Approaches for approximate additivity of the Holevo information of quantum channels

Abstract

We study quantum channels that are close to another channel with weakly additive Holevo information and derive upper bounds on their classical capacity. Examples of channels with weakly additive Holevo information are entanglement-breaking channels, unital qubit channels, and Hadamard channels. Related to the method of approximate degradability, we define approximation parameters for each class above that measure how close an arbitrary channel is to satisfying the respective property. This gives us upper bounds on the classical capacity in terms of functions of the approximation parameters, as well as an outer bound on the dynamic capacity region of a quantum channel. Since these parameters are defined in terms of the diamond distance, the upper bounds can be computed efficiently using semidefinite programming (SDP). We exhibit the usefulness of our method with two example channels: a convex mixture of amplitude damping and depolarizing noise, and a composition of amplitude damping and dephasing noise. For both channels, our bounds perform well in certain regimes of the noise parameters in comparison to a recently derived SDP upper bound on the classical capacity. Along the way, we define the notion of a generalized channel divergence (which includes the diamond distance as an example), and we prove that for jointly covariant channels these quantities are maximized by purifications of a state invariant under the covariance group. This latter result may be of independent interest.

Bounding the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels

“…As one of the last technical developments of our paper, we address the question of computing energyconstrained channel distances in a very broad sense, by considering the energy-constrained, generalized channel divergence of two quantum channels, as an extension of the generalized channel divergence developed in [LKDW18]. In particular, we prove that an optimal Gaussian input state for the energy-constrained, generalized channel divergence of two particular Gaussian channels is the two-mode squeezed vacuum state that saturates the energy constraint.…”

“…We finally used the generalized channel divergence from [LKDW18] to address the question of optimal input states for the energy-bounded diamond norm and other related divergences. In particular, we showed that for two Gaussian channels that are jointly phase and displacement covariant, the Gaussian energy-constrained generalized channel divergence is achieved by a two-mode squeezed vacuum state that saturates the energy constraint.…”

“…In this section, we address the question of computing the energy-constrained diamond norm of several channels of interest that have appeared in our paper. We provide a very general argument, based on some definitions and results in [LKDW18] and phrased in terms of the 'generalized channel divergence' as a measure of the distinguishability of quantum channels. We find that, among all Gaussian input states with a fixed energy constraint, the two-mode squeezed vacuum state saturating the energy constraint is the optimal state for the energy-constrained generalized channel divergence of two particular Gaussian channels.…”

Quantifying the magic of quantum channels

“…Recalling the definition of the channel mana in terms of the discrete Wigner function (see (61)) and abbreviating X CPWP as Ξ, consider that…”

“…Examples of generalized divergences, in addition to the trace distance and relative entropy, include the Petz-Rényi relative entropies [56], the sandwiched Rényi relative entropies [57,58], the Hilbert α-divergences [59], and the c 2 divergences [60]. One can then define the generalized channel divergence [61], as a way of quantifying the distinguishability of two quantum channels   A B and   A B , as follows:…”