Bell nonlocality with a single shot

Abstract

In order to reject the local hidden variables hypothesis, the usefulness of a Bell inequality can be quantified by how small a p-value it will give for a physical experiment. Here we show that to obtain a small expected p-value it is sufficient to have a large gap between the local and Tsirelson bounds of the Bell inequality, when it is formulated as a nonlocal game. We develop an algorithm for transforming an arbitrary Bell inequality into an equivalent nonlocal game with the largest possible gap, and show its results for the CGLMP and $I_{nn22}$ inequalities. We present explicit examples of Bell inequalities with gap arbitrarily close to one, and show that this makes it possible to reject local hidden variables with arbitrarily small p-value in a single shot, without needing to collect statistics. We also develop an algorithm for calculating local bounds of general Bell inequalities which is significantly faster than the naïve approach, which may be of independent interest.

11+7 pages, 1 figure, 3 computer programs. v3: minor improvements in presentation

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Cited by in corpus (17)

• Quantum theory based on real numbers can be experimentally falsified
• Experimental refutation of real-valued quantum mechanics under strict locality conditions
• Exponentially decreasing critical detection efficiency for any Bell inequality
• Quantum verification and estimation with few copies
• Classical Cost of Transmitting a Qubit
• Optimized detection of high-dimensional entanglement
• Quantum Preparation Games
• Distance-based resource quantification for sets of quantum measurements
• Bounding the detection efficiency threshold in Bell tests using multiple copies of the maximally entangled two-qubit state carried by a single pair of particles
• Distributed quantum incompatibility
• Self-testing tilted strategies for maximal loophole-free nonlocality
• Single-measurement Bell analysis
• Optimal conversion of Kochen-Specker sets into bipartite perfect quantum strategies
• Local hidden variable values without optimization procedures
• Beating one bit of communication with and without quantum pseudo-telepathy
• Constructive nonlocal games with very small classical values
• Classical bounds on two-outcome bipartite Bell expressions and linear prepare-and-measure witnesses: Efficient computation in parallel environments such as graphics processing units