, 11:014, 2015. doi: 10.1007/JHEP11(2015)014. 14

    work page doi:10.1007/jhep11(2015)014 2015