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Thermodynamics of the Page curve in Markovian open quantum systems

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A Markovian master equation makes the entanglement entropy of an excited open system follow a Page curve at zero temperature.

desk verdict A clean Markovian illustration of Page-curve-like entropy dynamics with a Landauer twist; the main caveat is the unexamined link from master-equation entropy to exact entanglement entropy. read the letter →

arxiv 2501.09082 v1 pith:476J5LBP submitted 2025-01-15 quant-ph cond-mat.stat-mechhep-th

Jonas Glatthard This is my paper · ORCID
classification quant-phcond-mat.stat-mechhep-th
keywords PagecurveentanglemententropyMarkovianopenquantumsystemsglobalmasterequationzero-temperaturereservoirinformation-erasureheatcostbalance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper argues that Page-curve-like entropy dynamics, normally associated with evaporating black holes, arises generically in ordinary open quantum systems described by Markovian master equations. It claims that when a system starts in a pure excited state weakly coupled to a zero-temperature reservoir, its von Neumann entropy first rises as system and reservoir entangle, then falls back to zero as the system relaxes to its ground state. The entropy decrease is thermodynamically constrained: by the entropy balance equation, it can happen only when heat flows out of the system, giving a differential version of the principle that erasing information costs heat. In two model systems—a two-level atom and a harmonic oscillator—the entropy peaks exactly when half the initial energy has left the system.

What carries the argument

The load-bearing object is the global (Davies-form) master equation, a Markovian GKLS semigroup generator whose rates obey detailed balance. It guarantees that the Gibbs state is the unique steady state and supplies an entropy balance equation with non-negative entropy production $\sigma = -\dot S(\rho\|\tau)$. At zero temperature the steady state is the pure ground state, so the long-time entropy vanishes; the same balance equation then forces $\dot S < 0$ to coincide with $\dot Q < 0$. In the examples the dynamics is carried by one jump operator at zero temperature, and the two-level populations or the oscillator covariances convert directly into explicit entropy curves.

What would settle it

Perform a numerically exact simulation of a two-level system coupled to a zero-temperature bosonic bath, without the Markovian or secular approximations, and check whether the entanglement entropy peaks exactly when half the initial energy has decayed; if the maximum occurs at a different time, the Markovian Page curve is an artifact of the approximation.

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Extended reading notes

Core claim

The central claim is that under the global master equation, the reduced state of a system initialized in a pure excited state and weakly coupled to a zero-temperature Markovian bath has an entanglement entropy that follows the Page curve: it starts at zero, grows to a maximum, and returns to zero at long times. The mechanism is the competition between entanglement generation and relaxation to the zero-temperature Gibbs state, which is pure for non-degenerate Hamiltonians. The entropy balance equation $\dot S = \beta \dot Q + \sigma$, with non-negative entropy production $\sigma \geq 0$, enforces that any entropy decrease ($\dot S < 0$) must be accompanied by heat leaving the system ($\dot Q < 0$); this is a differential form of the information-erasure heat cost. In the two-level example the entropy is given analytically by $S(t)=\gamma t e^{-\gamma t} - (1-e^{-\gamma t})\log(1-e^{-\gamma t})$, with maximum $\ln 2$ at $t^{*}=\ln 2/\gamma$, exactly when half the initial excitation energy has decayed; the oscillator example shows the same feature.

Load-bearing premise

The central assumption is that the global master equation, built from the Born, Markov, and secular approximations, faithfully reproduces the true entanglement dynamics of the system and bath, and that its entropy balance equation remains valid as the bath temperature approaches zero.

Editorial extensions

If this is right

  • If the paper is right, excited pure states weakly coupled to cold reservoirs routinely exhibit non-monotonic entanglement entropy, making the Page curve a generic open-quantum-system phenomenon rather than a black-hole special case.
  • The Page time in these models is determined by energy decay: the entropy maximum occurs when half the initial excitation energy has left the system.
  • Entropy decrease under the global master equation is always exothermic, so a heat-current measurement can signal when information is being erased from the system.
  • Standard open-quantum-system tools suffice to reproduce the Page-curve behavior seen in exact system-bath calculations, connecting entanglement dynamics to quantum thermodynamics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The author leaves implicit that the half-energy turnover could serve as a practical clock: monitoring energy decay alone would locate the entropy maximum without full state tomography.
  • The same entropy-balance argument should apply to any master equation whose zero-temperature steady state is pure, so Page-like curves are expected for other dissipative channels, not just amplitude damping.
  • A testable extension is to drive the system out of equilibrium and check whether every purification step obeys the differential heat bound $\dot Q \geq -T \dot S$ at all times.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the von Neumann entropy dynamics of a quantum system evolving under a global (Davies) Lindblad master equation. For a two-level system and a harmonic oscillator initialized in pure excited states and weakly coupled to a zero-temperature Markovian reservoir, the entropy first increases from zero to a maximum and then returns to zero, which the author identifies with Page-curve behavior. In both examples the turnover time (the Page time) coincides with the moment at which half the initial energy has left the system. The paper also uses the entropy balance equation to argue that an entropy decrease requires a heat flow out of the system, and interprets this as a differential version of Landauer's principle. The two-level example is solved analytically (Eq. (13)); the oscillator example is presented numerically.

