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arXiv:1707.05337 · Physical Review A

Continuous-variable supraquantum nonlocality

Andreas Ketterer, Adrien Laversanne-Finot, Leandro Aolita

Published Jan 31, 2018• 7 citations
arXivHTMLPDF

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.

DOI
10.1103/physreva.97.012133

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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.
Extremal non-signaling (ENS) boxes are correlations that correspond to vertices of the non-signaling polytope of a Bell scenario. Neither quantum theory nor any theory for ideal measurements allows for ENS boxes. That is, according to quantum theory, ENS boxes are nonphysical. Still, ENS boxes are crucial for addressing a number of problems in Bell nonlocality. Here, we obtain ENS boxes in arbitrary bipartite Bell scenarios and present the complete list of ENS boxes for several unexplored scenarios. Equipped with the boxes, we revisit several foundational questions. We find that already two copies of any ENS box violate the exclusivity (or local orthogonality) and Specker's principles. We provide the minimal decomposition of the magic square correlation - the simplest known perfect correlation in nature - in terms of ENS boxes. We identify the minimal scenario in which a dit of communication (with d < 6) is insufficient to simulate ENS boxes. Our results show that the ENS boxes approach leads to new results and opens new avenues for research.
Abstract Objectives: To derive assumptions of Bell Inequalities from one single postulate – Absolute Postulate of Universe – as mentioned by the author S.Bagaria. Method: It focuses on the assumptions – locality, reality, measurement independence and counter-factual definiteness - made in proving Bell’s Inequality and derive them from Absolute Postulate of the Universe. Then, its consequences are studied like PR Boxes being real, DI-QKD, quantum information theory, randomness and foundations of quantum mechanics. Findings: From Absolute Postulate of the Universe, it has been showed that there is absolute determinism in world, resulting in zero theoretical uncertainty. Thus, the algebraically maximal correlations (S = 4) are possible. Thus, the PR-boxes can be real. But, Tsirelson Bound can’t be violated in quantum computing as HUP can’t be violated in Quantum Physics. Novelty: The four assumptions of Bell Inequality are derived from Absolute Postulate of the Universe. The correlations in the universe can be maximal yet unverifiable experimentally. EPR paradox is resolved without assuming faster than light transfer of information as the world is shown to be “non-local with no-signalling”. Keywords: Bell inequality, Popescu–Rohrlich box, Tsirelson bound, Quantum nonlocality, No-signalling, CHSH inequality, Quantum violation, absolute determinism
What principle excludes superquantum correlations that don’t violate the nosignaling principle? In an argument using PR boxes in a 2009 Nature paper, Pawlowski and colleagues show that the Tsirelson bound – the limit achievable in a simulation of a Popescu–Rohrlich correlation with quantum resources – follows from an information-theoretic principle they call “information causality.” For three qubits, it is now known that there are superquantum correlations that cannot be excluded by a bipartite principle like information causality, so this principle does not completely characterize quantum correlations. An intriguing principle, developed by Cabello as the “exclusivity principle” or “Specker’s principle” (from Ernst Specker’s story about a seer from Nineva), and separately by Fritz and colleagues as the principle of “local orthogonality,” yields the Tsirelson bound, and also rules out correlations beyond the quantum limit for the Klyachko correlation and other cases where contextuality or nonlocality is relevant.
The so-called bipartite nonsignaling boxes are systems whose statistics is constrained solely by the principle of no instantaneous signaling between distant locations. Such systems can exhibit much stronger correlations than those admitted by quantum mechanics. Inspired by the quantum logic approach of Tylec and Ku\ifmmode \acute{s}\else \'{s}\fi{} [J. Phys. A: Math. Theor. 48, 505303 (2015)], we consider nonsignaling boxes with three inputs per party and extend the set of measurements with just a single global measurement---one that mimics a quantum two-party Bell measurement. We then show that this seemingly mild extension completely rules out supraquantum correlations: the resulting system admits precisely quantum-mechanical correlations of two qubits. We also consider nonmaximally entangled measurements, obtaining interpolation between quantum and full nonsignaling theory. Our study paves the way to a general program of amending nonsignaling theories with some measurements inherited from quantum mechanics, leading to various interpolations between nonsignaling boxes and quantum mechanics.
A longstanding question in Bell nonlocality is whether finite-alphabet nonsignalling resources can reproduce the continuum of measurement correlations of partially entangled qubits. Here we resolve this problem by showing that every fixed pure two-qubit state admits an exact simulation of all local projective-measurement statistics using shared randomness and finitely many Popescu-Rohrlich (PR) boxes, without communication or postselection. The boxes operate in parallel, with a worst-case bound independent of the measurement settings: at most 128cot⁡4θ128\cot^4θ128cot4θ calls for the Schmidt state cos⁡θ∣00⟩+sin⁡θ∣11⟩\cosθ|00\rangle+\sinθ|11\ranglecosθ∣00⟩+sinθ∣11⟩, 0<θ<π/40<θ<π/40<θ<π/4. The key is an exact finite-selection method that reduces input-dependent continuous sampling to a choice from a fixed-length shared random list, implemented through a finite PR-box network. The construction extends to arbitrary mixed two-qubit states and local binary-outcome POVMs. This reveals a fundamental separation between the continuity of quantum measurement correlations and the alphabet size of the resources needed to simulate them, shedding light on the resource-theoretic structure of Bell nonlocality.
Quantum correlations forms a subset of the set of nonsignaling boxes. This allows us to characterize quantum correlations as a convex combination of the extremal boxes of the nonsignaling polytope which are Popescu-Rohrlich boxes (maximally nonlocal boxes) and local deterministic boxes. There exists multiple decomposition of quantum correlations in the context of the nonsignaling polytope. I find that the existence of Popescu-Rohrlich box decomposition for local boxes associates two notions of discord which capture nonclassicality of quantum correlations originating from Bell nonlocality and EPR-steering. I introduce, Bell and Mermin discord, and demonstrate that any bipartite nonsignaling box admits a three-way decomposition. This decomposition allows us to isolate the origin of nonclassicality into three disjoint sources: a Popescu-Rohrlich box, a maximally local box that detects EPR-steering, and a classical correlation. Interestingly, I show that all non-null quantum discord states which are neither classical-quantum states nor quantum-classical states can give rise to nonclassical correlations which have non-null Bell and/or Mermin discord for suitable noncommuting measurements. I introduce two notions of genuine discord, which are the generalizations of Bell and Mermin discord to the multipartite boxes, to characterize the presence of genuine nonclassicality in multipartite quantum correlations.