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arXiv:2505.07474

Statistical analysis of Bell tests via generalized measurements

Alfredo Luis

Published May 12, 2025• 0 citations
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Abstract

We provide a fully statistical analysis of the results of a Bell test beyond mean values. This is possible in a practical scheme where all the observables involved in the test are simultaneously measured at the expense of unavoidably additional noise. To deal with this noise we can follow two strategies leading to the same results. These are to adapt the Bell bound to include the noise, or to remove the additional noise via suitable data inversion.

DOI
10.48550/arxiv.2505.07474

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We provide a fully statistical analysis of the results of a Bell test beyond mean values. This is possible in a practical scheme where all the observables involved in the test are simultaneously measured at the expense of unavoidably additional noise. To deal with this noise we can follow two strategies leading to the same results. These are to adapt the Bell bound to include the noise, or to remove the additional noise via suitable data inversion.
Bell tests are of profound statistical nature. Besides physical considerations, the proper understanding of their implications should involve detailed statistical analyses. In this regard, recent works have shown that their consequences and interpretations depend on the probability space adopted. Some other recent works have also shown that generalized measurements may allow to further exploit the statistics of Bell-like tests. Following these ideas, in this work we show that one and the same experimental arrangement can provide a practical scheme valid for two very different probability spaces. Moreover, we show that this allows the introduction of novel Bell tests that are not possible in more standard approaches.
Abstract A key objective in conducting a Bell test is to quantify the statistical evidence against a local-hidden variable model (LHVM) given that we can collect only a finite number of trials in any experiment. The notion of statistical evidence is thereby formulated in the framework of hypothesis testing, where the null hypothesis is that the experiment can be described by an LHVM. The statistical confidence with which the null hypothesis of an LHVM is rejected is quantified by the so-called P value, where a smaller P value implies higher confidence. Establishing good statistical evidence is especially challenging if the number of trials is small, or the Bell violation very low. Here, we derive the optimal P value for a large class of Bell inequalities. What is more, we obtain very sharp upper bounds on the P value for all Bell inequalities. These values are easily computed from the experimental data, and are valid even if we allow arbitrary memory in the devices. Our analysis is able to deal with imperfect random number generators, and event-ready schemes, even if such a scheme can create different kinds of entangled states. Finally, we review requirements for sound data collection, and a method for combining P values of independent experiments. The methods discussed here are not specific to Bell inequalities. For instance, they can also be applied to the study of certified randomness or to tests of noncontextuality.
The aim of this note is to attract once again attention of the quantum community to statistical analysis of data which was reported as violating Bell's inequality. This analysis suffers of a number of problems. And the main problem is that rough data is practically unavailable. However, experiments which are not followed by the open access to the rough data have to be considered as with no result. The absence of rough data generates a variety of problems in statistical interpretation of the results of Bell's type experiment. One may hope that this note would stimulate experimenters to create the open access data-base for, e.g., Bell tests. Unfortunately, recently announced experimental loophole-free violation of a Bell inequality using entangled electron spins separated by 1.3 km was not supported by open-access data. Therefore in accordance with our approach "it has no result." The promising data after publication is, of course, a step towards fair analysis quantum experiments. May be this is a consequence of appearance of this preprint, v1. But there are a few questions which would be interesting to clarify before publication (and which we shall discuss in this note).
A device-independent dimension test for a Bell experiment aims to estimate the underlying Hilbert space dimension that is required to produce given measurement statistical data without any other assumptions concerning the quantum apparatus. Previous work mostly deals with the two-party version of this problem. In this paper, we propose a very general and robust approach to test the dimension of any subsystem in a multiparty Bell experiment. Our dimension test stems from the study of a new multiparty scenario which we call prepare-and-distribute . This is like the prepare-and-measure scenario, but the quantum state is sent to multiple, noncommunicating parties. Through specific examples, we show that our test results can be tight. Furthermore, we compare the performance of our test to results based on known bipartite tests, and witness remarkable advantages, which indicates that our test is of a true multiparty nature. We conclude by pointing out that with some partial information about the quantum states involved in the experiment, it is possible to learn other interesting properties beyond dimension.
Quantum teleportation have a central role in quantum information science and allows transferring of an unknown quantum state through entanglement and classical communication. Unfortunately, the interaction with external and internal noise severely affects the quality of teleportation and poses limitations on practical applications of quantum communication networks. In this work, instead of conventional Bell teleportation, we introduce a Machine Learned adaptive protocol for optimizing multiple components of Quantum Teleportation in order to achieve higher fidelity in various noise environments. In order to demonstrate the performance of the proposed scheme, we study three different noise models, including bit-flip, amplitude damping, and depolarizing noise, both in case of single-qubit and two-qubit channels. As a result, we observe substantial improvement in the teleportation fidelity in comparison to the classical Bell-state teleportation protocol in certain noise conditions. Furthermore, the machine-learned protocol reveals a nontrivial strategy for compensation of decoherence and information losses. In addition, obtained results indicate the flexibility and reliability of the proposed framework for implementing various adaptive quantum communications while shedding light on possibilities of discovery of optimal quantum algorithms by means of automated approache