Significance. If the reduced-state entropy can be legitimately identified with the physical system-bath entanglement entropy, the paper provides a simple and elegant demonstration of Page-curve-like entropy dynamics in a standard Markovian setting, together with a clean thermodynamic inequality. The analytic expression (13) is exact for the given Lindblad equation and the half-energy turnover is a memorable, falsifiable prediction. The paper also correctly points out that the final zero entropy is a consequence of the zero-temperature fixed point of the Davies generator. However, the central physical interpretation as genuine entanglement entropy rests on assumptions about the underlying microscopic model that are not established in the manuscript, and the zero-temperature limit of the entropy balance equation requires care. With those points addressed, the paper would be a useful contribution to the literature on entropy dynamics and quantum thermodynamics.

major comments (3)
  1. [Examples / Fig. 1 caption / Eq. (13)] The identification of S(ρ(t)) with the system-bath entanglement entropy is not justified. The Davies master equation (2) is obtained by tracing out the bath under Born-Markov-secular assumptions and does not supply the joint system-bath state; the statement in the Fig. 1 caption that "the overall system-plus-environment state is and remains pure" is an assumption about the underlying unitary model, not a property of Eq. (2). For the microscopic couplings that motivate the jump operators (e.g., S ∝ σ_x or a+a†), counter-rotating terms dress the exact zero-temperature ground state of H_S+H_I+H_B, so the exact reduced steady state has nonzero entropy of order (coupling)^2 and residual correlations; the final S=0 in Eq. (13) and Fig. 2 is therefore a fixed-point artifact of the secular approximation. Unless the authors prove, or explicitly state, the microscopic model and the approximation under which the master-equation entropy coincides with the entanglement entropy, the central Page-curve claim remains conditional.
  2. [Entropy under the global master equation / Eqs. (5)-(8)] The use of Eq. (5) at zero temperature requires a limiting argument that is not given. For T→0, β→∞ and the reference state τ in Eq. (7) becomes the pure ground state; the relative entropy S(ρ||τ) diverges whenever ρ has any excited-state population, so σ in Eq. (5) is not well-defined by the standard formula. The implication \dot S<0 ⇒ \dot Q<0 and the Landauer bound (9) are therefore not direct consequences of Eq. (5) in the zero-temperature examples. In the two solved examples one can verify \dot Q<0 directly from the explicit decay, but the thermodynamic wording should be qualified or a regularized β→∞ limit should be supplied.
  3. [Examples / Fig. 2] The oscillator claim that the Page time coincides with half the initial energy for all displayed initial states is not derived. The text gives the energy decay E(t)-E0 ∝ e^{-γt} and a numerical illustration, but no evolution equations for Σ(t) and no proof that the von Neumann entropy maximum occurs at t*=ln2/γ for arbitrary initial squeezing. Since this is one of the two central examples and is advertised in the abstract, the authors should either provide the analytic derivation or explicitly state the result as a numerical observation for the particular parameters in Fig. 2.
minor comments (5)
  1. [Before Eq. (13)] The phrase "the overall dynamics Hermitian" should read "the overall dynamics unitary"; a Lindblad master equation does not generate Hermitian dynamics.
  2. [Eqs. (11) and (14)] The rate definitions γ± in Eqs. (11) and (14) are typeset ambiguously; as printed they appear to give a vanishing or negative downward rate at T=0, which contradicts the stated exponential decay. Please correct the expressions.
  3. [Fig. 1 and Fig. 2 captions] Fig. 1 caption contains "an the bath," which should be "and the bath"; Fig. 2 caption contains "hight temperatures," which should be "high temperatures."
  4. [References] Reference [60] is listed as "A VS Quantum Science"; the journal name is AVS Quantum Science.
  5. [Eq. (16) and figure axes] Eq. (16) uses log2 while Figs. 1 and 2 label the vertical axis S/kB; please state the logarithm convention used for the plotted entropy.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Page-time results are derived from explicit master-equation solutions and the standard Spohn entropy balance; self-citations are motivational and not load-bearing.

