REVIEW
Instability of Markov chains under infinitesimal coherent dissipation
Quantum Physics · Submitted 13 Oct 2026 · 32 references
Pith's verdict: Clean analytic proof that Markov length can diverge under infinitesimal local dissipation, with implications for mixed-state phases. The central claims are rigorous, but the generic relevance of the power-law tail remains an open question.
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Referee report
The paper
The paper claims
An arbitrarily short local Lindbladian instantly produces infinite Markov length from a product state.
The reading
What carries the argument
The argument rests on an exactly solvable free-fermion model with jump operators $F_j = c_j^\dagger + c_{j+1}^\dagger$. The covariance matrix evolves as $N(t) = I - e^{-t K}$ where $K$ has eigenvalues $\kappa(k) = 2+2\cos k$ that vanish quadratically at $k=\pi$. This spectral zero produces a nonanalyticity in the entropy function $h(n) = -n\log n - (1-n)\log(1-n)$ that survives in the conditional mutual information, giving a $b^{-4}$ lower bound at any $t>0$. An auxiliary-fermion construction then lifts the fermionic evolution to a strictly local spin Lindbladian via a Jordan-Wigner transformation that cancels nonlocal strings, preserving the conditional mutual information exactly.
Extended reading notes
Core claim
The central discovery is that an arbitrarily short, strictly local Lindbladian can generate a state with infinite Markov length from a product state. At any fixed positive time, the conditional mutual information between two small regions decays as a power law in the buffer size, ruling out any finite length scale. The same mechanism works for a class of rapidly relaxing Lindbladians whose stationary states have algebraic conditional mutual information for any nonzero coupling. These results show that finite Markov length is unstable under weak local dissipation, yet the states remain locally reversibly connected to a product state with sublinear light cones.
Load-bearing premise
The proof that the fermionic conditional mutual information lower bound holds uniformly for all buffer sizes beyond a threshold relies on the convergence of an infinite series and the positivity of a matrix resolvent, which is standard but could be sensitive to boundary effects on finite chains.
Editorial extensions
If this is right
- Finite Markov length is not a robust property of local dissipative dynamics: an infinitesimal amount of local Lindbladian evolution can destroy it instantly.
- Rapid relaxation (mixing) alone does not guarantee finite Markov length; the Lindbladian must also be frustration-free to achieve exponential decay, as shown by contrasting with Ref. [11].
- Infinite Markov length does not imply slow dynamics or an obstruction to phase equivalence: the constructed states are connected to a product state by locally reversible circuits with sublinear light cones and inverse-polynomial error.
- Nonlinear correlation measures such as the square-root density correlator $\operatorname{Tr}(\sqrt{\rho}\,n_0\sqrt{\rho}\,n_r)$ also acquire algebraic tails, while ordinary correlation functions remain short-ranged.
- The algebraic conditional mutual information appears in the von Neumann entropy but is absent from integer-order R\'enyi conditional mutual information, which decays exponentially.
- In the infrared limit where all regions are macroscopic, the conditional mutual information vanishes, so the algebraic tail can be viewed as a UV correction.
Reading between the lines
- This instability suggests that Markov length is a more fragile measure than the spectral gap for ground states, and the analogy between Markov length and gap is limited to frustration-free Lindbladians.
- The results sharpen the criteria for defining mixed-state phases: if superpolynomial accuracy is required, these states leave the trivial phase immediately; if polynomial accuracy is allowed, they remain connected. The choice of accuracy threshold becomes a physical assumption.
- The algebraically decaying conditional mutual information may impose limits on local error correction: decoding a region may require information from a growing buffer that scales polynomially rather than exponentially with distance.
- The mechanism (a zero in the eigenvalue spectrum of the Lindbladian generator) could appear in other settings, such as driven systems or non-Markovian baths, where analogous nonanalyticities in entropy might produce long-range conditional correlations.
Original abstract
Under local Lindbladian evolution, the growth of conditional mutual information and its associated Markov length scale is not limited by Lieb-Robinson bounds, yet it has been unclear how quickly the Markov length can increase. We show that an arbitrarily short, strictly local Lindbladian can generate infinite Markov length from a product state. We also construct strictly local Lindbladians that converge rapidly to steady states with algebraically decaying conditional mutual information. Despite their long-range conditional correlations, the states we construct are local reversibly connected to a product state by sublinear depth circuits. Thus infinite Markov length does not imply slow relaxation nor obstruct phase stability when suitably defined.
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.
The abstract, sentence by sentence
Under local Lindbladian evolution, the growth of conditional mutual information and its associated Markov length scale is not limited by Lieb-Robinson bounds, yet it has been unclear how quickly the Markov length can increase. We show that an arbitrarily short, strictly local Lindbladian can generate infinite Markov length from a product state. We also construct strictly local Lindbladians that converge rapidly to steady states with algebraically decaying conditional mutual information. Despite their long-range conditional correlations, the states we construct are local reversibly connected to a product state by sublinear depth circuits. Thus infinite Markov length does not imply slow relaxation nor obstruct phase stability when suitably defined.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Under local Lindbladian evolution, the growth of conditional mutual information and its associated Markov length scale is not limited by Lieb-Robinson bounds, yet it has been unclear how quickly the Markov length can increase. | gap 0.74 | 0.06 | 0.04 | asserted 1.00 |
| s1 | We show that an arbitrarily short, strictly local Lindbladian can generate infinite Markov length from a product state. | result 0.94 | 0.96 | 0.68 | asserted 1.00 |
| s2 | We also construct strictly local Lindbladians that converge rapidly to steady states with algebraically decaying conditional mutual information. | result 0.72 | 0.95 | 0.09 | asserted 1.00 |
| s3 | Despite their long-range conditional correlations, the states we construct are local reversibly connected to a product state by sublinear depth circuits. | result 0.93 | 0.93 | 0.11 | asserted 1.00 |
| s4 | Thus infinite Markov length does not imply slow relaxation nor obstruct phase stability when suitably defined. | interpretation 1.00 | 0.82 | 0.77 | asserted 0.99 |
What was examined, and what was not
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Figures
Showing 1 of the paper's 2 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:2610.18983 by the authors.
References
Discussion
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