Bell’s theorem addresses the statistical predictions of quantum mechanics and the concept of local realism. When testing Bell’s theorem using entangled particles, such as photons with polarization states, one typically looks at the correlations between measurements made by two detectors set at different orientations.

Let’s denote the angle between the orientations of the two detectors as $\theta$. The probability that the two detectors will flash different colors (i.e., they will measure different outcomes) is given by the quantum mechanical prediction for entangled particles.

For entangled photons, the probability $P(\text{different outcomes})$ that the two detectors will register different outcomes (flash different colors) is:

P(different outcomes)=12(1−cos⁡(2θ))P(\text{different outcomes}) = \frac{1}{2} \left( 1 – \cos(2\theta) \right)P(different outcomes)=21​(1−cos(2θ))

Here’s a step-by-step explanation:

1. Entangled Photons and Measurement:
   • When entangled photons are measured with detectors set at angles $\alpha$ and $\beta$, the measurement outcomes are correlated.
   • The angle $\theta$ is the difference between the angles of the two detectors: $\theta =\alpha -\beta$.
2. Quantum Mechanical Prediction:
   • According to quantum mechanics, the correlation between the measurement outcomes is given by $\cos (2\theta )$.
3. Calculating Probability:
   • The probability that the detectors show the same outcome is 12(1+cos⁡(2θ))\frac{1}{2} \left( 1 + \cos(2\theta) \right)21​(1+cos(2θ)).
   • Conversely, the probability that the detectors show different outcomes is: P(different outcomes)=1−P(same outcome)=1−12(1+cos⁡(2θ))P(\text{different outcomes}) = 1 – P(\text{same outcome}) = 1 – \frac{1}{2} \left( 1 + \cos(2\theta) \right)P(different outcomes)=1−P(same outcome)=1−21​(1+cos(2θ))
   • Simplifying this, we get: P(different outcomes)=12(1−cos⁡(2θ))P(\text{different outcomes}) = \frac{1}{2} \left( 1 – \cos(2\theta) \right)P(different outcomes)=21​(1−cos(2θ))

Therefore, the probability that the two detectors will flash different colors when their orientation settings differ by an angle $\theta$ is 12(1−cos⁡(2θ))\frac{1}{2} \left( 1 – \cos(2\theta) \right)21​(1−cos(2θ)).

This result is crucial in demonstrating the violation of Bell’s inequalities, providing evidence against local hidden variable theories and supporting the non-local predictions of quantum mechanics.

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