Diverging conditional correlation lengths in the approach to high temperature

Dmitry A. Abanin; Jerome Lloyd; Sarang Gopalakrishnan

REVIEW

Diverging conditional correlation lengths in the approach to high temperature

Jerome Lloyd, Dmitry A. Abanin, Sarang Gopalakrishnan

Quantum Physics · Submitted 4 Aug 2025

Pith's verdict: A clean and potentially important caution about the Markov-length diagnostic, but the load-bearing convergence evidence is unreadable in the supplied text, so the exponential-growth claim needs referee scrutiny.

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arXiv ID2508.02567 (version 1)

Pith IDTPYJY2WD, Pith's permanent record of this paper and its review

CitationDmitry A. Abanin, Jerome Lloyd, Sarang Gopalakrishnan. "Diverging conditional correlation lengths in the approach to high temperature." arXiv:2508.02567 (2025). https://pith.science/paper/TPYJY2WD

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Editor's note

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Bottom line: this paper makes a genuinely useful observation if the numerics hold. The Markov length, recently proposed as a diagnostic for quantum mixed-state phase transitions, grows exponentially in a purely classical 1D Ising quench into the high-temperature phase, while ordinary correlation functions show no growing length scale. That is a real constraint on the Sang–Hsieh diagnostic and worth the attention of anyone working on mixed-state phases or thermalization.

The abstract is clear and internally consistent. The Markov length is computed from the time-evolved state, the exponential growth is presented as an observed numerical fact, and the information-theoretic arguments are explanations rather than fitted parameters. I see no circularity at the abstract level. The MPS-based numerical technique is a reasonable contribution, and the authors are honest about the parent-Hamiltonian conclusion being inherited from the Sang–Hsieh framework.

Where I hesitate: the full text I was given is garbled at the character level, so I cannot verify the definitions, derivations, or numerical convergence checks. The stress-test worry is fair: if the Markov length is defined through a conditional-mutual-information cutoff, then finite bond dimension can suppress small CMI values and produce apparent exponential growth; finite system size also caps the accessible Markov length. The abstract itself does not show threshold dependence, system-size extrapolation, or time-window checks. That does not mean the result is wrong, but the central claim is exactly the kind of thing that needs those checks, and I could not see them. The parent-Hamiltonian interpretation is also a step beyond what the numerics directly show, though the paper does flag it as an inference.

This paper deserves a serious referee, not a desk reject. Send it to someone who can check the threshold and bond-dimension convergence, and ask them to verify that the exponential growth is not a finite-entanglement or finite-size artifact. If that holds up, the result is solid and citable. The audience is small but active: people building information-theoretic diagnostics for mixed-state transitions should know about this caution.

Referee report

Major objections (4)

[Full text / numerical sections] The supplied full text is garbled to the point that the numerical sections cannot be read, so I cannot verify the definition of the Markov-length extraction threshold, the bond dimensions, the system sizes, or any convergence/extrapolation procedure. Because the Markov length is defined as the smallest buffer length at which the conditional mutual information falls below a cutoff, the reported exponential growth could in principle be an artifact of a fixed threshold or of finite bond-dimension truncation. Please provide a clean manuscript and explicitly demonstrate that the exponential growth is robust under variation of the threshold, bond dimension, and system size, with any finite-size extrapolation clearly described.
[Abstract / conclusion] The claim that growth of the Markov length implies divergence of the range of a putative parent Hamiltonian is asserted as a direct consequence, but no derivation of this inference is visible in the readable parts of the text. Since the Sang–Hsieh framework was developed for quantum mixed states, its transfer to classical probability distributions over spin configurations is a substantive step. Please provide an explicit argument, or a precise reference to a theorem, establishing this implication for the classical setting.
[Abstract / comparison with conventional observables] The statement that 'conventional observables do not exhibit growing length scales' is made only at the abstract level; the readable text does not show the correlation function, its correlation length, or the time window used for comparison. Please specify the observable and present its time dependence alongside the Markov length so that the absence of a conventional growing length scale is directly checkable.
[Information-theoretic arguments] The paper promises 'simple information-theoretic arguments that explain our results' for all cases, but these arguments are not readable in the submitted full text. To make the explanation checkable, please state at least one such argument explicitly, ideally as a bound or scaling statement for the conditional mutual information that yields exponential Markov-length growth, rather than as a heuristic remark.

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The paper

The paper claims

Under a quench into the high-temperature phase of a 1D classical Ising model, the Markov length grows exponentially even though ordinary correlation lengths do not.

