An Instability of the Quantum Markov Length

David C. Newsom; Michael Levin

REVIEW

An Instability of the Quantum Markov Length

David C. Newsom, Michael Levin

Quantum Physics · Submitted 9 Oct 2026

Pith's verdict: Strong explicit counterexamples to Markov length stability, but the claimed generalization to all rank-deficient states has a gap.

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arXiv ID2610.12929 (version 3)

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

CitationDavid C. Newsom, Michael Levin. "An Instability of the Quantum Markov Length." arXiv:2610.12929 (2026). https://pith.science/paper/UYZPLEGZ

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Editor's note

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This paper shows something real and surprising: the Markov length can jump from zero to infinite under arbitrarily weak local channels. The core mechanism—a dark mode that makes the lower correlation band touch zero—is clean and the explicit fermionic, bosonic, and qubit constructions are solid. The algebraically decaying CMI is derived rigorously (Euler-Maclaurin, Toeplitz asymptotics) and the connection to the nonanalyticity of von Neumann entropy at rank-deficient points is well explained. This is a significant result that refutes the natural stability conjecture and challenges recent definitions of mixed-state phases. I would absolutely cite this paper for the concrete examples.

The soft spot is the generalization section ("Instability beyond product states"). The argument that any translation-invariant state with a rank-deficient reduced density operator can be decomposed as σ⊗|0⟩⟨0| on each unit cell is not proved and is likely false in general. The stress-test note is correct: a Bell state on two qubits has rank 1 ≤ 2 but is not a product state under any decomposition of the Hilbert space as a tensor product. The paper gives no proof that such a factorization exists, only a claim. The rest of the section depends on this factorization. This is a load-bearing gap in the claimed generality. The authors acknowledge the restriction to rank-deficient states, but within that set the condition is not sufficient. I think the authors would need to assume something like classicality or separability of the reduced state, which they do not.

That said, the main examples stand. The paper would be better if it either proves the factorization (maybe using the fact that the state is translation invariant and the reduced state is not just rank-deficient but also has some Markov property from the finite Markov length? No, they start from a state with finite Markov length, but that state is general.) or, more honestly, weakens the claim and keeps only the product-state and explicitly constructed examples. The qubit example via Jordan-Wigner is already a nontrivial non-Gaussian example.

For peer review: yes, this deserves a serious referee. The central examples are correct and important. The generalization needs to be tightened, but it is not the heart of the paper. I would accept it with major revision requiring either a proof or a retraction of the unsupported generalization claim.

Referee report

Significance

The main conclusion—that a finite Markov length is not generally stable under weak local channels—is significant because it poses a direct challenge to recent proposals that define mixed-state phases of matter through the Markov length or local reversibility. The paper's explicit, analytically tractable examples (fermionic, bosonic, qubit, Lindbladian) make the mechanism concrete and falsifiable. If the generalization to arbitrary rank-deficient states could be rigorously justified, the result would rule out a large class of candidate definitions for mixed-state phases. The paper also correctly identifies the origin of the instability in the non-analyticity of the von Neumann entropy, which is absent for integer-order Rényi entropies—a physically important nuance.

Major objections (1)

[Instability beyond product states (pp. 4–5)] The claim that any translation-invariant state with at least one rank-deficient reduced density operator exhibits the same instability rests on an unsubstantiated decomposition. The text asserts that from rank(ρ_R) ≤ (1/2) dim(H_R) one can write H_R ≅ \tilde{H}_R ⊗ ℂ² and ρ_R ≅ σ_R ⊗ |0⟩⟨0|, and moreover that the same identification can be made in every unit cell so that ρ ≅ σ ⊗ (|0⟩⟨0|)^{⊗ Z^d}. This decomposition does not follow from the rank condition alone: for example, a Bell state on a two-qubit unit cell has rank 1 yet cannot be expressed as a tensor product of a mixed state and a pure state on a virtual qubit. Even if each unit cell admitted such a local factorization, there is no argument that the virtual qubits are uncorrelated across cells. The generalization is therefore not established by the reasoning given. The core product-state examples remain valid, but this broader claim must either be given a rigorous proof or be removed (or explicitly stated as a conjecture with the gap acknowledged).

