Device-independent quantum key distribution with asymmetric CHSH inequalities

Abstract

The simplest device-independent quantum key distribution protocol is based on the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality and allows two users, Alice and Bob, to generate a secret key if they observe sufficiently strong correlations. There is, however, a mismatch between the protocol, in which only one of Alice's measurements is used to generate the key, and the CHSH expression, which is symmetric with respect to Alice's two measurements. We therefore investigate the impact of using an extended family of Bell expressions where we give different weights to Alice's measurements. Using this family of asymmetric Bell expressions improves the robustness of the key distribution protocol for certain experimentally-relevant correlations. As an example, the tolerable error rate improves from 7.15% to about 7.42% for the depolarising channel. Adding random noise to Alice's key before the postprocessing pushes the threshold further to more than 8.34%. The main technical result of our work is a tight bound on the von Neumann entropy of one of Alice's measurement outcomes conditioned on a quantum eavesdropper for the family of asymmetric CHSH expressions we consider and allowing for an arbitrary amount of noise preprocessing.

Violation of Bell's inequality with quantum-dot single-photon sources

“…It is therefore very likely that a similar advantage can be gained for this system. It has, however, recently been proven that the security of some generalised DIQKD pro-tocols involving non-maximally entangled states can be bounded to a similar threshold as the CHSH inequality [26,27]. It is thus unclear if a similar advantage can also be obtained for DIQKD.…”

Simple and practical DIQKD security analysis via BB84-type uncertainty relations and Pauli correlation constraints

“…This again follows by identifying A 1 and Ā1 with the Z and X operators and reusing a one-sided deviceindependent bound known for BB84 with noisy preprocessing [12,18].…”

“…The two above bounds were used in [12] to analyze the security of a family of CHSH-based DIQKD protocols. But more generally, it is also possible to obtain other bounds, such as the two ones below, which we will apply to other variants of CHSH-based DIQKD protocols in Section III.…”

“…In this work, we present a generic approach to bound the conditional entropy in the 2-input/2-output deviceindependent setting that is conceptually and technically relatively simple. It is a generalization of the approach in [12] that was used to derive an analytical bound on the conditional entropy for a family of asymmetric CHSH inequalities. As we explain here, the main conceptual steps of this security analysis are not specific to the protocol considered in [12] but can actually be easily adapted to most 2-input/2-output device-independent protocols.…”

“…It is a generalization of the approach in [12] that was used to derive an analytical bound on the conditional entropy for a family of asymmetric CHSH inequalities. As we explain here, the main conceptual steps of this security analysis are not specific to the protocol considered in [12] but can actually be easily adapted to most 2-input/2-output device-independent protocols.…”