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Quantum Nonlocal Boxes Exhibit Stronger Distillability

Peter Høyer, Jibran Rashid

Published Aug 15, 2016• 1 citation
OpenAlex

Abstract

The hypothetical nonlocal box (NLB) proposed by Popescu and Rohrlich allows two spatially sepa-rated parties, Alice and Bob, to exhibit stronger than quantum correlations. If the generated correlations are weak, they can sometimes be distilled into a stronger correlation by repeated applications of the NLB. Motivated by the limited distillability of NLBs, we initiate here a study of the distillation of correlations for nonlocal boxes that output quantum states rather than classical bits (qNLBs). We propose a new pro-tocol for distillation and show that it asymptotically distills a class of correlated quantum nonlocal boxes to the value 12 (3 3 + 1) ≈ 3.098076, whereas in contrast, the optimal non-adaptive parity protocol for classical nonlocal boxes asymptotically distills only to the value 3.0. We show that our protocol is an optimal non-adaptive protocol for 1, 2 and 3 qNLB copies by constructing a matching dual solution for the associated primal semidefinite program (SDP). We conclude that qNLBs are a stronger resource for nonlocality than NLBs. The main premise that develops from this conclusion is that the NLB model is not the strongest resource to investigate the fundamental principles that limit quantum nonlocality. As such, our work provides strong motivation to reconsider the status quo of the principles that are known

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Non-Locality distillation in tripartite NLBs

