Upper bounds on secure key against non-signaling adversary via non-signaling squashed secrecy monotones
Marek Winczewski, Tamoghna Das, Karol Horodecki · arXiv (Cornell University) · 2019
We initiate a systematic study to provide upper bounds on device-independent key, secure against a non-signaling adversary (NSDI), distilled by a wide class of operations, currently used in both quantum and non-signaling device-independent protocols. These operations consist of a direct measurements on the devices followed by Local Operations and Public Communication (MDLOPC). We employ the idea of squashing on the secrecy monotones, which provide upper bounds on the key rate in secret key agreement (SKA) scenario, and show that secrecy monotones are the upper bounds on NSDI key. As an important instance, an upper bound on NSDI key rate called squashed non-locality, has been constructed. It exhibits several important properties, including convexity, monotonicity, additivity on tensor products, and asymptotic continuity. Using this bound, we identify numerically a domain of two binary inputs and two binary outputs non-local devices for which the non-locality is zero, and therefore one can not distil key from them via MDLOPC operations. These are mixtures of Popescu-Rohrlich (PR) and anti-PR box with the weight of PR box less than $80\%$. This example confirms the intuition that non-locality need not imply secrecy in the non-signaling scenario. The approach is general, describing how to construct other tighter yet possibly less computable upper bounds. Our technique for obtaining upper bounds is based on the non-signaling analog of quantum purification: the complete extension, which yields equivalent security conditions as previously known in the literature.