Relative Entropy Bounds on Quantum, Private and Repeater Capacities

TL;DR

This paper establishes a strong-converse bound on the private capacity of quantum channels with unlimited two-way classical communication, using max-relative entropy of entanglement and new divergence inequalities, with implications for quantum repeaters.

Contribution

Introduces a novel strong-converse bound on private quantum capacity based on max-relative entropy, improving existing bounds and analyzing repeater capacities.

Findings

Bound surpasses previous transposition and squashed entanglement bounds

Provides explicit examples of channels with negligible private repeater capacity

Uses new inequalities for sandwiched Rényi divergences

Abstract

We find a strong-converse bound on the private capacity of a quantum channel assisted by unlimited two-way classical communication. The bound is based on the max-relative entropy of entanglement and its proof uses a new inequality for the sandwiched R\'{e}nyi divergences based on complex interpolation techniques. We provide explicit examples of quantum channels where our bound improves both the transposition bound (on the quantum capacity assisted by classical communication) and the bound based on the squashed entanglement introduced by Takeoka et al.. As an application we study a repeater version of the private capacity assisted by classical communication and provide an example of a quantum channel with negligible private repeater capacity.