Tamoghna Das

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 formulate a security condition for the considered class of protocols, that is based on the newly introduced non-signaling norm. This norm takes supremum over certain operations, that can be used to discriminate devices. It is shown that the security condition based on this norm, is equivalent to two security conditions present in the literature. 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 squashed 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, and additivity on tensor products. Using this bound, we identify numerically a domain of two binary inputs and two binary outputs non-local devices for which the squashed non-locality is zero. Therefore one can not distill key from them via MDLOPC operations. These are mixtures of Popescu-Rohrlich (PR) and anti-PR box with the weight of PR 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. This extension provides the ultimate eavesdropping power with the minimal consumption of eavesdropper's memory and, as we prove, yields equivalent security conditions as previously known in the literature.