Homogeneous product-noise violations of the Rényi hierarchy

Determine whether homogeneous product Pauli noise of the form \(\nu^{\otimes N}\) can violate the monotonic Rényi coherent-information hierarchy for some stabilizer code.

Background

The paper proves that Rényi coherent information is nondecreasing in the integer Rényi order for finite stabilizer codes when the Pauli noise is generated by independent Bernoulli events. It also constructs inhomogeneous product Pauli-noise counterexamples showing that the hierarchy can fail outside this independent-event class.

The authors note that their counterexample uses inhomogeneous single-qubit noise, whereas numerical checks for certain homogeneous product channels did not reveal a violation. They therefore leave unresolved whether a homogeneous product Pauli channel $\nu^{\otimes N}$ can produce a hierarchy violation for any stabilizer code.