The reading

The paper sets out to show that the Markov length, an information-theoretic quantity proposed as a diagnostic for quantum mixed-state phase transitions, diverges in a setting with no conventional phase transition at all. Under classical stochastic dynamics in one dimension, a quench from an ordered low-temperature state into the high-temperature phase produces a Markov length that grows exponentially in time, while standard correlation functions exhibit no growing length scale. The authors conclude that the state becomes increasingly non-Gibbsian and that the range of any parent Hamiltonian representing it must diverge with the Markov length. A sympathetic reader should care because this means the information-theoretic view of thermalization has a singular late-time limit even in a purely classical problem, and because the paper supplies a matrix-product-state technique for measuring the Markov length in such settings.

What carries the argument

The central object is the Markov length, defined from the decay of conditional mutual information: for a tripartition $A-B-C$, $I(A:C|B)$ measures how much information about $A$ remains in $C$ after conditioning on $B$, and the Markov length is the distance over which this quantity falls below a cutoff. The computational engine is a matrix-product-state representation of the classical probability distribution, whose bond dimension controls the accessible Markov length. The mechanism is that heating from an ordered initial state creates long-range conditional dependencies, so information about distant spins remains hidden even after local unconditional correlations have relaxed, and the minimal memory required to describe the state grows without bound.

Extended reading notes

Core claim

The paper's central claim is that the Markov length, the distance over which conditional mutual information $I(A:C|B)$ decays when a middle region $B$ is conditioned on, is not a clean diagnostic of mixed-state phase transitions. In the one-dimensional classical Ising model, a quench from an ordered low-temperature state into the high-temperature phase produces a probability distribution whose Markov length grows exponentially in time, $\xi_M(t)\sim e^{ct}$, while conventional correlation functions show no growing length scale. The authors argue that this makes the state increasingly non-Gibbsian and forces the range of any putative parent Hamiltonian to diverge together with the Markov length. From this information-theoretic viewpoint, the late-time limit of thermalization is singular, even in a purely classical stochastic setting. They support the claim with a matrix-product-state technique for computing the Markov length and with simple information-theoretic arguments for general thermal quenches.

Load-bearing premise

The load-bearing premise is that the numerical matrix-product-state extraction of the Markov length, with finite bond dimension, finite system size, and a cutoff below which conditional mutual information is treated as zero, is converged enough that the exponential growth belongs to the thermodynamic-limit state; a second interpretive premise is that a diverging Markov length forces the parent-Hamiltonian range to diverge.

Editorial extensions

If this is right

The Markov length cannot, by itself, certify a mixed-state phase transition, because it diverges in an ordinary classical heating process where no thermodynamic transition occurs.
A system quenched into the high-temperature phase becomes increasingly non-Gibbsian, so no finite-range Gibbs distribution can reproduce its conditional correlations at late times.
The range of any parent Hamiltonian that represents the state must grow with time, meaning the late-time limit has no finite-range local parent Hamiltonian.
The same exponential growth is expected for general thermal quenches, since the authors provide information-theoretic arguments covering all studied cases.
The matrix-product-state method offers a practical route to measuring Markov lengths in one-dimensional stochastic dynamics.

Reading between the lines

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

Editorial inference: the same mechanism should appear in quantum thermalization, for instance under dephasing or weak coupling to a bath, so information-theoretic mixed-state phase-diagram proposals will need to subtract this dynamical divergence as a baseline.
Editorial inference: the exponential growth suggests a rare-event or domain-wall-counting origin, such as $\xi_M \sim e^{t/\tau}$ with $\tau$ set by the energy gap to the ordered state, and a direct Monte Carlo measurement of conditional mutual information could test this.
Editorial inference: if no finite-range parent Hamiltonian exists at any late time, the usual picture of approach to a local Gibbs ensemble is incomplete, and the paper's singular late-time limit may be the generic rule rather than an exception.
Editorial inference: the result connects to the minimal-memory complexity of sampling the time-evolved distribution, which appears to grow exponentially and may have consequences for classical simulation algorithms.

