On Distinguishing of Non-Signaling Boxes via Completely Locality Preserving Operations

We introduce a scenario of discrimination between bipartite boxes and apply it to boxes with two binary inputs and two binary outputs (2 × 2). We develop the analogy between the theory of Bell non-locality and the theory of entanglement by considering the class of completely locality preserving (CLP) operations. A CLP operation satisfies two conditions: 1) transforms boxes with local hidden variable model (LHVM) into boxes with LHVM (i.e., is locality preserving) and 2) when tensored with an identity operation, forms a new operation, which is also locality preserving. We derive linear program, which gives an upper bound on the probability of success of discrimination between different isotropic boxes using this class of operations. In particular, we provide an upper bound on the probability of success of discrimination between isotropic boxes with the same mixing parameter. As a counterpart of entanglement monotone, we use the non-locality cost. Discrimination is restricted by the fact that non-locality cost does not increase under considered class of operations and geometry of 2 × 2 boxes. We provide an example of CLP operations, which are called comparing operations (COP). The latter operations consist of direct measurement of the shared box by both the parties, followed by a predefined strategy in order to establish the guess, which may depend on the obtained outcomes. We then show that with the help of the COP operations, one can distinguish perfectly any two extremal boxes in 2 × 2 case and any local extremal box from any other extremal box in case of two inputs and two outputs of arbitrary cardinalities.