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arXiv:1610.01833 · Journal of Physics A Mathematical and Theoretical

On the inequivalence of the CH and CHSH inequalities due to finite statistics

Marc-Olivier Renou, D Rosset, Anthony Martin, N Gisin

Published May 30, 2017• 12 citations
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Abstract

Abstract Different variants of a Bell inequality, such as CHSH and CH, are known to be equivalent when evaluated on nonsignaling outcome probability distributions. However, in experimental setups, the outcome probability distributions are estimated using a finite number of samples. Therefore the nonsignaling conditions are only approximately satisfied and the robustness of the violation depends on the chosen inequality variant. We explain that phenomenon using the decomposition of the space of outcome probability distributions under the action of the symmetry group of the scenario, and propose a method to optimize the statistical robustness of a Bell inequality. In the process, we describe the finite group composed of relabeling of parties, measurement settings and outcomes, and identify correspondences between the irreducible representations of this group and properties of outcome probability distributions such as normalization, signaling or having uniform marginals.

DOI
10.1088/1751-8121/aa6f78

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Abstract Different variants of a Bell inequality, such as CHSH and CH, are known to be equivalent when evaluated on nonsignaling outcome probability distributions. However, in experimental setups, the outcome probability distributions are estimated using a finite number of samples. Therefore the nonsignaling conditions are only approximately satisfied and the robustness of the violation depends on the chosen inequality variant. We explain that phenomenon using the decomposition of the space of outcome probability distributions under the action of the symmetry group of the scenario, and propose a method to optimize the statistical robustness of a Bell inequality. In the process, we describe the finite group composed of relabeling of parties, measurement settings and outcomes, and identify correspondences between the irreducible representations of this group and properties of outcome probability distributions such as normalization, signaling or having uniform marginals.
Different variants of a Bell inequality, such as CHSH and CH, are known to be equivalent when evaluated on nonsignaling outcome probability distributions. However, in experimental setups, the outcome probability distributions are estimated using a finite number of samples. Therefore the nonsignaling conditions are only approximately satisfied and the robustness of the violation depends on the chosen inequality variant. We explain that phenomenon using the decomposition of the space of outcome probability distributions under the action of the symmetry group of the scenario, and propose a method to optimize the statistical robustness of a Bell inequality. In the process, we describe the finite group composed of relabeling of parties, measurement settings and outcomes, and identify correspondences between the irreducible representations of this group and properties of outcome probability distributions such as normalization, signaling or having uniform marginals.
Quantum non-locality and quantum contextuality result from the impossibility of explain- ing the probabilities of measurement outcomes in quantum systems using theoretical mod- els that assume that observables have well-defined values prior to the measurement, and that these values do not depend on which other compatible measurements are performed simultaneously to the measurement. These two features are not only of foundational im- portance for quantum theory, but are also recognized as resources for quantum computing and other applications. The experimental verification of contextuality has garnered significant attention, and its recognition as an operational resource for quantum information is expected to further fuel this interest. Developing a robust theoretical framework for contextuality that is both operationally grounded and experimentally applicable is crucial. Particularly, real-world experiments often violate the non-disturbance assumption, requiring alternative frame- works that remain effective under such conditions. This dissertation builds upon a recent work by Renou et.al, which explores how dif- ferent variants of a Bell inequality, despite being equivalent under the condition of no- signaling, are not equally robust in real experiments. In order to do so, they propose a method which employs the group of symmetries of a Bell scenario to analyze and op- timize the robustness of Bell inequalities. The primary goal of this dissertation is to generalize this method to contextuality scenarios, exploring the role of group representa- tion theory in optimizing inequalities, and enhancing both theoretical understanding and experimental applicability.
A number of papers have suggested that it is inappropriate to combine data from different experiments when undertaking experimental tests of Bell's inequalities. It has been suggested that a correct analysis, using a single probability space, leads to inequalities which are not violated by experiment. If correct, this would be contrary to the normal interpretation of such experimental data. However, in this note, a generalised Clauser-Horne-Shimony-Holt (CHSH) inequality is derived for a system of four experiments constructed on a single probability space which combines the data from the four experiments. It is shown that this leads to the standard CHSH inequality which is normally used to interpret experimental data. Thus the commonly accepted conclusion that experimental violations of Bell's inequality imply that local realistic models are inconsistent with the predictions of quantum mechanics has not been challenged.
Bell-test data are typically used to certify nonlocal correlations, while quantum metrology is concerned with the sensitivity of measurement statistics to unknown parameters. Here we connect these tasks at the level of experimentally accessible Bell statistics. Using standard moment-based Fisher-information inequalities, we derive lower bounds on the classical Fisher information directly from Bell scores and their parameter dependence, without requiring full state tomography. For CHSH measurements we establish a hierarchy between the information contained in the full outcome record, the set of individual correlators, and the compressed Bell parameter, thereby quantifying the metrological cost of data compression. In polarization-phase sensing, this hierarchy reveals a Bell-metrology tradeoff: the point of maximal CHSH violation need not maximize, and can even suppress, the Fisher information carried by the Bell score itself. We further derive finite-sample lower-confidence bounds from coincidence counts and analyze the effects of biased marginals, accidental coincidences, visibility loss, and multipair emission. The framework provides a practical route for extracting metrological guarantees from Bell-type experiments using only standard correlation measurements.
The majority of recent works investigating the link between non-locality and randomness, e.g. in the context of device-independent cryptography, do so with respect to some specific Bell inequality, usually the CHSH inequality. However, the joint probabilities characterizing the measurement outcomes of a Bell test are richer than just the degree of violation of a single Bell inequality. In this work we show how to take this extra information into account in a systematic manner in order to optimally evaluate the randomness that can be certified from non-local correlations. We further show that taking into account the complete set of outcome probabilities is equivalent to optimizing over all possible Bell inequalities, thereby allowing us to determine the optimal Bell inequality for certifying the maximal amount of randomness from a given set of non-local correlations.