Original abstract

The Markov length was recently proposed as an information-theoretic diagnostic for quantum mixed-state phase transitions [Sang & Hsieh, Phys. Rev. Lett. 134, 070403 (2025)]. Here, we show that the Markov length diverges even under classical stochastic dynamics, when a low-temperature ordered state is quenched into the high temperature phase. Conventional observables do not exhibit growing length scales upon quenching into the high-temperature phase; however, the Markov length grows exponentially in time. Consequently, the state of a system as it heats becomes increasingly non-Gibbsian, and the range of its putative "parent Hamiltonian" must diverge with the Markov length. From this information-theoretic point of view the late-time limit of thermalization is singular. We introduce a numerical technique for computing the Markov length based on matrix-product states, and explore its dynamics under general thermal quenches in the one-dimensional classical Ising model. For all cases, we provide simple information-theoretic arguments that explain our results.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 5,664 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.14 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.27 · compares with prior results 0.79 · mentions code or data availability 0.03.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The Markov length was recently proposed as an information-theoretic diagnostic for quantum mixed-state phase transitions [Sang & Hsieh, Phys. Rev. Lett. 134, 070403 (2025)]. Here, we show that the Markov length diverges even under classical stochastic dynamics, when a low-temperature ordered state is quenched into the high temperature phase. Conventional observables do not exhibit growing length scales upon quenching into the high-temperature phase; however, the Markov length grows exponentially in time. Consequently, the state of a system as it heats becomes increasingly non-Gibbsian, and the range of its putative "parent Hamiltonian" must diverge with the Markov length. From this information-theoretic point of view the late-time limit of thermalization is singular. We introduce a numerical technique for computing the Markov length based on matrix-product states, and explore its dynamics under general thermal quenches in the one-dimensional classical Ising model. For all cases, we provide simple information-theoretic arguments that explain our results.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The Markov length was recently proposed as an information-theoretic diagnostic for quantum mixed-state phase transitions [Sang & Hsieh, Phys. Rev. Lett. 134, 070403 (2025)].	background 1.00	0.04	0.03	asserted 1.00
s1	Here, we show that the Markov length diverges even under classical stochastic dynamics, when a low-temperature ordered state is quenched into the high temperature phase.	result 0.90	0.93	0.63	asserted 1.00
s2	Conventional observables do not exhibit growing length scales upon quenching into the high-temperature phase; however, the Markov length grows exponentially in time.	result 1.00	0.86	0.33	asserted 1.00
s3	Consequently, the state of a system as it heats becomes increasingly non-Gibbsian, and the range of its putative "parent Hamiltonian" must diverge with the Markov length.	interpretation 0.99	0.78	0.31	asserted 0.99
s4	From this information-theoretic point of view the late-time limit of thermalization is singular.	interpretation 1.00	0.63	0.27	asserted 1.00
s5	We introduce a numerical technique for computing the Markov length based on matrix-product states, and explore its dynamics under general thermal quenches in the one-dimensional classical Ising model.	method 0.79	0.96	0.04	asserted 0.98
s6	For all cases, we provide simple information-theoretic arguments that explain our results.	contribution 0.80	0.89	0.04	asserted 0.98

What kind of paper the classifier reports this is: simulation study 0.51 · evidence from simulation or computation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Forward citations

Cited by 4 Pith papers

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

Universal Scaling of Conditional Mutual Information in Classical Stochastic Dynamics
cond-mat.stat-mech 2026-10 accept novelty 7.0 of 10

Conditional mutual information in one-dimensional classical stochastic dynamics obeys a universal scaling form with an inverse-square prefactor when the Markov length diverges with time.
Markov length can diverge in systems whose universal physics is spatially Markovian
cond-mat.stat-mech 2026-10 accept novelty 7.0 of 10

Irrelevant detailed-balance-breaking perturbations on critical Gibbs states generate power-law conditional mutual information that is cutoff-suppressed, leading to infinite Markov length at finite cutoff but not in th...
Renormalization group and long-range conditional mutual information in hierarchical models
quant-ph 2026-09 accept novelty 7.0 of 10

Existence of an ideal (locally reversible) renormalization group forces the conditional mutual information to be UV-finite, and two hierarchical models demonstrate that long-range CMI and simple RG flow can coexist.
Non-perturbative constraints on stability and renormalization group flows in nonequilibrium matter
cond-mat.stat-mech 2026-02 conditional novelty 7.0 of 10

CMI scaling function is monotonically non-increasing along RG flows in nonequilibrium systems, forbidding flows from lower to higher CMI fixed points.

How the literature uses it

2 arXiv papers cite this in 3 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 1 sentence without a confident label

Builds on it 1 paper

We also consider a more general channel that flips a $↓$($↑$) spin with probability $p$($q$) [ref] ; the case $p=q$ was also studied previously in Ref. [ref] .

Non-perturbative constraints on stability and renormalization group flows in nonequilibrium matter · Introduction · 0.89

Cites it as background 1 paper

Nevertheless, in recent years, there have been various attempts to provide a comprehensive classification of such phases [ref] , and one of the key objects that has emerged in this discussion is the Markov length.

Information Critical Phases under Decoherence · Introduction · 0.96

Discussion

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