Stability of Markov length for full-rank initial states

Determine whether the Markov length remains stable under sufficiently weak finite-depth circuits of local channels when the initial state has full-rank reduced density operators on every finite region, including whether stability holds for the maximally mixed state or finite-temperature Gibbs states.

Background

The paper demonstrates that arbitrarily weak finite-depth local channels can transform a pure product state with zero Markov length into a state with algebraically decaying conditional mutual information and infinite Markov length. The mechanism depends on rank-deficient reduced density operators, which create entropy nonanalyticities that generate algebraic finite-size corrections.

The authors explicitly leave unresolved whether this instability persists for states whose reduced density operators are all full rank. They identify the maximally mixed state and finite-temperature Gibbs states as important candidate classes for possible stability, noting that existing partial stability results provide some evidence in that direction.

References

One question raised by our work is whether the Markov length can be stable for other choices of the initial state \rho_{0}. The instability we have demonstrated here relies on \rho_{0} having at least one rank-deficient reduced density operator. We therefore cannot rule out the possibility that the Markov length is stable for initial states whose reduced density operators are all full rank. In particular, the Markov length may be stable for the maximally mixed state or, more generally, for finite-temperature Gibbs states.

— An Instability of the Quantum Markov Length  (2610.12929 - Newsom et al., 9 Oct 2026) in Discussion section, paragraph beginning “One question raised by our work…”

Another question is whether the instability of the Markov length found here is intrinsically quantum. Specifically, consider a lattice of classical spins with finite local state spaces, initially described by a product probability distribution. Must the distribution produced by any finite-depth circuit of sufficiently weak local stochastic maps have exponentially decaying CMI? More generally, understanding the stability of the Markov length in classical many-body systems is an interesting direction for future work.

— An Instability of the Quantum Markov Length  (2610.12929 - Newsom et al., 9 Oct 2026) in Discussion section, paragraph beginning “Another question is whether the instability…”