Continuous-variable supraquantum nonlocality

Andreas Ketterer, Adrien Laversanne-Finot, and Leandro Aolita

Phys. Rev. A 97, 012133 – Published 31 January 2018

Abstract

Supraquantum nonlocality refers to correlations that are more nonlocal than allowed by quantum theory but still physically conceivable in postquantum theories, in the sense of respecting the basic no-faster-than-light communication principle. While supraquantum correlations are relatively well understood for finite-dimensional systems, little is known in the infinite-dimensional case. Here, we study supraquantum nonlocality for bipartite systems with two measurement settings and infinitely many outcomes per subsystem. We develop a formalism for generic no-signaling black-box measurement devices with continuous outputs in terms of probability measures, instead of probability distributions, which involves a few technical subtleties. We show the existence of a class of supraquantum Gaussian correlations, which violate the Tsirelson bound of an adequate continuous-variable Bell inequality. We then introduce the continuous-variable version of the celebrated Popescu–Rohrlich (PR) boxes, as a limiting case of the above-mentioned Gaussian ones. Finally, we characterize the geometry of the set of continuous-variable no-signaling correlations. Namely, we show that that the convex hull of the continuous-variable PR boxes is dense in the no-signaling set. We also show that these boxes are extreme in the set of no-signaling behaviors and provide evidence suggesting that they are indeed the only extreme points of the no-signaling set. Our results lay the grounds for studying generalized-probability theories in continuous-variable systems.

Received 12 September 2017
Revised 7 December 2017

DOI:https://doi.org/10.1103/PhysRevA.97.012133

Physics Subject Headings (PhySH)

Nonlocality Quantum information processing with continuous variables

Quantum Information, Science & Technology

Authors & Affiliations

Andreas Ketterer^1,2,*, Adrien Laversanne-Finot^2,†, and Leandro Aolita^3,4,‡

¹Naturwissenschaftlich-Technische Fakultät, Universität Siegen, Walter-Flex-Str. 3, 57068 Siegen, Germany
²Laboratoire Matériaux et Phénomènes Quantiques, Sorbonne Paris Cité, Université Paris Diderot, CNRS UMR 7162, 75013 Paris, France
³Instituto de Física, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972 Rio de Janeiro, RJ, Brazil
⁴ICTP South American Institute for Fundamental Research Instituto de Física Teórica, UNESP-Universidade Estadual Paulista R. Dr. Bento T. Ferraz 271, Bl. II, São Paulo 01140-070, SP, Brazil

^*andreas.ketterer@uni-siegen.de
^†adrien.laversanne-finot@univ-paris-diderot.fr
^‡aolita@if.ufrj.br

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 97, Iss. 1 — January 2018

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Schematic representation of a bipartite Bell experiment with continuous measurement outcomes in the so-called device-independent scenario of black-box measurement instruments. Two spacelike separated observers, Alice (A) and (Bob), perform local measurements on their subsystems with dichotomic measurements choices (inputs) $x$ and $y$ , respectively, and obtain continuous-variable measurement outcomes (outputs) $a$ and $b$ .
Reuse & Permissions
Figure 2
Pictorial (not rigorous) geometrical representation of the (possible) inner structure of the set $M_{NS}$ of CV no-signaling behaviors in the Bell scenario of Fig. 1. $M_{NS}$ contains the set $M_{Q}$ of quantum behaviors, which contains, in turn, the set $M_{L}$ of local behaviors. All three sets are generic convex sets with infinitely many extreme points, delimited by facets as well as curved hypersurfaces. This is in contrast with the finite-dimensional case, where both $M_{NS}$ and $M_{L}$ are convex polytopes, delimited exclusively by facets that can be characterized by a finite number of linear Bell inequalities. In the plot, an example of a linear Bell inequality is represented as the straight line LI. Such linear inequality can, e.g., correspond to a Bell inequality for finite-dimensional systems, which can be violated by CV quantum correlations using so-called binning procedures [20, 21, 22, 23, 24, 27, 28] (see also references in Ref. [19]). Besides this, a hypothetical quantum extreme point is shown in the figure (light-blue corner). While such points are in principle possible, no explicit example thereof is known. In this paper we consider a nonlinear Bell inequality, the CFRD inequality [25], represented as a curve in the plot. This inequality applies in the genuinely CV scenario of our interest and has, additionally, the appealing feature of admitting violations only by supraquantum behaviors (see Sec. 3). Finally, four exemplary CV PR boxes are represented as extreme points of $M_{NS}$ (black dots).
Reuse & Permissions
Figure 3
(a) Density plots of a Gaussian PR box of order two, with center vector characterized by $a = (ℓ, - ℓ) = b$ and width vector $σ = (ℓ / 5, ℓ / 5)$ , for the inputs $(x, y) = (0, 0), (0, 1)$ , or $(1, 0)$ (left) and $(x, y) = (1, 1)$ (right). Note that, for both plots, the projections onto the horizontal as well as vertical axes coincide, reflecting the fact that the behavior is no-signaling. Each center point may also have a different width (or squeezing), but we do not consider that here for simplicity. (b) Violation of the CFRD inequality, normalized by the factor $ℓ^{4}$ , by the Gaussian behavior in question as a function of the parameter $ℓ / σ$ . The CFRD inequality certifies that the Gaussian PR box is supraquantum for the parameter region with $ℓ / σ \geq {(1 + \sqrt{2})}^{1 / 2} \approx 1.55$ .
Reuse & Permissions

Physical Review A

covering atomic, molecular, and optical physics and quantum science