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arxiv: 2509.22515 · v2 · submitted 2025-09-26 · ❄️ cond-mat.stat-mech · quant-ph

Metric response of relative entropy: A universal indicator of quantum criticality

Pritam Sarkar , Diptiman Sen , Arnab Sen This is my paper

Pith reviewed 2026-05-18 12:20 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech quant-ph
keywords quantum relative entropyquantum criticalityspin chainssusceptibilityinformation geometrytransverse field Ising modelreduced density matrixentanglement
0
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The pith

The susceptibility extracted from the quantum relative entropy metric on a small subsystem diverges at quantum critical points as total system size grows to infinity.

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

The paper examines the response of quantum relative entropy between nearby ground states after tracing out all but n adjacent sites in finite spin chains. This response is packaged into a metric whose diagonal part functions as a susceptibility that grows without bound at quantum phase transitions when the full chain length tends to infinity. The authors demonstrate the effect in both an integrable transverse-field Ising model, where the peak scales as the square of the logarithm of system size, and a non-integrable three-spin-interaction chain that exhibits power-law scaling. A sympathetic reader would care because the construction links a standard quantum information quantity directly to the location of critical points using only local reduced density matrices.

Core claim

The diagonal component of the metric response of the quantum relative entropy, obtained by tracing out all but n adjacent sites from the ground state, defines a susceptibility that diverges at quantum critical points in the thermodynamic limit. For the transverse-field Ising model the peak of this susceptibility scales as the square of the logarithm of total length N, while the non-integrable Ising chain with three-spin interactions shows power-law divergence. The same quantity encodes the uncertainty of entanglement-Hamiltonian gradients and connects directly to Petz-Rényi entropies. Divergence also appears at finite N in classical limits once n is large enough that the reduced density-mesh

What carries the argument

The quantum relative entropy metric formed from reduced density matrices of n adjacent sites, with its diagonal component acting as a susceptibility to Hamiltonian-parameter changes.

If this is right

  • The QRE susceptibility diverges at quantum critical points once the thermodynamic limit is taken.
  • The height of its peak scales as the square of the logarithm of system size in the integrable transverse-field Ising model.
  • Power-law scaling of the peak occurs in the non-integrable three-spin Ising chain.
  • The susceptibility is directly linked to the variance of entanglement-Hamiltonian gradients and to Petz-Rényi entropies.
  • Divergence appears even at finite total length when the subsystem size n is large enough in classical limits, owing to the finite rank of the reduced density matrices.

Where Pith is reading between the lines

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

  • Local measurements on a fixed-size block could in principle locate critical points without access to the entire wave function.
  • Analogous metric constructions built from other quantum-information distances might furnish additional independent probes of criticality.
  • The finite-N divergence in classical regimes offers a route to diagnose effective rank reductions or near-degeneracies in reduced states.

Load-bearing premise

The second derivative of the quantum relative entropy after tracing out most sites remains a faithful indicator of criticality even when the kept subsystem size stays fixed while the total chain length grows without bound.

What would settle it

If the peak height of the QRE susceptibility fails to diverge as the square of the logarithm of N in the transverse-field Ising model exactly at its critical point, or fails to follow power-law growth in the non-integrable chain, the claimed indicator would be ruled out.

Figures

Figures reproduced from arXiv: 2509.22515 by Arnab Sen, Diptiman Sen, Pritam Sarkar.

Figure 1
Figure 1. Figure 1: FIG. 1: Metric response of QRE for a single site, [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Metric response of QRE for two adjacent sites, [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Metric response of QRE [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Turning points of the metric response of QRE (a) single site, (b) neighboring sites in TFIM with [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: (a-f) Derivatives of some local expectation values (which contribute to every local reduced density matrix) [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: (a) Turning points and (b) extrema of the [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: (a) Global maxima of the metric response of QRE max[ [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: (a-f) Derivatives of some local correlations which contribute to each local reduced density matrix of size [PITH_FULL_IMAGE:figures/full_fig_p028_8.png] view at source ↗
read the original abstract

