Secure key distribution based on Popescu-Rohrlich box fraction of dimensionally restricted nonlocality
Chellasamy Jebarathinam
TL;DR
The paper introduces dimensionally restricted nonlocality (DRNL) in the 2×2 Bell scenario and a nonlinear witness NL that detects DRNL. It defines the PR box fraction of DRNL and the Gamma measure, establishing that NL = 0 for low-dimensional Bell-local models while Gamma captures the PR-box component in broad mixtures. It then demonstrates that any DRNL correlation contains secrecy against dimensionally restricted Eve and shows that, in a CHSH-based QKD protocol, the key rate can be K^→ ≥ I(A:B) because I(A:E) = 0 under DRNL; in particular, a noisy PR box with any p_PR > 0 yields positive secrecy against such an Eve. These results indicate practical secure key distribution opportunities using DRNL as a resource, even when entanglement is not certified, under dimensionally restricted adversaries.
Abstract
For the bipartite Bell scenario with two inputs and two outputs, a nonlinear witness of dimensionally restricted nonlocality is introduced. Popescu-Rohrlich (PR) box fraction of dimensionally restricted nonlocality is then introduced and studied using the aforementioned witness and a nonlinear measure of correlations. This PR box fraction is also nonzero for certain Bell-local correlations. It is shown that any nonsignaling correlation shared by Alice and Bob that has dimensionally restricted nonlocality contains secrecy against any third party, Eve, who is also dimensionally restricted. In this context, for the specific Bell scenario considered, it is demonstrated that the PR box fraction of dimensionally restricted nonlocality can be used as a resource for secure quantum key distribution, even if entanglement is not certified.