Continuity of certain quantum channel divergences
Abstract
The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities $0$ and $1$. The remedy is the Arveson--Radon--Nikodym derivative T of one map with respect to the sum Ω, whose transpose is the environment effect selecting that map in a Stinespring dilation of Ω. In T, these quantities and all maximal f-divergences of operator convex f are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most 2|T1−T2|∞1/2, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.
Related
- Continuity of Regularized Channel Rényi Divergences
- Divergences in Nonpolynomial Quantum Field Theories
- On Estimating Quantum Divergences with p-Power Means
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- Counterexamples to the Tensorization Property of Strong Data Processing Inequalities for Quantum Divergences
BibTeX
@misc{jones2026continuity,
title = {{Continuity of certain quantum channel divergences}},
author = {Morgan Jones},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16270},
eprint = {2610.16270},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16270},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Continuity of certain quantum channel divergences
AU - Morgan Jones
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.16270
UR - https://arxiv.org/abs/2610.16270
AB - The geometric channel fidelity, the geometric Rényi divergences and the Belavkin--Staszewski relative entropy give converse bounds for capacities, discrimination and estimation. Channels are known only approximately, yet two pairs of channels with equal sums can be arbitrarily close in the diamond norm and have fidelities $0$ and $1$. The remedy is the Arveson--Radon--Nikodym derivative T of one map with respect to the sum Ω, whose transpose is the environment effect selecting that map in a Stinespring dilation of Ω. In T, these quantities and all maximal f-divergences of operator convex f are continuous, with explicit dimension-free moduli; for two pairs of channels with the same sum the fidelities differ by at most 2|T_1−T_2|_∞^1/2, which gives Hölder continuity in the diamond norm on such splittings. The fidelity is continuous at a pair of completely positive maps exactly when the Choi matrix of the sum is invertible or the fidelity vanishes: noisy channels of full Kraus rank are continuity points, unitary gates are not.
ER -