Bell–CHSH Inequality

Equivalent forms

|S|_{\mathrm{QM}} \le 2\sqrt{2} \;\; (\text{Tsirelson bound})

E(a,b) = -\cos(a-b) \;\; (\text{singlet state})

A one-line inequality settles a philosophical debate that ran for half a century — by experiment.

Symbol	Name	SI unit	Dimension	Range
$S$	CHSH correlation sumoutput Combination of four two-setting correlations; \|S\| > 2 rules out local realism	dimensionless	1	-2.828 – 2.828
$a$	Alice setting 1 First analyzer angle on Alice's side	rad	1	0 – 3.14159
$a_prime$	Alice setting 2 Second analyzer angle on Alice's side	rad	1	0 – 3.14159
$b$	Bob setting 1 First analyzer angle on Bob's side	rad	1	0 – 3.14159
$b_prime$	Bob setting 2 Second analyzer angle on Bob's side	rad	1	0 – 3.14159

Unit systems

Symbol	SI	CGS	Imperial
$S$	dimensionless	dimensionless	dimensionless
$a$	rad	rad	deg
$a_prime$	rad	rad	deg
$b$	rad	rad	deg
$b_prime$	rad	rad	deg

Where it holds

The bound |S| <

= 2

holds for all theories satisfying locality and measurement independence. The quantum prediction requires entangled pairs and space-like separated, freely chosen settings; detection and locality loopholes were closed simultaneously in 2015.

Dimensional analysis

S -> dimensionless (sum of four correlation coefficients in [-1, 1])

bound 2 (or 2*sqrt(2)) -> dimensionless

Pure numbers on both sides.

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Discovery

John Bell; Clauser, Horne, Shimony, Holt · 1964

Bell, working at CERN as an accelerator physicist with foundations as a hobby, showed in 1964 that Einstein's local realism makes testable predictions. CHSH made it lab-ready in 1969. Aspect's 1982 photon tests, loophole-free experiments in 2015, and the 2022 Nobel Prize closed the case: nature violates the inequality exactly as quantum mechanics predicts.

Try this

Can you prove the universe isn't secretly classical — with four numbers?

Einstein insisted entangled particles must carry hidden instructions. Bell found an inequality every local hidden-variable theory must obey — and quantum mechanics breaks it. Experiments agree with quantum mechanics. Reality is provably non-classical.

Research status: stable

Real-world applications

Device-independent quantum key distribution (security certified by Bell violation itself)
Certified quantum random-number generators (NIST randomness beacon)
Entanglement verification benchmarks for quantum networks and repeaters
Self-testing of quantum devices: maximal violation certifies the state and measurements

Common misconceptions

Violation does NOT enable faster-than-light signaling — each side's local outcomes remain perfectly random
It does not prove 'spooky action': it rules out local hidden variables; which assumption fails (locality or realism) is interpretation-dependent
Entanglement correlations do not weaken with distance — photons violate the bound across hundreds of kilometers of fiber

Experimental verification

Freedman-Clauser 1972 (first violation), Aspect 1982 (time-varying analyzers), Hensen et al. 2015 in Delft (loophole-free, entangled electron spins 1.3 km apart), with NIST and Vienna closing both loopholes the same year using photons. The 2022 Nobel Prize went to Aspect, Clauser, and Zeilinger. Typical loophole-free values:

S \sim 2.4

with overwhelming statistical significance.

Derivation

Local realism: each pair carries values A(a), A(a'

) = +/-1

for Alice and B(b),

B(b') = +/-1

for Bob, fixed before measurement.

Algebra: A(a)[B(b)-B(b')] + A(a'

)[B(b)+B(b')] = +/-2

always, since one bracket vanishes and the other is +/-2.

Averaging gives |S| <

= 2

.

Quantum mechanically, for the singlet E(a,

b) = -\cos

(a-b); choosing

a=0

, a'

=pi/2

,

b=pi/4

,

b'=3pi/4

gives four correlations of magnitude 1/sqrt(2) that add to |

S| = 2*\sqrt{2}

.

Limiting cases

Any local hidden-variable model⟶ |S| <

= 2

Each particle carrying predetermined answers caps the sum at 2 — Bell's bound.

Singlet state, optimal angles (0, pi/2; pi/4, 3pi/4)⟶ |

S| = 2*\sqrt{2} \sim 2.828

Quantum mechanics' maximum — the Tsirelson bound, saturated by maximally entangled states.

Separable (non-entangled) quantum states⟶ |S| <

= 2

Entanglement is necessary for violation — product states behave classically here.

What if…

What if detectors miss most of the pairs?

The detection loophole opens: a clever local model can fake violation using the missed events. Closing it needs > 82.8% efficiency — achieved with ions, superconducting circuits, and finally photons in 2015.

What if the settings are chosen before the particles separate?

Locality is no longer enforced — hidden variables could know the settings. Real tests use fast quantum random number generators, and 'cosmic Bell tests' used light from distant quasars.

What if |S| could reach the algebraic maximum of 4?

That world (Popescu-Rohrlich boxes) would obey no-signaling yet make communication complexity trivial — one reason physicists suspect deep principles pin nature to Tsirelson's 2sqrt(2).

1

Maximal quantum violation

Given ·

a:: 0
a prime:: 1.5708
b:: 0.7854
b prime:: 2.3562

Find · S for the singlet state

Steps

E(a, $b) = -\cos (0 - pi/4) = -0.707$ ; E(a, $b') = -\cos (0 - 3pi/4) = +0.707$
E(a', $b) = -\cos (pi/2 - pi/4) = -0.707$ ; E(a', $b') = -\cos (pi/2 - 3pi/4) = -0.707$
$S = -0.707 - 0.707 - 0.707 - 0.707 = -2.828$ , so | $S| = 2\sqrt(2)$

Result · Each correlation has magnitude

\cos (pi/4) = 0.707

; they combine to |

S| = 2*\sqrt{2} \sim 2.828

> 2.

2

Classical strategy ceiling

Given ·

strategy:: pre-agreed answers A, $B = +/-1$

Find · maximum |S|

Steps

B(b) and B(b') are each +/-1: either $B(b)-B(b') = 0$ and $B(b)+B(b') = +/-2$ , or vice versa
The expression equals +/-2 pair-by-pair
Averaging over any distribution of hidden variables keeps |S| < $= 2$

Result · A(a)[B(b)-B(b')] + A(a'

)[B(b)+B(b')] = +/-2

for every pair, so no local strategy beats |

S| = 2

.

Density Matrix→Spin-1/2 Operators→Stern–Gerlach Deflection→Born Rule→