Researchers from the Institute for Quantum Optics and Quantum Information (IQOQI), Austria, derived the exact mathematical relationship between measurement dependence, Bell inequality violation, and detector efficiency in CHSH experiments. Aaron Alai used linear programming to prove that the minimum measurement dependence required to reproduce quantum correlations with 90% detector efficiency is ‘(27√2 − 33)/100’. The study unifies these three factors within a single analytical framework, providing exact trade-off surfaces and experimentally testable predictions for distinguishing local hidden-variable models from quantum mechanics.

An exact mathematical connection exists between how much measurements must depend on each other within certain theories, the extent to which they can violate Bell’s inequality, a key test of quantum mechanics, and detector efficiency during experiments. This pinpointed connection was previously missing from studies offering only approximations or focusing solely on single factors. The team also identified a local mechanism called moiré phase-locking that mimics quantum behaviour with detectable characteristics, providing another avenue for examining fundamental physics principles.

The minimal level of measurement dependence required to account for results from a CHSH experiment, designed to determine if particles are connected in ways not allowed by classical physics, is (27sqrt{2}-33)/100 at nine-tenths detector efficiency. Earlier research offered approximations or focused solely on individual parameters rather than all three simultaneously; this work reveals the relationship between these elements.

Precise quantification of measurement interdependence at varying detector efficiencies

A theoretical minimum for measurement dependence in certain experiments has been reduced by over twenty-seven percent. Prior approximations only considered individual parameters or relied on incomplete models of interconnected measurements; this new calculation offers greater precision. A linear programming approach establishes an exact mathematical relationship between measurement dependence, violations of Bell’s inequality, a key test of quantum mechanics, and overall detection efficiency during experimentation.

Analysis reveals a minimum dependence value of (27sqrt{2}-33)/100 at nine-tenths detector efficiency, refining our understanding of the necessary interconnectedness in experiments testing quantum mechanics. The team also determined that deterministic sign models reproduce singlet correlations with a unique profile where detection scales proportionally to the square root of measured values. Furthermore, any local explanation for these results requires trade-offs involving modulation of coincidence sums, a measurable effect up to 12.4% in amplitude which vanishes at specific angles but has remained poorly bounded for decades.

Determining Minimal Interdependence via Linear Programming of Local Hidden Variable Theories

Linear programming proved key to this work; it is a mathematical technique optimising outcomes given multiple constraints, similar to finding the best route on a map with restrictions. Formulating minimal measurement dependence as such a linear program required defining ‘faithfulness’, the requirement that solutions accurately reproduce experimental observations like detection rates and correlations. By systematically exploring all possible local hidden variable theories within these constraints, they pinpointed the absolute minimum interdependence needed to explain CHSH experiment results, effectively identifying the lowest cost in terms of interconnectedness matching observed quantum behaviour.

Calculations were employed at thirty-two grid points spanning S ∈[2.1, 3.4] and ηeff ∈[0.85, 1], achieving machine precision. The analysis explored scenarios with detector efficiency, denoted as ηeff, and values of ‘faithfulness’, ensuring accurate matches between proposed solutions and experimental data.

Constraining interdependence strengthens benchmarks separating quantum mechanics from local realism

Long demanding increasingly precise boundaries for local realism has been the pursuit of definitively distinguishing quantum mechanics from alternative explanations; this refers to the idea that objects possess definite properties independent of measurement. This work delivers a remarkably tight constraint on how much ‘interdependence’ measurements must exhibit to mimic quantum behaviour. However, such precision reveals an intriguing tension between theoretical completeness and practical application. Despite raising doubts about fully differentiating these areas in practice, discovering a local mechanism capable of mimicking quantum behaviour with up to ten per cent fidelity remains significant.

This establishes a benchmark for any experiment attempting to disprove local realism by defining how closely alternative theories must match quantum predictions to be considered viable. The team’s research determines the minimum level of measurement dependence required to achieve a CHSH value, establishing a mathematical relationship linking measurement dependence, experimental violation of tests like Bell’s inequality and detector efficiency, the ability to reliably register particles.

The researchers determined a precise limit on the interdependence measurements require to replicate outcomes from the CHSH experiment, a test of local realism versus quantum mechanics. This work establishes that a local account could reproduce observed results with up to 12.4% fidelity under specific conditions involving detector efficiencies between 0.85 and 1. Calculations at thirty-two grid points defined a mathematical connection between this interdependence, values obtained in experiments such as Bell’s inequality, and detection capabilities. The authors suggest further investigation may focus on settings-torus protocols reaching $5σ$ sensitivity within hours.

👉 More information
🗞 The exact price of local realism in CHSH experiments: a measurement-dependence-detection trade-off surface, a moiré phase-locking mechanism that saturates it, and an unmeasured fringe in the fourfold coincidence sum
✍️ Aaron Alai
🧠 ArXiv: https://arxiv.org/abs/2608.18886

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