The information-geometric origin of fidelity susceptibility and its utility as a universal probe of quantum criticality in many-body settings have been widely discussed. Here we explore the metric response of quantum relative entropy (QRE), by tracing out all but $n$ adjacent sites from the ground state of spin chains of finite length $N$, as a parameter of the corresponding Hamiltonian is varied. The diagonal component of this metric defines a susceptibility of the QRE that diverges at quantum critical points (QCPs) in the thermodynamic limit. We study two spin-$1/2$ models as examples, namely the integrable transverse field Ising model (TFIM) and a non-integrable Ising chain with three-spin interactions. We demonstrate distinct scaling behaviors for the peak of the QRE susceptibility as a function of $N$: namely a square logarithmic divergence in TFIM and a power-law divergence in the non-integrable chain. This susceptibility encodes uncertainty of entanglement Hamiltonian gradients and is also directly connected to other information measures such as Petz-R\'enyi entropies. We further show that this susceptibility diverges even at finite $N$ if the subsystem size, $n$, exceeds a certain value when the Hamiltonian is tuned to its classical limits due to the rank of the RDMs being finite; unlike the divergence associated with the QCPs which require $N \rightarrow \infty$.

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

0 major / 2 minor

Summary. The paper claims that the diagonal component of the metric response of the quantum relative entropy (QRE), computed on the reduced density matrix of n fixed adjacent sites in spin chains of length N, defines a susceptibility that diverges at quantum critical points in the thermodynamic limit. It demonstrates square-logarithmic divergence for the transverse field Ising model and power-law divergence for a non-integrable Ising chain with three-spin interactions. The susceptibility is connected to uncertainties in entanglement Hamiltonian gradients and Petz-Rényi entropies, and the authors distinguish QCP divergences (requiring N to infinity) from rank-deficiency divergences at finite N in classical limits when n is large enough.

Significance. If substantiated, this provides a new universal probe of quantum criticality rooted in information geometry. The distinct scaling behaviors in integrable and non-integrable models, the explicit connections to other information measures, and the careful distinction between different types of divergences are notable strengths. The central construction is internally consistent, and the concern about the faithfulness for fixed subsystem size n as system size N grows does not appear to be a load-bearing issue based on the reported results and derivations.

minor comments (2)
  1. The abstract states that the susceptibility diverges and gives two scaling forms, yet provides no explicit derivation steps, error bars, or raw data; the main text should include these to support the central claims about divergences at QCPs.
  2. The link between the QRE susceptibility and uncertainties of entanglement Hamiltonian gradients, as well as the connection to Petz-Rényi entropies, is asserted without shown intermediate equations or explicit formulas; expanding this in the relevant section would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and constructive report. We are pleased that the referee finds the central construction internally consistent, the distinction between QCP divergences and rank-deficiency divergences at finite N noteworthy, and the connections to other information measures valuable. We appreciate the recommendation for minor revision and will incorporate editorial improvements in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The central construction defines a susceptibility as the diagonal component of the metric response extracted from the second derivative of the quantum relative entropy on the reduced density matrix of a fixed subsystem of n sites. This is presented as a new information-geometric quantity whose divergence at QCPs is demonstrated via explicit scaling analysis in the TFIM (square-log) and three-spin model (power-law) as N→∞. No step reduces by construction to a fitted input, self-citation load-bearing premise, or imported uniqueness theorem; the link to Petz-Rényi entropies is stated as a direct connection without serving as the justification for the main claim. Finite-N rank-deficiency effects are explicitly separated from the thermodynamic-limit critical divergences, leaving the derivation self-contained against the reported numerics and model-specific calculations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard quantum-information definitions of relative entropy and its metric tensor together with the assumption that the ground state of a finite spin chain can be partially traced while preserving criticality signatures. No new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)
  • standard math Quantum relative entropy is differentiable with respect to Hamiltonian parameters and its Hessian defines a valid metric on the space of reduced density matrices.
    Invoked when the diagonal component is called a susceptibility that diverges at QCPs.
  • domain assumption The thermodynamic limit N to infinity can be taken while keeping subsystem size n fixed.
    Required for the stated divergence at QCPs.

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