Optimal bounds for correlations via semidefinite programming

Nonlocal boxes (NLBs) are devices that are used to generate correlated bits between two spatially separated parties. If the generated correlation is weak, it can sometimes be distilled into a stronger correlation by repeated applications of the nonlocal box. We initiate here a study of the distillation of correlations for nonlocal boxes that output quantum states rather than classical bits. We give a semidefinite programming formulation of when distillation is possible under non-adaptive protocols. We propose a protocol for distillation and show that it asymptotically distills a class of correlated quantum nonlocal boxes to the value 12 (3 √ 3 + 1) ≈ 3.09876, whereas in contrast, the optimal non-adaptive parity protocol for classical nonlocal boxes asymptotically distills only to the value 3.0. We then show that our protocol is an optimal non-adaptive protocol for 1, 2 and 3 copies by proving a matching upper bound. Our work demonstrates that some quantum nonlocal boxes exhibit stronger distillability than their classical counterpart and provides evidence that qNLBs are a stronger resource for nonlocality than NLBs. 1 Nonlocality distillation Consider two parties, Alice and Bob, spatially separated and isolated, interested in jointly computing some boolean function f(·, ·). A third party, Charlie, provides Alice with an input wA (unbeknown to Bob) and Bob with an input wB (unbeknown to Alice) and challenges them to compute the bit f(wA, wB). Charlie allows Alice and Bob to communicate, but charges for each and every bit communicated between Alice and Bob, whom therefore have pre-agreed upon a protocol that minimizing the amount of communication required for them to compute the bit f(wA, wB). This is what we know as communication complexity. It seems entirely impossible to jointly compute a non-trivial function if no information can be interchanged between Alice and Bob, and it is indeed one of the first results typically shown in any introduction to communication complexity. But as soon as one tweaks the models ever so slightly, surprising results are possible. This paper is about understanding one such tweaking and its implications for our understanding of computational resources and of the physical world. A nonlocal box is a device shared between two parties that, when defined as we do here, in itself is incapable of transfering any information from Alice to Bob, or vice-versa, from Bob to Alice. Yet it renders all of communication complexity trivial. A nonlocal box takes as input two bits, a bit x from Alice and a bit y from Bob, and outputs two bits, a bit a provided to Alice (and only Alice) and a bit b provided to Bob (and only Bob). If the two input bits x and y from Alice and Bob equal (0, 0), (0, 1), or (1, 0), the box