Introduction

We approach stronger that quantum correlations with a new perspective. The nonlocal box model (NLB) allows two spatially separated parties, Alice and Bob, to exhibit stronger than quantum correlations. Rather than considering a hypothetical box resource, we allow the spatially separated parties Alice and Bob, access to a trusted third party Charlie. Charlie is allowed to communicate with Alice and Bob without allowing communication between Alice and Bob. A natural generalization is to consider the case when Alice and Bob utilize quantum states for communicating with Charlie and he communicates back using quantum states as well.

We model Charlie as a quantum nonlocal box, abbreviated qNLB, which takes as input a joint quantum state and outputs a joint quantum state. A priori, such a model may not obey our non-signalling requirement since any unitary U_AB not of the form U_A⊗U_B allows for signalling. It thus may appear that a quantum generalization of the NLB model would always allow for signalling, but this only holds true if we restrict the maps to be unitary. Quantum nonlocal boxes that satisfy the non-signalling requirement and allow for quantum states as output are possible when we drop the requirement of the box being unitary. Such boxes have previously been studied under the notion of causal maps, completely positive trace-preserving maps, and non-signalling operations. Here we initiate a systematic study of such boxes in terms of nonlocality.

Nonlocality distillation occurs if it is possible for Alice and Bob to concentrate the nonlocality in n copies of an imperfect NLB/qNLB to form a stronger NLB/qNLB. Given the apparent limited distillability of NLBs, we propose a new non-adaptive protocol for nonlocality distillation of qNLBs. As our main result, we show that qNLBs exhibit strictly stronger nonlocality distillation than NLBs, for non-adaptive distillation protocols. We prove our main result by setting up a semi-definite programming framework for analyzing non-adaptive protocols for qNLB distillation. We then use this framework to define and give a protocol for qNLB distillation and show that it asymptotically distills the class of correlated quantum nonlocal boxes to the value 3.098, whereas in contrast, the optimal non-adaptive parity protocol for classical correlated nonlocal boxes asymptotically distills to the value 3.0 (Figure 1). The protocol is also proven to be an optimal non-adaptive protocol for 1, 2 and 3 copies of qNLBs by constructing a matching dual solution for the semi-definite program.

Figure 1: (a) Distillation achievable for correlated NLBs by our protocol for qNLBs and the parity protocol for NLBs. (a) Value attained by a single copy of qNLB (solid line) and NLB (dotted line). (b) Distilled value attained for n copies of qNLBs (solid lines) and NLBs (dotted lines), for n = 2 and n = 100, respectively

Motivation

A major component of the research on nonlocality can be linked to identifying a set of restrictions that produce physical theories of varying strength in terms of their correlations. From quantum strategies that violate Bell inequalities to the no-signalling principle, information causality, local quantum measurements and macroscopic locality, one of the goals is to obtain a useful understanding of the conditions that imply quantum correlations. These conditions serve as fundamental physical principles that guide the development of physical theories. One attempt to develop our understanding of the limits on nonlocality is via nonlocality distillation protocols.

The class of correlated NLBs are already known to be asymptotically distillable to a perfect NLB. This is only achieved by an adaptive protocol. In our current work we have shown that even if we restrict out attention to non-adaptive protocols, qNLBs offer improved distillation over NLBs. A generalization of our SDP approach that allows for adaptive protocols may reveal a similar improvement for adaptive protocols. This may imply distillability for correlations that are currently not known to be distillable and at the same time an increased understanding of correlations that violate principles such as information causality.

As a consequence of the work nonlocality distillation we propose a two-pronged approach for classifying correlations which are not known to satisfy the principle of macroscopic locality. We provide numerical evidence that correlations with non-trivial marginals which are not known to satisfy the macroscopic locality principle may be distillable even when corresponding correlations with trivial marginals are not. On the flip side, we argue that if reality does allow correlations that violate the principle, then nonlocality distillation could still be utilized to improve the possibility of detecting such a violation in the lab.