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The hypothetical nonlocal box (NLB) proposed by Popescu and Rohrlich allows two spatially sepa-rated parties, Alice and Bob, to exhibit stronger than quantum correlations. If the generated correlations are weak, they can sometimes be distilled into a stronger correlation by repeated applications of the NLB. Motivated by the limited distillability of NLBs, we initiate here a study of the distillation of correlations for nonlocal boxes that output quantum states rather than classical bits (qNLBs). We propose a new pro-tocol for distillation and show that it asymptotically distills a class of correlated quantum nonlocal boxes to the value 12 (3 3 + 1) ≈ 3.098076, whereas in contrast, the optimal non-adaptive parity protocol for classical nonlocal boxes asymptotically distills only to the value 3.0. We show that our protocol is an optimal non-adaptive protocol for 1, 2 and 3 qNLB copies by constructing a matching dual solution for the associated primal semidefinite program (SDP). We conclude that qNLBs are a stronger resource for nonlocality than NLBs. The main premise that develops from this conclusion is that the NLB model is not the strongest resource to investigate the fundamental principles that limit quantum nonlocality. As such, our work provides strong motivation to reconsider the status quo of the principles that are known
The hypothetical nonlocal box (NLB) proposed by Popescu and Rohrlich allows two spatially separated parties, Alice and Bob, to exhibit stronger than quantum correlations. If the generated correlations are weak, they can sometimes be distilled into a stronger correlation by repeated applications of the NLB. Motivated by the limited distillability of NLBs, we initiate here a study of the distillation of correlations for nonlocal boxes that output quantum states rather than classical bits (qNLBs). We propose a new protocol for distillation and show that it asymptotically distills a class of correlated quantum nonlocal boxes to the value [Formula: see text], whereas in contrast, the optimal non-adaptive parity protocol for classical nonlocal boxes asymptotically distills only to the value 3.0. We show that our protocol is an optimal non-adaptive protocol for 1, 2 and 3 qNLB copies by constructing a matching dual solution for the associated primal semidefinite program (SDP). We conclude that qNLBs are a stronger resource for nonlocality than NLBs. The main premise that develops from this conclusion is that the NLB model is not the strongest resource to investigate the fundamental principles that limit quantum nonlocality. As such, our work provides strong motivation to reconsider the status quo of the principles that are known to limit nonlocal correlations under the framework of qNLBs rather than NLBs.
We first present a protocol for deterministically distilling nonlocality, building upon a recent result of Forster et al. [Phys. Rev. Lett. 102, 120401 (2009)10.1103/PhysRevLett.102.120401]. Our protocol, which is optimal for two-copy distillation, works efficiently for a specific class of postquantum nonlocal boxes, which we term correlated nonlocal boxes. In the asymptotic limit, all correlated nonlocal boxes are distilled to the maximally nonlocal box of Popescu and Rohrlich. Then, taking advantage of a result of Brassard et al. [Phys. Rev. Lett. 96, 250401 (2006)10.1103/PhysRevLett.96.250401] we show that all correlated nonlocal boxes make communication complexity trivial, and therefore appear very unlikely to exist in nature. Astonishingly, some of these nonlocal boxes are arbitrarily close to the set of classical correlations. This result therefore gives new insight to the problem of why quantum nonlocality is limited.
We prove that the PARITY protocol is optimal for a general class of non-adaptive distillation protocols of all nnn player nonlocal boxes (NLBs) based on XOR games. The conditional distributions generated by these NLBs are assumed to have trivial local marginals. We also show that already for n=2n=2n=2, PARITY is no longer optimal if the local marginals are non-trivial. The OR protocol is shown to perform better and in the process also slightly extend the known correlations that collapse communication complexity. This emphasizes again the need to understand the local properties of nonlocal systems in order to obtain a better characterization of the global behavior. We conclude by showing an equivalence between adaptive distillation protocols that use identical NLBs and PARITY protocol using nonidentical NLBs.
Forster et al. recently showed that weak nonlocality can be amplified by giving the first protocol that distills a class of nonlocal boxes (NLBs) [Phys. Rev. Lett. 102, 120401 (2009)] We first show that their protocol is optimal among all nonadaptive protocols. We next consider adaptive protocols. We show that the depth-2 protocol of Allcock et al. [Phys. Rev. A 80, 062107 (2009)] performs better than previously known adaptive depth-2 protocols for all symmetric NLBs. We present a depth-3 protocol that extends the known region of distillable NLBs. We give examples of NLBs for which each of the Forster et al., the Allcock et al., and our protocols perform best. The understanding we develop is that there is no single optimal protocol for NLB distillation. The choice of which protocol to use depends on the noise parameters for the NLB.
Non-local correlations are one of the most fascinating consequences of quantum physics from the point of view of information: Such correlations, although not allowing for signaling, are unexplainable by pre-shared information. The correlations have applications in cryptography, communication complexity, and sit at the very heart of many attempts of understanding quantum theory - and its limits - better in terms of classical information. In these contexts, the question is crucial whether such correlations can be distilled, i.e., whether weak correlations can be used for generating (a smaller amount of) stronger. Whereas the question has been studied quite extensively for bipartite correlations (yielding both pessimistic and optimistic results), only little is known in the multi-partite case. We show that a natural generalization of the well-known Popsecu-Rohrlich box can be distilled, by an adaptive protocol, to the algebraic maximum. We use this result further to show that a much bigger class of correlations, including all purely three-partite correlations, can be distilled from arbitrarily weak to maximal strength with partial communication, i.e., using only a subset of the channels required for the creation of the same correlation from scratch. In other words, we show that arbitrarily weak non-local correlations can have a “communication value” in the context of the generation of maximal non-locality.
In quantum mechanics some spatially separated sub-systems behave as if they are part of a single system, the superposition of states of this single system cannot be written as products of states of individual sub-systems,we say that the state of such system is entangled, such systems give rise to non-local correlations between outcomes of measurements. The non-local correlations are conditional probability distributions of some measurement outcomes given some measurement settings and cannot be explained by shared information.These correlations can be studied using a non-local box(NLB) which can be viewed as a quantum systema. A NLB is an abstract object which has number of inputs(measurement settings) and number of outputs(outcomes), such NLBs can be both quantum and super-quantum. The correlations are of use in quantum information theory, the stronger the correlations the more useful they are, hence we study protocols that have multiple weaker non-local systems, application of these protocols to weaker systems may result in stronger non-local correlations, we call such protocols non-locality distillation protocols. In our work here we present non-locality distillation protocols for tripartite NLBs specifically GHZ box and class 44,45 and 46 of no-signalling polytope.