Minor objections (2)

[Main text, footnote 16] The statement that algebraic decay in the one-sided A-B-C geometry implies algebraic decay in the two-sided C-B-A-B-C geometry via 'finite-depth factorization and data processing' is given without justification. A brief derivation or a reference to a standard property would improve readability.
[Appendix I, construction of qubit channel] The argument that the parity strings can be dropped because they act trivially on ρ_q0 is clear. However, the proof that the dropped strings do not affect the action on the entire circuit (Eq. I14–I16) could benefit from a short remark on why the commutativity relation (Eq. I12) suffices when the gates are applied in a fixed order.

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The paper

The paper claims

The quantum Markov length is unstable under arbitrarily weak local channels, even for a product state.

The reading

The paper proves that the Markov length—the length scale of exponential decay of conditional mutual information (CMI) in a many-body state—is not generally stable under weak local channels. The instability already occurs for a pure product state: a depth-2 circuit of nearest-neighbor Gaussian channels, each arbitrarily close to the identity, transforms the vacuum into a state whose CMI decays algebraically and whose Markov length is therefore infinite. The mechanism relies on a dark mode that creates a zero in the correlation matrix spectrum, making the von Neumann entropy non-analytic at the Fermi point. The authors show that the same instability exists on an ordinary qubit chain and, more generally, for any translation-invariant initial state with at least one rank-deficient reduced density operator. These results challenge recent definitions of mixed-state phase equivalence that are based on the Markov length.

What carries the argument

The central object is the correlation matrix of the Gaussian state produced by a depth-2 circuit of two-site Gaussian channels. The spectrum has two bands λ_±(k); the lower band λ_-(k) vanishes quadratically at k=0 because of a dark mode. This zero makes the entropy-sum function g(k)=h(λ_+(k))+h(λ_-(k)) non-analytic (its second derivative has a logarithmic singularity). The non-analyticity causes the Fourier coefficients ˆg(m) to decay algebraically (∼ 2γ/|m|³), which via the Euler–Maclaurin formula gives an $N^{{-2}}$ term in the ring entropy and an analogous algebraic term in the interval entropy, ultimately producing an algebraically decaying CMI.

Extended reading notes

Core claim

The authors construct a one-dimensional fermionic chain where a depth-2 circuit of nearest-neighbor Gaussian channels, each controlled by a parameter that can be made arbitrarily small, maps the vacuum to a state whose conditional mutual information decays algebraically with the buffer width, implying infinite Markov length. The key feature is a dark mode (the operator cπ = Σ_j (-1)^j c_j) that is orthogonal to all the modes coupled to ancillas and therefore remains unoccupied. This creates a zero in the lower eigenvalue band of the correlation matrix at k=0, which induces a logarithmic singularity in the function g(k)=h(λ_+(k))+h(λ_-(k)) that sums the von Neumann entropies of the two bands. The singularity causes the Fourier coefficients of g to decay as 2γ/|m|³, which in turn produces an $N^{{-2}}$ correction to the entropy of a large ring and an algebraic CMI. The same algebraic decay survives in bosonic and qubit analogs, and the mechanism is shown to be general: any translation-invariant state with a local rank-deficient reduced density operator can be dressed by a circuit of weak local channels to yield infinite Markov length.

Load-bearing premise

The initial state has at least one finite region where the reduced density operator is not full rank (rank-deficient); for a pure product state this holds, and the general proof assumes it.

Editorial extensions

If this is right

Mixed-state phase definitions that rely on finite Markov length (e.g., via local reversibility) become too restrictive: they would place a product state and the state obtained by an arbitrarily weak local channel in different phases, contradicting the natural expectation that the trivial phase should be stable under weak noise.
The instability implies that a bounded Markov length is not a robust diagnostic for topological order in mixed states: states with identical long-distance topological properties can have drastically different Markov lengths.
Integer-order Rényi CMIs (n≥2) remain exponentially decaying for the same construction, providing a way to distinguish between the instability from rank deficiency and other sources of algebraic decay.
The result identifies a concrete fine-tuning condition (a dark mode) that must be avoided if one wishes to guarantee stability of the Markov length under weak decoherence.
It sharpens the open problem of whether Markov length is stable for states with all reduced density operators full rank, such as finite-temperature Gibbs states.

Reading between the lines

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

The instability may be avoidable in classical stochastic processes: classical product distributions evolve under local maps with full-rank local marginals, and the analogous logarithmic singularity may be absent. Testing whether a classical analogue exists would clarify whether the phenomenon is intrinsically quantum.
The fine-tuning required (a dark mode) suggests that generic noisy channels without such symmetry will not produce algebraic CMI. If realistic decoherence models lack the relevant dark modes, the Markov length may still be a useful diagnostic in practice.
The result opens the possibility that some topologically ordered states (e.g., toric code under Pauli noise) might eventually develop algebraic CMI at finite noise strength if the noise can create a rank deficiency, though current evidence [6,7,12] indicates stability up to a finite threshold.

