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

TL;DR

This paper derives new upper bounds on the energy-constrained quantum and private capacities of phase-insensitive bosonic Gaussian channels, using data-processing and approximate degradability techniques, and explores optimal input states for channel divergence.

Contribution

It introduces several novel upper bounds on capacities of bosonic Gaussian channels, including the data-processing and ε-degradable bounds, and analyzes optimal Gaussian inputs for channel divergence.

Findings

Data-processing bound is within 1.45 bits of known lower bounds.

Strong limitations on superadditivity of coherent information in low-noise regimes.

Gaussian states, specifically two-mode squeezed vacuum, are optimal for certain channel divergence measures.

Abstract

We establish several upper bounds on the energy-constrained quantum and private capacities of all single-mode phase-insensitive bosonic Gaussian channels. The first upper bound, which we call the "data-processing bound," is the simplest and is obtained by decomposing a phase-insensitive channel as a pure-loss channel followed by a quantum-limited amplifier channel. We prove that the data-processing bound can be at most 1.45 bits larger than a known lower bound on these capacities of the phase-insensitive Gaussian channel. We discuss another data-processing upper bound as well. Two other upper bounds, which we call the "$\varepsilon$-degradable bound" and the "$\varepsilon$-close-degradable bound," are established using the notion of approximate degradability along with energy constraints. We find a strong limitation on any potential superadditivity of the coherent information of any phase-insensitive Gaussian channel in the low-noise regime, as the data-processing bound is very near to a known lower bound in such cases. We also find improved achievable rates of private communication through bosonic thermal channels, by employing coding schemes that make use of displaced thermal states. We end by proving 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. What remains open for several interesting channel divergences, such as the diamond norm or the Renyi channel divergence, is to determine whether, among all input states, a Gaussian state is optimal.