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AIP conference proceedings

Towards a Graph-Theoretic Characterization of Popescu-Rohrlich Box Simulation Strategies

Samuel C. Fletcher, Alexander I. Lvovsky

Published Jan 1, 2009• 0 citations
Publisher

Abstract

The propensity for certain quantum system to consistently violate Bell inequalities—dubbed quantum nonlocality—has recently been found to be an information theoretic resource in quantum communication that is distinct from entanglement. Accordingly, a useful measurement of “how much” of this resource is present in a given system is needed. One candidate, the PR‐box (sometimes called a nonlocal machine), has been suggested as a possible benchmark, but one of the difficulties in using it has been to determine exactly what kinds of quantum systems are simulable by it. The present work is a first step towards a graph‐theoretic understanding of PR‐boxes that may prove useful in this endeavor.

DOI
10.1063/1.3131284

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The propensity for certain quantum system to consistently violate Bell inequalities—dubbed quantum nonlocality—has recently been found to be an information theoretic resource in quantum communication that is distinct from entanglement. Accordingly, a useful measurement of “how much” of this resource is present in a given system is needed. One candidate, the PR‐box (sometimes called a nonlocal machine), has been suggested as a possible benchmark, but one of the difficulties in using it has been to determine exactly what kinds of quantum systems are simulable by it. The present work is a first step towards a graph‐theoretic understanding of PR‐boxes that may prove useful in this endeavor.
We present bipartite Bell-type inequalities which allow the two partners to use some nonlocal resource. Such inequalities can only be violated if the parties use a resource which is more nonlocal than the one permitted by the inequality. We introduce a family of N-input nonlocal machines, which are generalizations of the well-known PR (Popescu-Rohrlich) box. Then we construct Bell-type inequalities that cannot be violated by strategies that use one of these new machines. Finally we discuss implications for the simulation of quantum states.
A set of nonlocal correlations that have come to be known as a Popescu-Rohrlich (PR) box suggest themselves as a natural unit of nonlocality, much as a singlet is a natural unit of entanglement. We present two results relevant to this idea. One is that a wide class of multipartite correlations can be simulated using local operations on PR boxes only. We show this with an explicit scheme, which has the interesting feature that the number of PR boxes required is related to the computational resources necessary to represent a function defining the multipartite box. The second result is that there are quantum multipartite correlations, arising from measurements on a cluster state, that cannot be simulated with n PR boxes, for any n.
One of the most intriguing features of quantum physics is the non‐locality of correlations that can be obtained by measuring entangled particles. Recently, it has been noticed that non‐locality can be studied without reference to the Hilbert space formalism. I review here the properties of the basic mathematical tool used for such studies, the so called Popescu‐Rohrlich‐box, in short PR‐box. Among its feats, are the simulation of the correlations of the singlet and of other non‐local probability distributions. Among its features, the “anomaly of non‐locality” and a great power for information‐theoretical tasks. Among its failures, the impossibility of reproducing all multi‐partite distributions and the triviality of the allowed dynamics.
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.
FINDING: The Popescu-Rohrlich (PR) box defines the maximal nonlocal vertex of the Bell polytope; its extremal structure is characterized by convex geometry, and recent work shows PR-box fraction cannot uniquely single out quantum theory among generalized nonsignaling theories. | MATH: PR box correlations: P(a,b∣x,y)=12P(a,b|x,y) = \frac{1}{2}P(a,b∣x,y)=21​ if a⊕b=xya \oplus b = xya⊕b=xy, else 0. Bell polytope B\mathcal{B}B: convex hull of local deterministic strategies; its extremal points in (2,2,2) scenario are the 8 local vertices plus 4 PR-box vertices (the "nonlocal" extreme points). The PR box saturates the CHSH inequality: ⟨A0B0⟩+⟨A0B1⟩+⟨A1B0⟩−⟨A1B1⟩=4\langle A_0B_0\rangle + \langle A_0B_1\rangle + \langle A_1B_0\rangle - \langle A_1B_1\rangle = 4⟨A0​B0​⟩+⟨A0​B1​⟩+⟨A1​B0​⟩−⟨A1​B1​⟩=4 (Tsirelson bound for quantum: 22≈2.8282\sqrt{2} \approx 2.82822​≈2.828). | CONNECTION: The ratio 22/4=1/2≈0.70712\sqrt{2}/4 = 1/\sqrt{2} \approx 0.707122​/4=1/2​≈0.7071 — not directly in the golden set, but note 0.70712=0.50.7071^2 = 0.50.70712=0.5, and the quantum-to-PR gap 4−22≈1.17164 - 2\sqrt{2} \approx 1.17164−22​≈1.1716 relates to \(2(\sqrt{2}-1) \approx 0.82 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We quantify the amount of non-locality contained in n noisy versions of the so-called Popescu–Rohrlich boxes ( PRBs ), i.e. bipartite systems violating the CHSH Bell inequality maximally. Following the approach by Elitzur, Popescu and Rohrlich, we measure the amount of non-locality of a system by representing it as a convex combination of a local behaviour, with maximal possible weight, and a non-signalling system. We show that the local part of n systems, each of which approximates a PRB with probability 1 − ε, is of order Θ(ε ⌈ n /2⌉ ) in the isotropic, and equal to (3ε) n in the maximally biased case.