Joint small-margin dependence for the noisy addressed phase family

Determine the optimal dependence of the sequential response capacity of the measured-address quantum phase family with known dephasing noise on the probability margin, beyond the fixed-margin bounds established for <\gamma\leq1/16.

Background

The paper establishes matching upper and lower bounds of order RTlog⁡2(1+min⁡{T,1/p})RT\log_2(1+\min\{T,1/p\}) for the measured-address family of independently varying phase channels subject to known dephasing, but only for a fixed small probability margin 0<γ≤1/160<\gamma\leq1/16. The authors explicitly leave unresolved the optimal joint dependence on the margin γ\gamma and the duration, address dimension, and noise strength.

Resolving this problem would refine the fixed-margin capacity law into a sharp small-margin characterization for noisy coherent phase processes, including the transition between the coherence-limited and noiseless regimes.

References

Theorem~\ref{main:noisy-law} holds at fixed margin; the optimal joint small-margin dependence remains open.

— Sequential Capacity of Quantum Processes with Finite Memory  (2610.02068 - Wang, 1 Oct 2026) in Section 3, paragraph following Table 1 (theorem on capacity with known dephasing)