full rationale

The central derivation is self-contained. For the two-level system, the entropy formula (13) is obtained by solving the Lindblad equation (11) with the stated initial state; no parameter is fitted to the entropy curve, and the Page time t* = ln(2)/gamma follows from differentiating S(t). The equality between the maximum-entropy time and the half-energy time is a derived identity because both are determined by e^{-gamma t} = 1/2. The oscillator example is illustrated from the Gaussian evolution of the master equation (14) and is not fitted to any target curve. The implication Sdot < 0 implies Qdot < 0 in Eq. (8) is a direct consequence of the Spohn entropy balance (5) and the non-negativity of entropy production sigma; it is not inserted as an assumption. The differential Landauer inequality (9) is explicitly a restatement of Eq. (5) under the identification Q_B = -Q, so the paper's wording acknowledges the relation as a restatement rather than a new independent derivation. The cited prior work by the same author (Ref. [39]) is used as motivation and as a qualitative comparison target, but the examples and inequalities in this paper stand on the solved master equations and the standard Davies/Spohn framework; no uniqueness theorem from the author's own papers is invoked to rule out alternatives, and no fitted parameter is renamed as a prediction. The skeptical issues of Markovian closure and the T to 0 limit of the entropy balance concern whether the master equation faithfully models the exact unitary dynamics; those are validity assumptions, not circular reductions of the paper's stated results.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fitted parameters; gamma, epsilon0, omega0, delta are physical model inputs. The main axioms are standard for Markovian open quantum systems; the T to 0 limit of the entropy balance equation is the least standard item.

assumptions (4)
  • domain assumption The global master equation in Davies form accurately describes the reduced system dynamics under weak coupling, Markov, and secular approximations.
    Taken from Refs [55-58]; the paper's entire analysis operates within this approximation.
  • domain assumption The system and bath start in a factorized state rho ⊗ tau_B.
    Stated in Section 'Entropy under the global master equation'.
  • standard math Quantum relative entropy production is non-negative and the Gibbs state is the unique steady state.
    Spohn's results [58,59], used for Eqs. (5)-(7).
  • ad hoc to paper The entropy balance equation Sdot = beta Qdot + sigma remains valid in the T to 0 limit.
    The paper applies Eq. (5) at T=0 without deriving the beta to infinity limit.

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Cite this review

Pith. "Pith review of Thermodynamics of the Page curve in Markovian open quantum systems." pith.science (2026). https://pith.science/paper/476J5LBP

@misc{pith2026250109082,
  author       = {Pith},
  title        = {Pith review of: Thermodynamics of the Page curve in Markovian open quantum systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/476J5LBP}},
  note         = {Machine review of arXiv:2501.09082}
}
read the original abstract

Typically, the von Neumann entropy of a subsystem increases until it plateaus at the thermal value. Under some circumstances, however, the intermediate value can dwarf the final value, even if the subsystem starts in a pure state. A famous example in the context of the black hole information paradox is the entropy of the Hawking radiation, where this behaviour is dubbed the Page curve. More generally, this is the case for excited systems weakly coupled to cold reservoirs. Here we study the entropy dynamics for Lindbladian evolution, i.e. open quantum systems in weak contact with Markovian reservoirs. This allows us to study the non-equilibrium thermodynamics of the subsystem entropy decrease and link it to Landauer's principle: the entropy decrease must be accompanied by a heat flow out of the system. We give an analytic expression of the entanglement dynamics for a decaying excitation in a two-level system and study it under equilibration of a localised oscillator. In both cases the Page time occurs when half the initial energy has left the system.

Figures

Figures reproduced from arXiv: 2501.09082 by the authors.

Figure 1
Figure 1. FIG. 1. The entropy dynamics of an excited two-level system in a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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

  1. Sharp Page transitions in generic Hamiltonian dynamics

    quant-ph 2025-02 conditional novelty 6.0 of 10

    Numerical and hydrodynamic evidence that the min-entropy Page peak becomes a sharp first-order transition in a generic non-integrable spin chain coupled to a cold bath.

  2. Quantum trajectories and Page-curve entanglement dynamics

    cond-mat.stat-mech 2025-01 conditional novelty 6.0 of 10

    In a dephased fermionic chain connected to an empty reservoir, entanglement follows a Page curve whose growth, decay, and peak scaling depend on the noise protocol and probe geometry.

  3. Emergent boundary-memory from unitarity constraints in a minimal two interacting quantum particles

    quant-ph 2026-07 reject novelty 4.0 of 10

    A two-particle walk with motion-only interactions is claimed to require a boundary-memory ancilla for unitarity and to show Page-curve-like entropy, but Eq. 5 is demonstrably non-unitary.

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