Versatile relative entropy bounds for quantum networks

TL;DR

This paper introduces a flexible upper bound on the entanglement shared over quantum networks, allowing channel-specific measures and applying to a broad class of channels, including phase-invariant ones.

Contribution

It generalizes previous bounds by enabling channel-by-channel entanglement measures, notably using the relative entropy of entanglement for Choi-simulable channels.

Findings

Provides a versatile upper bound on entanglement in quantum networks.

Develops tools for bounding max-relative entropy of entanglement for phase-invariant channels.

Derives an analytical formula for the max-relative entropy of entanglement of the qubit amplitude damping channel.

Abstract

We provide a versatile upper bound on the number of maximally entangled qubits, or private bits, shared by two parties via a generic adaptive communication protocol over a quantum network when the use of classical communication is not restricted. Although our result follows the idea of Azuma et al. [Nat. Comm. 7, 13523 (2016)] of splitting the network into two parts, our approach relaxes their strong restriction, consisting of the use of a single entanglement measure in the quantification of the maximum amount of entanglement generated by the channels. In particular, in our bound the measure can be chosen on a channel-by-channel basis, in order to make it as tight as possible. This enables us to apply the relative entropy of entanglement, which often gives a state-of-the-art upper bound, on every Choi-simulable channel in the network, even when the other channels do not satisfy this property. We also develop tools to compute, or bound, the max-relative entropy of entanglement for channels that are invariant under phase rotations. In particular, we present an analytical formula for the max-relative entropy of entanglement of the qubit amplitude damping channel.