Limitations on a device-independent key secure against a nonsignaling adversary via squashed nonlocality

We initiate a systematic study to provide upper bounds on device-independent keys, secure against a nonsignal-ing adversary (NSDI). We employ the idea of "squashing" on the secrecy monotones and show that squashed secrecy monotones are the upper bounds on the NSDI key. Our technique for obtaining upper bounds is based on the nonsignaling analog of quantum purification: the complete extension. As an important instance of an upper bound, we construct a measure of nonlocality called "squashed nonlocality." Using this bound, we identify numerically a certain domain of two binary inputs and two binary outputs of nonlocal devices for which the squashed nonlocality is zero. Therefore one can not distill a secure key from these nonlocal devices via a considered (standard) class of operations. Showing a connection of our approach to the one in New J. Phys. 8, 126 (2006), we provide, to our knowledge, the tightest known upper bound in the (3,2,2,2) scenario. Moreover, we formulate a security condition, equivalent to known ones, for the considered class of protocols. To achieve this, we introduce a nonsignaling norm that constitutes an analogy to the trace norm used in the security condition of the quantum key distribution.