Symmetry-Enforced Quadratic Degradability Beyond Low Dimensions

Abstract

Approximate degradability provides a powerful framework for bounding the quantum and private capacities of noisy quantum channels in regimes where exact degradability fails. While generic low-noise channels exhibit a non-degradability parameter that decays as a fractional power of the noise strength, certain symmetric channels are known to display an enhanced quadratic suppression. In this work, we investigate the structural origin of this phenomenon through a family of high-dimensional, rotationally symmetric noise models constructed from angular momentum operators. We first establish that the pure noise component of these channels is maximally distinguishable from the identity channel in diamond norm, revealing a geometric orthogonality between signal and noise. Building on this structure, we construct an explicit symmetric degrading map and prove that the approximate degradability parameter scales quadratically with the noise parameter for all system dimensions. To clarify the mechanism behind this behavior, we identify algebraic conditions on the noise operators that guarantee the cancellation of leading-order non-degradability terms. These conditions apply not only to the rotationally symmetric model studied here, but also to a distinct family of high-dimensional depolarizing channels based on discrete unitary operator bases. Numerical evaluations of capacity lower bounds further illustrate the practical impact of the quadratic suppression. Together, these results demonstrate that enhanced approximate degradability arises from symmetry-induced orthogonality and invariance properties, rather than from low-dimensional or model-specific effects.