Protocols for nonlocality distillation

Popescu and Rohrlich proposed in 1994 a hypothetical nonlocal box (NLB) that attains the maximum value for the CHSH inequality without allowing for communication between two spatially separated parties, Alice and Bob. Their seminal work has have long-lasting impact on how we study quantum correlations and significantly increased our understanding of why certain correlations are not allowed by quantum physics. A hypothesized world in which nonlocal boxes are available have profound implications. van~Dam showed that perfect nonlocal boxes imply trivial communication complexity for boolean functions, i.e. any boolean function may be computed by a single bit of communication between Alice and Bob. This was extended by Brassard, Buhrman, Linden, Methot, Tapp, and Unger to include nonlocal boxes that work correctly with probability greater than 0.908. Pawlowski, Paterek, Kaszlikowski, Scarani, Winter, and Zukowski showed that all strategies that violate Tsirelson’s bound also violate the principle of information causality which states that the transmission of $n$ classical bits can cause an information gain of at most $n$ bits. It is unclear if such results hold for all non-quantum correlations. Is the nonlocal box introduced by Popescu and Rohrlich a representative for all non-quantum correlations, or are there foundational differences between non-quantum correlations? In this talk, I will address these questions through the study of distillation of nonlocal boxes. A distillation process for nonlocal boxes takes a non-perfect nonlocal box and makes it more perfect. I will formalize this notion, prove the optimality of a distillation process for oblivious distillation processes, introduce a new distillation process that distill a class of non-perfect nonlocal boxes better than any previously known protocol, and present results on the possible non-existence of a single optimal distillation process for all non-perfect boxes.