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.
We find an upper bound for the minimum output entropy of a unital quantum channel, and obtain an exact formula for general qubit channels. Our techniques incorporate the Rényi entropies, particularly, with Rényi parameter α = 2. Moreover, since our upper bound is additive under tensor product, we get as a corollary an upper bound for the classical capacity of unital quantum channels. Interestingly, our upper bound for the classical capacity depends only on the operator norm of matrix representations of channels on the space of traceless Hermitian operators, and is tight in the sense that it gives the precise quantity of classical capacity of the Werner-Holevo channel. As an example, we study quantum channels with operator sum representation that is made of the discrete Weyl operators (generalized Pauli operators), and explain how our formula works in this case. Finally, we find new examples for which the minimum output Rényi 2-entropy is additive.
Additivity of the Holevo capacity is proved for product channels, under the condition that one of the channels is a unital qubit channel, with the other completely arbitrary. As a byproduct this proves that the Holevo bound is the classical information capacity of such qubit channels, and provides an explicit formula for this classical capacity. Additivity of minimal entropy and multiplicativity of p-norms are also proved under the same assumptions. The proof relies on a new bound for the p-norm of an output state from the half-noisy phase-damping channel.
The classical capacity of a quantum channel generally requires regularization because input codewords may be entangled across channel uses. We remove this regularization for generalized amplitude-damping channels (GADCs) and determine their exact strong-converse exponent throughout the full parameter range. The exponent is a single-letter transform of the sandwiched Rényi Holevo information and is attained by product input codewords with collective decoding. The key ingredient is a decomposition of the joint output into positive operators whose triangular factors have identical scalar Gram matrices after norm compression. A sharp triangular Schatten-norm compression inequality then establishes maximum output norm multiplicativity for diagonally sandwiched GADCs tensored with arbitrary completely positive maps. This yields additivity of the sandwiched Rényi Holevo information at every order above one. Consequently, the classical capacity equals the known one-copy Holevo information and admits a scalar optimization. Thus, entangled input codewords improve neither the capacity nor the optimal exponential decay of decoding success above capacity.