Markov length can diverge in systems whose universal physics is spatially Markovian

summary

The gist

The gist Markov length can diverge in systems whose universal physics is spatially Markovian Conditional Mutual Information and Phase Distinction Conditional mutual information (CMI) plays a central

In short

The paper investigates Conditional Mutual Information (CMI) to distinguish between different mixed-state phases based on local reversibility. It shows that states flowing to the same renormalization group (RG) fixed point can have qualitatively different CMI, specifically generating cutoff-suppressed power-law tails. This means Markov length can diverge even when the universal physics is spatially Markovian, challenging conventional RG universality notions.

Key concepts

Conditional Mutual Information (CMI): CMI measures how much knowing one variable helps predict another variable, given a third. In this context, it probes spatial non-Markovianity—whether the system's future depends only on its immediate past or on a longer history of interactions.
Markov Length: The Markov length refers to the distance beyond which a system's state becomes independent of its initial conditions, meaning it behaves like a Markov process. The paper shows that CMI can be algebraically decaying (power-law), which prevents the system from having a finite Markov length, even if the underlying physics is spatially Markovian.
Renormalization Group (RG) Fixed Point: The RG describes how physical properties change across different length scales. The paper explores states that flow to the same fixed point under RG but exhibit different CMI behaviors. This highlights a tension between the universal description provided by RG and the specific information-theoretic properties of a state.
Cutoff-Suppressed Power-Law CMI: This refers to algebraic decay in CMI that is suppressed by the microscopic lattice cutoff. While this decay suggests infinite Markov length at any fixed lattice scale, it vanishes in the continuum limit where the ratio of lattice size to macroscopic scales goes to zero.

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This episode discusses

The paper

Markov length can diverge in systems whose universal physics is spatially Markovian · Read on arXiv

Yu-Hsueh Chen, Tarun Grover

Department of Physics, University of California at San Diego · Kavli Institute for Theoretical Physics, University of California, Santa Barbara

Conditional mutual information (CMI) probes spatial non-Markovianity and plays a central role in a recently developed framework for defining mixed-state phases based on local reversibility. This framework sharply distinguishes exponentially decaying from algebraically decaying CMI, since the latter obstructs equivalence to a state with finite Markov length. Here we show that states that flow to the same renormalization-group (RG) fixed point can nevertheless exhibit qualitatively different CMI. Starting from a critical Gibbs state of a finite-range local classical Hamiltonian, whose CMI vanishes beyond the interaction range, we show that an RG-irrelevant perturbation that breaks detailed balance generically generates algebraically decaying CMI. Crucially, these power-law tails are cutoff-suppressed: although they lead to an infinite Markov length at every fixed lattice cutoff, they vanish as a positive power of the cutoff in the continuum limit. We construct solvable models that exhibit this phenomenon and predict that the Ising-symmetric critical point of Toom's cellular automata generically exhibits such behavior. We also find analogous cutoff-suppressed CMI in the high-temperature paramagnetic phase of the long-range Ising paramagnet, in contrast with the genuine power-law CMI at the finite-temperature critical point in this same model that survives the continuum limit. We also derive a general result that in one spatial dimension, CMI decays faster than the inverse square of the buffer length if and only if the long-distance physics is Markovian.

Transcript

Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Markov length can diverge in systems whose universal physics is spatially Markovian".

Mira: The gist Markov length can diverge in systems whose universal physics is spatially Markovian Conditional Mutual Information and Phase Distinction Conditional mutual information (CMI) plays a central role in defining…

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So, looking at the whole "Markov length can diverge in systems whose universal physics is spatially Markovian," what does this paper really suggest about the relationship between universality and phase definition?

Mira: It highlights a tension between conventional renormalization group notions of universality and channel-based notions of mixed-state phase equivalence <ref:2610.11235#pg2>.

Lev: If we boil it down, the paper predicts a cutoff-suppressed power-law CMI tail at the critical point on Toom’s symmetric line, even though that system is expected to have Markovian universal physics <ref:2610.11235#pg4>.

Kai: So, the implication for us listening here is that we have to be careful how we define what a phase actually is based on local reversibility if we are looking at systems with long-range interactions <ref:2610.11235#pg2>.

Mira: I think it points toward needing to relax the channel range in mixed-state phase equivalence from polylogarithmic to sublinear, or perhaps including short-depth channels with algebraic spatial tails if their strength is cutoff-suppressed <ref:2610.11235#pg4>.

Lev: And they also mentioned that slow relaxation dynamics can generate these cutoff-suppressed power-law mutual information and CMI even when the steady state itself has short-ranged correlations <ref:2610.11235#pg4>.

Kai: So, this paper is showing us that the divergence of Markov length in systems with Markovian physics exposes a gap between what we think universality means and what happens when you look at it through the lens of conditional mutual information <ref:2610.11235#pg1>.

Conclusion: Kai: So we're wrapping up on this paper about "Markov length can diverge in systems whose universal physics is spatially Markovian." It basically tackles that weird tension between what renormalization group theory predicts for long-range systems and what happens when you look at them through the lens of conditional mutual information.

Mira: Yeah, it’s about how CMI, which measures spatial memory and reversibility, can behave in ways that don't match the expected Markovian behavior of the underlying physics. The authors show that even if a system flows to a standard fixed point under renormalization group rules, you can still get different kinds of CMI depending on the starting point or the specific way you perturb it.

Lev: From an error correction standpoint, that distinction between exponentially decaying and algebraically decaying CMI is huge because those two types of decay tell us fundamentally different things about how long-range correlations persist in a state. One implies local reversibility, the other doesn't.

Kai: So what does this mean for the actual physical systems we’re trying to build or measure? The paper suggests that even when the universal physics *is* Markovian—meaning you expect everything to eventually become memoryless—you can still have these power-law tails in your information measures, especially at critical points.

Mira: Exactly. They show that an RG-irrelevant perturbation, something small you might ignore in a simple model, can generically generate this algebraically decaying CMI. The crucial part is that this tail is cutoff-suppressed; it vanishes as you look at the continuum limit of the lattice, even though it seems like there's an infinite Markov length at every finite lattice size.

Lev: That cutoff suppression is the real engineering hurdle, though. If we were trying to run a simulation on a finite grid and we only saw that tail, we’d think the system is non-Markovian forever. But because it vanishes in the continuum limit, it doesn't actually mess with the long-distance physics in a practical sense.

Kai: That’s interesting because it forces us to rethink how we define "phase equivalence" when using information measures rather than just looking at correlation functions. It suggests that what looks like non-Markovian behavior on a small scale might just be a finite lattice artifact that smooths out in the real world.

Mira: Right, so the main implication is that you can't automatically equate different mixed-state phases just because their RG flow ends at the same point. You need to look at the specific details of how those information measures decay.

Lev: It’s a warning for anyone trying to map out phase diagrams using these tools; you have to be careful about what assumptions you make about locality and reversibility when things get long-range.

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