Device-independent lower bounds on the conditional von Neumann entropy

TL;DR

This paper introduces a numerical method to compute lower bounds on the rates of device-independent quantum protocols, significantly improving accuracy and applicability, including finite round security proofs.

Contribution

The authors develop a convergent hierarchy of optimization problems relaxed to semidefinite programs for lower bounding the conditional von Neumann entropy in device-independent protocols.

Findings

Achieves higher rate bounds for DI-RE and DI-QKD protocols.

Demonstrates minimal detection efficiency threshold within current experimental capabilities.

Rapid convergence to known analytical bounds up to several decimal places.

Abstract

The rates of several device-independent (DI) protocols, including quantum key-distribution (QKD) and randomness expansion (RE), can be computed via an optimization of the conditional von Neumann entropy over a particular class of quantum states. In this work we introduce a numerical method to compute lower bounds on such rates. We derive a sequence of optimization problems that converge to the conditional von Neumann entropy of systems defined on general separable Hilbert spaces. Using the Navascu\'es-Pironio-Ac\'in hierarchy we can then relax these problems to semidefinite programs, giving a computationally tractable method to compute lower bounds on the rates of DI protocols. Applying our method to compute the rates of DI-RE and DI-QKD protocols we find substantial improvements over all previous numerical techniques, demonstrating significantly higher rates for both DI-RE and DI-QKD. In particular, for DI-QKD we show a minimal detection efficiency threshold which is within the realm of current capabilities. Moreover, we demonstrate that our method is capable of converging rapidly by recovering all known tight analytical bounds up to several decimal places. Finally, we note that our method is compatible with the entropy accumulation theorem and can thus be used to compute rates of finite round protocols and subsequently prove their security.