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arxiv: 2507.20914 · v4 · submitted 2025-07-28 · 🪐 quant-ph · cond-mat.stat-mech

Planckian bound on quantum dynamical entropy

Xiangyu Cao This is my paper

Pith reviewed 2026-05-19 02:33 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum dynamical entropyPlanckian boundmany-body systemsthermal fluctuationsentropy ratecorrelation functionsquantum monitoringpurification rate
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0 comments X p. Extension

The pith

Monitoring thermal fluctuations in many-body quantum systems yields an entropy growth rate bounded by a universal Planckian value.

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

The paper defines a simplified version of quantum dynamical entropy that measures the information gained by continuously monitoring a single observable. For generic many-body systems this entropy grows steadily when the observable captures thermal fluctuations of an extensive quantity, and the growth rate is expressed explicitly through two-point correlation functions once the thermodynamic and long-time limits are taken. The central claim is a conjecture that this rate obeys a universal upper bound set by the Planckian time scale. A sympathetic reader would care because the bound would place a fundamental limit on how rapidly quantum information can be extracted from thermal states outside classical or large-N regimes.

Core claim

A simplified quantum dynamical entropy is introduced to quantify information gain from continuous monitoring of an observable. In the thermodynamic and long-time limits the entropy rate for generic many-body systems is computed explicitly from the two-point correlation functions of the monitored observable. This computation supports the conjecture of a universal Planckian bound on the entropy rate, together with related results on the purification rate.

What carries the argument

Simplified Connes-Narnhofer-Thirring quantum dynamical entropy, which quantifies information gain from monitoring thermal fluctuations of an extensive observable via its two-point correlation functions.

If this is right

  • The entropy rate is determined solely by two-point correlation functions without requiring full knowledge of the many-body state.
  • A nonzero entropy growth rate appears in generic many-body systems once thermal fluctuations of extensive observables are monitored outside classical and large-N limits.
  • The same framework yields related bounds on the purification rate of quantum states.
  • The conjectured bound would apply universally to the information extraction rate from thermal states in the stated limits.

Where Pith is reading between the lines

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

  • The bound may connect to other Planckian limits appearing in quantum chaos and hydrodynamic descriptions of many-body systems.
  • It supplies a concrete, testable prediction for the information gain rate that could be checked in quantum simulators or cold-atom experiments.
  • Extensions of the same monitoring construction might apply to driven or open quantum systems where correlation functions remain accessible.

Load-bearing premise

The entropy rate and its conjectured bound are obtained only after taking the thermodynamic limit and the long-time limit for a generic many-body system in which thermal fluctuations of an extensive observable produce nonzero information gain away from classical or large-N regimes.

What would settle it

An explicit calculation of the entropy rate for a concrete model such as the transverse-field Ising chain in the thermodynamic limit that either exceeds or saturates the conjectured Planckian bound.

Figures

Figures reproduced from arXiv: 2507.20914 by Xiangyu Cao.

Figure 1
Figure 1. Figure 1: FIG. 1. The purification setup. Two identical systems [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Exact brute-force numerical calculation of the quan [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Keldysh correlation function [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We introduce a simplified version of Connes-Narnhofer-Thirring's quantum dynamical entropy for quantum systems. It quantifies the amount of information gained about the initial condition from continuously monitoring an observable. A nonzero entropy growth rate can be obtained by monitoring the thermal fluctuation of an extensive observable in a generic many-body system, away from classical or large $N$ limits. We explicitly compute the entropy rate in the thermodynamic and long-time limit, in terms of the two-point correlation functions. We conjecture a universal Planckian bound for the entropy rate. Related results on the purification rate are also obtained.

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

2 major / 2 minor

Summary. The paper introduces a simplified version of the Connes-Narnhofer-Thirring quantum dynamical entropy that quantifies information gained by continuously monitoring an observable. It derives an explicit expression for the entropy rate in the thermodynamic and long-time limits in terms of two-point correlation functions for thermal fluctuations of extensive observables. The central claim is a conjecture of a universal Planckian bound on this rate for generic many-body quantum systems away from classical or large-N regimes, with related results on purification rates.

