On distinguishing of non-signaling boxes via completely locality preserving operations

We consider discriminating between bipartite boxes with 2 binary inputs and 2 binary outputs (2x2) using the class of completely locality preserving operations i.e. those, which transform boxes with local hidden variable model (LHVM) into boxes with LHVM, and have this property even when tensored with identity operation. Following approach developed in entanglement theory we derive linear program which gives an upper bound on the probability of success of discrimination between different isotropic boxes. 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. We also show that with help of allowed class of operations one can distinguish perfectly any two extremal boxes in 2x2 case and any local extremal box from any other extremal box in case of two inputs and two outputs of arbitrary cardinalities.