Bell Inequality

Mathematical inequalities that distinguish quantum mechanics from classical hidden-variable theories.

Bell inequalities are constraints that any local hidden-variable theory must satisfy. Quantum mechanics violates these inequalities, proving that nature is fundamentally non-classical.

The Question

Can quantum correlations be explained by hidden classical information? Einstein believed quantum randomness might reflect our ignorance of “hidden variables.” Bell’s 1964 theorem showed this is impossible (under reasonable assumptions).

Local Hidden Variables

A local hidden-variable theory assumes:

1. Realism: Measurement outcomes are predetermined by hidden variables
2. Locality: Measurements on one particle can’t instantly affect another

Bell proved: Any such theory satisfies certain inequalities that quantum mechanics violates.

The CHSH Inequality

The most commonly used Bell inequality (CHSH):

For two parties (Alice and Bob) with two measurement settings each (a, a’ and b, b’):

$$|E(a,b)-E(a,b^{\prime })+E(a^{\prime },b)+E(a^{\prime },b^{\prime })|\le 2$$

where $E(x,y)$ is the correlation between measurements.

Classical Bound

Any local hidden-variable theory: $\le 2$

Quantum Bound

Quantum mechanics allows up to: $2\sqrt{2}\approx 2.83$ (Tsirelson’s bound)

Experimental Tests

To test Bell inequalities:

1. Prepare entangled particles
2. Separate them (space-like separation ideally)
3. Choose measurement settings randomly
4. Check if correlations violate the inequality

Milestone Experiments

• 1982 (Aspect): First convincing violation
• 2015 (Loophole-free): Closed major loopholes simultaneously
• 2022 (Nobel Prize): Awarded to Aspect, Clauser, Zeilinger

Loopholes

Early experiments had potential issues:

Modern experiments close all known loopholes.

Implications

Bell inequality violations prove:

1. No local hidden variables: Nature isn’t locally deterministic
2. Entanglement is real: Not just a mathematical artifact
3. Quantum non-locality: Correlations can’t be explained classically

This doesn’t allow faster-than-light communication (results still look random locally).

Device-Independent Applications

Bell violations can certify quantum properties without trusting the devices:

• Device-independent QKD
• Certified randomness generation
• Self-testing quantum states

See also: Entanglement, Bell State, Quantum Key Distribution