Device-independent lower bounds on the conditional von Neumann entropy

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.

Device-independent Lower Bounds on the Conditional von Neumann Entropy

The paper titled "Device-independent lower bounds on the conditional von Neumann entropy" introduces a novel numerical method for computing lower bounds on the rates of several device-independent (DI) quantum protocols. The authors focus on protocols such as quantum key distribution (QKD) and randomness expansion (RE), which are fundamental to quantum cryptography. These protocols are distinctive because they rely on minimal assumptions about the devices' underlying quantum systems, providing robust security even when devices are untrusted.

Methodology

The essential development in this work is the derivation of a sequence of optimization problems that approximate the conditional von Neumann entropy. The convergence to this entropy is established for systems defined on general separable Hilbert spaces. To achieve practical computation, these problems are relaxed into semidefinite programs (SDPs) via the Navascués-Pironio-Acin (NPA) hierarchy, making the approach computationally feasible.

Numerical Results and Implications

The authors apply their method to calculate the rates for DI-RE and DI-QKD protocols, reporting substantial improvements over previous techniques. Notably, for DI-QKD, the approach determines a minimally required detection efficiency within current experimental capabilities. This result is significant as it indicates the potential for implementing DI-QKD protocols with existing technology.

Furthermore, the paper demonstrates the method's rapid convergence by effectively reproducing known analytical bounds to high precision. This rapid convergence is a crucial advantage, providing confidence in the method's robustness and accuracy.

Theoretical Contributions

The authors extend the analysis beyond finite-dimensional systems, making it applicable to infinite-dimensional ones, thus broadening the scope of potential applications. The method's compatibility with the entropy accumulation theorem also facilitates the evaluation of finite-round protocol rates, enabling security proofs for practical implementations.

Future Directions

This paper sets the stage for several future research directions. One potential avenue is the further optimization of the SDP relaxations for increased computational efficiency. Another interesting research domain would be applying this method to protocols with more complex interaction structures or mixed dimensions in quantum systems. Additionally, exploring practical implementations of the derived minimal detection efficiencies in real-world DI-QKD setups would be a valuable step towards advancing quantum cryptography's operational reach.

Overall, this research provides a robust, scalable framework for advancing DI quantum protocols by leveraging mathematical rigor and computational innovation, aligning with the ongoing evolution of secure quantum communications.