Original abstract

The Markov length, the length scale governing the exponential decay of conditional mutual information (CMI) in a quantum many-body state, has proven useful for characterizing mixed-state phases of matter. Nevertheless, we show that a state's Markov length is not generally stable under weak local channels. The failure of stability occurs already at a pure product state. We demonstrate this explicitly in a one-dimensional fermionic chain, where a fine-tuned depth-two circuit of nearest-neighbor Gaussian channels maps the vacuum to a state whose CMI decays \emph{algebraically} and whose Markov length is therefore infinite, even when the channels are taken arbitrarily close to the identity. The algebraic decay is rooted in the nonanalyticity of the von Neumann entropy at rank-deficient density operators. We exhibit an analogous instability on an ordinary qubit chain. More generally, we show that the Markov length is unstable in any translation-invariant state with even a single rank-deficient reduced density operator. These results pose a challenge to recent definitions of mixed-state phase equivalence based on the Markov length.

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, 2026-10-12: 1 request, 6,219 input tokens.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.31 · human participants 0.01 · controlled comparison 0.10 · randomized assignment 0.01 · states quantitative result 0.25 · compares with prior results 0.50 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

The Markov length, the length scale governing the exponential decay of conditional mutual information (CMI) in a quantum many-body state, has proven useful for characterizing mixed-state phases of matter. Nevertheless, we show that a state's Markov length is not generally stable under weak local channels. The failure of stability occurs already at a pure product state. We demonstrate this explicitly in a one-dimensional fermionic chain, where a fine-tuned depth-two circuit of nearest-neighbor Gaussian channels maps the vacuum to a state whose CMI decays \emph{algebraically} and whose Markov length is therefore infinite, even when the channels are taken arbitrarily close to the identity. The algebraic decay is rooted in the nonanalyticity of the von Neumann entropy at rank-deficient density operators. We exhibit an analogous instability on an ordinary qubit chain. More generally, we show that the Markov length is unstable in any translation-invariant state with even a single rank-deficient reduced density operator. These results pose a challenge to recent definitions of mixed-state phase equivalence based on the Markov length.

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, the length scale governing the exponential decay of conditional mutual information (CMI) in a quantum many-body state, has proven useful for characterizing mixed-state phases of matter.	background 1.00	0.06	0.03	asserted 0.99
s1	Nevertheless, we show that a state's Markov length is not generally stable under weak local channels.	result 0.92	0.95	0.87	asserted 1.00
s2	The failure of stability occurs already at a pure product state.	result 1.00	0.79	0.27	asserted 1.00
s3	We demonstrate this explicitly in a one-dimensional fermionic chain, where a fine-tuned depth-two circuit of nearest-neighbor Gaussian channels maps the vacuum to a state whose CMI decays \emph{algebraically} and whose Markov length is therefore infinite, even when the channels are taken arbitrarily close to the identity.	result 0.79	0.96	0.20	asserted 1.00
s4	The algebraic decay is rooted in the nonanalyticity of the von Neumann entropy at rank-deficient density operators.	interpretation 0.90	0.80	0.06	asserted 1.00
s5	We exhibit an analogous instability on an ordinary qubit chain.	result 0.84	0.94	0.06	asserted 1.00
s6	More generally, we show that the Markov length is unstable in any translation-invariant state with even a single rank-deficient reduced density operator.	result 0.97	0.95	0.41	asserted 1.00
s7	These results pose a challenge to recent definitions of mixed-state phase equivalence based on the Markov length.	interpretation 1.00	0.38	0.24	asserted 0.79

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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

Figures

**Figure 1.** FIG. 1. Geometry used to define the Markov length in two dimensions.

**Figure 2.** FIG. 2. Gate structure of the depth-

**Figure 3.** FIG. 3. Channel structure of a finite open chain containing the causal cone of

Figures reproduced from arXiv:2610.12929 by the authors.

References

Forward citations

Cited by 2 Pith papers

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

Instability of Markov chains under infinitesimal coherent dissipation
quant-ph 2026-10 accept novelty 8.0 of 10

Strictly local Lindbladian dynamics can generate infinite Markov length from a product state, even when the dynamics is arbitrarily weak.
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.

Discussion

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