Sublinear-scaling approximation parameters for the amplitude damping channel

Develop approximation parameters for the amplitude damping channel A_p—measuring proximity to properties such as covariance, entanglement-breaking, or Hadamardness—that scale sublinearly in the damping parameter p, so that continuity-based capacity bounds become nontrivial and tangent to the Holevo information near p=0.

Background

The continuity-based upper bounds in this work depend on diamond-norm approximation parameters, and the error term involves functions with infinite slope at zero when the parameter scales linearly in p. For A_p, all considered parameters (covariance, entanglement-breaking, Hadamard) scale at least linearly, rendering the resulting bounds trivial for small p.

Identifying approximation parameters with sublinear behavior in p would yield tighter, nontrivial bounds that improve on existing strong-converse bounds for the amplitude damping channel.