Efficiently Computable Limits on EPR Pair Generation in Quantum Broadcast Channels

TL;DR

This paper develops a framework for analyzing and bounding the capacity of quantum broadcast channels to generate EPR pairs, using semidefinite programming and causal constraints, with implications for quantum communication and entanglement distribution.

Contribution

It introduces a hierarchy of semidefinite programming bounds for quantum broadcast channel capacities and characterizes the role of no-signalling and PPT-preserving codes in this context.

Findings

Derived semidefinite programs for achievable entanglement fidelity.

Established hierarchy of converse bounds for quantum broadcast channel capacity.

Revealed connections between PPT-preserving codes and entanglement combing schemes.

Abstract

We investigate the generation of EPR pairs between three observers in a general causally structured setting, where communication occurs via a noisy quantum broadcast channel. The most general quantum codes for this setup take the form of tripartite quantum channels. Since the receivers are constrained by causal ordering, additional temporal relationships naturally emerge between the parties. These causal constraints enforce intrinsic no-signalling conditions on any tripartite operation, ensuring that it constitutes a physically realizable quantum code for a quantum broadcast channel. We analyze these constraints and, more broadly, characterize the most general quantum codes for communication over such channels. We examine the capabilities of codes that are fully no-signalling among the three parties, positive partial transpose (PPT)-preserving, or both, and derive simple semidefinite programs to compute the achievable entanglement fidelity. We then establish a hierarchy of semidefinite programming converse bounds -- both weak and strong -- for the capacity of quantum broadcast channels for EPR pair generation, in both one-shot and asymptotic regimes. Notably, in the special case of a point-to-point channel, our strong converse bound recovers and strengthens existing results. Finally, we demonstrate how the PPT-preserving codes we develop can be leveraged to construct PPT-preserving entanglement combing schemes, and vice versa.