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arXiv:0901.4070 · Physical Review Letters

Nonlocality Distillation and Postquantum Theories with Trivial Communication Complexity

Nicolas Brunner, Paul Skrzypczyk

Published Apr 24, 2009• 118 citations
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Abstract

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.

DOI
10.1103/physrevlett.102.160403

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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.
Via nonlocality distillation, a number of copies of a given nonlocal correlation can be turned into a new correlation displaying a higher degree of nonlocality. Apart from its clear relevance in situations where nonlocality is a resource, distillation protocols also play an important role in the understanding of information-theoretical principles for quantum theory. Here, we derive a necessary condition for nonlocality distillation from two copies and apply it, among other results, to show that one-dimensional (1D1\mathrm{D}1D) and 2D2\mathrm{D}2D quantum voids---faces of the nonlocal simplex set with no quantum realization---can be distilled up to PR boxes. With that, we generalize previous results in the literature, for instance, showing a broad class of postquantum correlations that make communication complexity trivial and violate the information causality principle.
Nonlocal correlations are a central feature of quantum theory, and understanding why quantum theory has a limited amount of nonlocality is a fundamental problem. Since nonlocality also has technological applications, e.g., for device-independent cryptography, it is useful to understand it as a resource and, in particular, whether and how different types of nonlocality can be interconverted. Here we focus on nonlocality distillation which involves using several copies of a nonlocal resource to generate one with more nonlocality. We introduce several distillation schemes which distill an extended part of the set of nonlocal correlations including quantum correlations. Our schemes are based on a natural set of operational procedures known as wirings that can be applied regardless of the underlying theory. Some are sequential algorithms that repeatedly use a two-copy protocol, while others are genuine three-copy distillation protocols. In some regions we prove that genuine three-copy protocols are strictly better than two-copy protocols. By applying our new protocols we also increase the region in which nonlocal correlations are known to give rise to trivial communication complexity. This brings us closer to an understanding of the sets of nonlocal correlations that can be recovered from information-theoretic principles, which, in turn, enhances our understanding of what is special about quantum theory.
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.
An optical scheme for simulating adaptive nonlocality distillation is proposed. In the asymptotic limit, arbitrary correlated nonlocal boxes can be distilled to the maximally nonlocal one, i.e., the Popescu-Rohrlich box [S. Popescu and D. Rohrlich, Found Phys. 24, 379 (1994)]. The major obstacle in simulating the adaptive nonlocality distillation scheme, proposed by Brunner and Skrzypczyk [N. Brunner and P. Skrzypczyk, Phys. Rev. Lett. 102, 160403 (2009)], is how to simulate the nonlocal boxes with a quantum system and to get the output of each box without destroying the quantum state simulating it. Here the nonlocal boxes are simulated by appropriately pre- and postselected polarization entangled photon pairs. The output of the first box is measured without destroying the quantum state, simulating it by exploiting a quantum nondemolition measurement, which is realized by a standard quantum teleportation process. This scheme provides a feasible tool for the experimental study of the manipulation of nonlocality in the postquantum regime and the exact physical principle precisely distinguishing physically realizable correlations from nonphysical ones.
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.