Bell inequalities are one of the most famous tests of the foundations of quantum mechanics.
They are designed to distinguish between two very different ideas about physical reality.
In a classical local-hidden-variable picture, the results of measurements are determined by pre-existing properties carried by the particles, and no influence can travel faster than light between distant detectors.
Quantum entanglement challenges that picture by predicting correlations that are stronger than any such local classical model can allow.
In the polarization version of an entangled-photon Bell experiment, two distant observers—traditionally called Alice and Bob—measure the polarization of each member of an entangled pair along chosen analyzer angles.
For the ideal maximally entangled state, the quantum correlation is
\[
E(a,b)=-\cos\!\bigl(2(a-b)\bigr),
\]
where \(a\) and \(b\) are the analyzer angles.
The factor of 2 appears because linear polarization is periodic under a rotation of
\(180^\circ\),
not
\(360^\circ\).
This is the standard correlation law for polarization-entangled photon pairs.
Bell’s idea is to combine four such correlations, using two possible settings for Alice
(\(a\) and \(a'\))
and two for Bob
(\(b\) and \(b'\)).
The most common form is the CHSH combination
\[
S=E(a,b)-E(a,b')+E(a',b)+E(a',b').
\]
Any local realistic theory must satisfy the Bell–CHSH inequality
\[
|S|\le 2.
\]
Quantum mechanics, however, predicts that entangled states can exceed this classical limit.
In fact, the largest possible quantum value is the Tsirelson bound,
\[
|S|_{\max}=2\sqrt{2}\approx 2.828.
\]
This is one of the clearest demonstrations that quantum entanglement cannot be reproduced by an ordinary local-hidden-variable model.
For the polarization correlation used in this calculator, the standard maximal-violation angles are
\[
a=0^\circ,\qquad a'=45^\circ,\qquad b=22.5^\circ,\qquad b'=67.5^\circ.
\]
With these choices,
\[
E(a,b)=-\cos 45^\circ=-\frac{\sqrt{2}}{2},
\]
\[
E(a,b')=-\cos 135^\circ=+\frac{\sqrt{2}}{2},
\]
\[
E(a',b)=-\cos 45^\circ=-\frac{\sqrt{2}}{2},
\]
\[
E(a',b')=-\cos 45^\circ=-\frac{\sqrt{2}}{2}.
\]
Therefore the CHSH expression becomes
\[
S
=-\frac{\sqrt{2}}{2}
-\frac{\sqrt{2}}{2}
-\frac{\sqrt{2}}{2}
-\frac{\sqrt{2}}{2}
=-2\sqrt{2}.
\]
Taking the absolute value gives
\[
|S|=2\sqrt{2}\approx 2.828,
\]
which clearly violates the classical Bell limit of 2.
A subtle but important point is that the specific sample angle set
\(0^\circ,45^\circ,90^\circ,135^\circ\)
does not give the maximal CHSH violation for the polarization correlation
\(E=-\cos(2\Delta\theta)\).
That angle set is often confused with other conventions, but for polarization-entangled photons the optimal settings involve
\(22.5^\circ\)
offsets.
Physically, Bell violation does not mean that information is being sent faster than light in a usable way.
Instead, it means that the joint correlations between distant outcomes cannot be explained by assigning fixed local properties to the particles in advance.
The entangled state behaves as a single nonclassical system, even when the measurements are performed far apart.
At a more advanced university level, Bell tests are discussed in terms of detector efficiency, locality loopholes, freedom-of-choice loopholes, and loophole-free experimental designs.
One also studies density matrices, mixed states, visibility thresholds, and the role of noise.
But even at an introductory level, the CHSH form already captures the central lesson:
quantum entanglement predicts correlations stronger than any local classical model allows.
\[
E(a,b)=-\cos\!\bigl(2(a-b)\bigr),
\qquad
S=E(a,b)-E(a,b')+E(a',b)+E(a',b'),
\qquad
|S|_{\text{classical}}\le 2.
\]
These formulas let you compare the classical Bell limit with the quantum prediction and visualize why entangled pairs can violate Bell’s inequality.