Significance. If the conjecture is substantiated, the result would link quantum information gain rates to a fundamental Planckian scale, offering a potential diagnostic for thermalization and chaos in many-body systems. The explicit reduction to two-point functions is a positive feature that enables direct comparison with correlation data from numerics or experiments.

major comments (2)
  1. [Abstract and §4 (Conjecture)] The central Planckian bound is introduced only as a conjecture following the entropy-rate computation; no derivation is supplied that starts from the two-point correlation expression and arrives at the bound without additional unstated steps or limits. This makes it impossible to assess whether the bound follows rigorously or rests on the generic positivity assumption.
  2. [§3 (Entropy rate computation) and discussion after the main formula] The claim that a nonzero entropy rate is obtained generically for thermal fluctuations of extensive observables in the thermodynamic and long-time limits (away from classical/large-N regimes) is load-bearing for universality. The manuscript does not supply a proof or counter-example analysis showing that the two-point functions alone guarantee positivity independent of higher-order correlations or model-specific details.
minor comments (2)
  1. [§2] Notation for the simplified entropy should be introduced with a clear comparison to the original Connes-Narnhofer-Thirring definition to avoid ambiguity.
  2. [§3] The thermodynamic-limit procedure and the precise order of limits (thermodynamic before long-time) should be stated explicitly with any required uniformity conditions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and insightful comments on our manuscript. We address each major comment below, clarifying the status of our results and indicating where we will revise the text for improved transparency.

read point-by-point responses

  1. Referee: [Abstract and §4 (Conjecture)] The central Planckian bound is introduced only as a conjecture following the entropy-rate computation; no derivation is supplied that starts from the two-point correlation expression and arrives at the bound without additional unstated steps or limits. This makes it impossible to assess whether the bound follows rigorously or rests on the generic positivity assumption.

    Authors: We agree that the Planckian bound is presented strictly as a conjecture and is not derived rigorously from the two-point correlation formula alone. The explicit entropy-rate expression we obtain is an integral over the connected two-point functions of the monitored observable. The conjecture arises by combining this expression with standard assumptions on the decay of correlations in generic many-body systems (finite correlation length or diffusive scaling) together with the positivity of the entropy production rate and the existence of a Planckian time scale set by ħ/k_B T. We do not claim a model-independent lower bound that follows solely from the two-point formula without these inputs. In the revised manuscript we will (i) restate the conjecture with an explicit list of the additional physical assumptions, (ii) add a short paragraph explaining the heuristic steps that motivate the Planckian scale, and (iii) include a brief discussion of why a fully rigorous proof appears to require techniques beyond the present reduction to two-point functions. revision: partial

  2. Referee: [§3 (Entropy rate computation) and discussion after the main formula] The claim that a nonzero entropy rate is obtained generically for thermal fluctuations of extensive observables in the thermodynamic and long-time limits (away from classical/large-N regimes) is load-bearing for universality. The manuscript does not supply a proof or counter-example analysis showing that the two-point functions alone guarantee positivity independent of higher-order correlations or model-specific details.

    Authors: We accept that the generic positivity statement is central and that the manuscript does not contain a general proof that the two-point functions alone force a strictly positive rate for every possible dynamics. The entropy-rate formula depends only on the two-point correlator, yet the sign of the resulting integral is determined by the long-time behavior of that correlator, which in turn is constrained by the underlying many-body dynamics. Higher-order correlations enter indirectly through the equations of motion that govern the two-point functions. In the revised version we will add a dedicated subsection that (a) recalls the explicit integral expression, (b) states the minimal conditions on the two-point functions (e.g., exponential or power-law decay with a finite correlation time) that guarantee positivity, and (c) briefly contrasts the classical and large-N limits where these conditions fail. We will also include a short numerical illustration for a non-integrable spin chain to support the generic claim, while acknowledging that a complete classification of all possible two-point functions lies outside the scope of the paper. revision: yes

Circularity Check

0 steps flagged

No circularity: entropy rate derived from two-point functions; Planckian bound is explicit conjecture

full rationale

The paper defines a simplified quantum dynamical entropy from observable monitoring and derives its rate explicitly in the thermodynamic/long-time limits as a functional of independent two-point correlation functions. The universal Planckian bound is stated as a conjecture based on the positivity of this rate for thermal fluctuations of extensive observables in generic many-body systems (away from classical/large-N limits). This is an assumption supporting the conjecture, not a reduction of the bound to the paper's own fitted inputs or self-definitional equations. No self-citation chains, ansatzes smuggled via prior work, or renaming of known results appear in the load-bearing steps. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard quantum mechanics for the definition of monitoring and entropy, plus domain assumptions about the existence of thermodynamic and long-time limits in generic many-body systems. No free parameters or new postulated entities are introduced beyond the new entropy definition itself.

axioms (2)
  • standard math Standard framework of quantum mechanics and statistical mechanics for open quantum systems
    Used to define continuous monitoring of an observable and the resulting information gain.
  • domain assumption Existence of well-defined thermodynamic and long-time limits for generic many-body systems
    Invoked to obtain a nonzero entropy growth rate from thermal fluctuations of an extensive observable.
invented entities (1)
  • Simplified quantum dynamical entropy no independent evidence
    purpose: Quantify information gained about the initial condition from continuous monitoring of an observable
    New definition introduced to simplify the Connes-Narnhofer-Thirring construction for this monitoring scenario.

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