Nonlocality Distillation and Postquantum Theories with Trivial Communication Complexity

We first present a protocol for deterministically distilling nonlocality, building upon a recent result of Forster et al. [Phys. Rev. Lett. 102, 120401 (2009)]. 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)] 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.

Figure 1

The set of quantum nonlocal correlations $\mathcal{Q}$ is a strict subset of the correlations constrained solely by the nonsignaling principle $\mathcal{N}\mathcal{L}$. In particular, there are postquantum correlations above, but also below Tsirelson’s bound ${\mathcal{B}}_Q=2\sqrt{2}$. Postquantum correlations violating the CHSH inequality by more than ${\mathcal{B}}_{cc}\approx 3.266$, are very unlikely to exist, since they make communication complexity trivial. Here we show that a similar conclusion holds for correlated nonlocal boxes (bold line). The dashed vertical line represents isotropic nonlocal boxes.

Figure 2

Protocol for two-copy deterministic nonlocality distillation.

Figure 3

Nonlocality distillation of correlated nonlocal boxes with our protocol. The graph shows the ${\mathrm{CHSH}}_f$ value of the final box as a function of its initial ${\mathrm{CHSH}}_i$ value. The dashed straight line, ${\mathrm{CHSH}}_f={\mathrm{CHSH}}_i$, is given as a reference. In the asymptotic regime, any nonlocal box $P_{ϵ}^{\mathrm{PR}}$ is distilled arbitrarily close to the PR box ($\mathrm{CHSH}=4$). The thin line (steps) shows the distillation of an initial box under successive iterations of the protocol.

Figure 4

Section of the nonsignaling polytope given by nonlocal boxes $P_{\xi ,\gamma }^{\mathrm{PR}}$. The blue line and the shaded (blue) areas represent the set of postquantum boxes that collapse communication complexity. The area shaded in light blue was identified in 10, while the area in dark blue has been identified in the present Letter. Astonishingly, some of the correlated nonlocal boxes are arbitrarily close to the local state $P^c$, yet they still collapse communication complexity. Nonlocal boxes above the dashed (red) curve are distilled in one iteration of the protocol. The solid curve is the quantum boundary [16].