Entanglement

A source in the middle fires entangled pairs, one particle to Alice on the left, one to Bob on the right. Each runs into an analyzer set to some angle, and each gives a plain ±1: a coin-flip, 50/50, no matter the angle. Nothing you measure on one side tells you anything on its own.

The story is in the correlation between the two sides. The graph along the bottom plots that correlation, E = ⟨A·B⟩, against the angle between the analyzers. For real entangled particles it traces a smooth cosine, E(θ) = −cos θ (the cyan curve). The dashed line is the best a "local-realist" world could ever do: particles carrying secret instructions agreed on at the source. The cosine bows outside that line, and that gap is the whole game.

The gauge on the left turns the gap into one number: the CHSH parameter S. Any world where the particles had definite, locally-carried properties is stuck at |S| ≤ 2. Quantum mechanics, and our universe, runs all the way up to 2√2 ≈ 2.83. Flip the toggle between the quantum source and a local hidden-variable source and watch one sail past 2 while the other can't. Drag the analyzer angle, add detector noise, sweep the full curve, or take the guided tour.

This is the second of the site's quantum pieces, the partner to the Double-Slit. The slit showed a single particle behaving like a wave of possibility. Entanglement asks the next question: when that possibility is shared between two particles, how real are their properties before you look? Einstein's answer was the sane one: they must be real and local, fixed at the source, and quantum mechanics is just too coarse to see them. He was, as far as any experiment can tell, wrong.

That is why it sits at the heart of the site's recurring question, what is actually real, alongside the spacetime pieces. Where Black Hole and Wormhole bend the stage, entanglement bends our grip on the actors: it says the world cannot be both local (no instantaneous influence across distance) and real (properties that exist before measurement). One of those has to go. And, unlike most foundational arguments, this one was settled in a laboratory, and won the 2022 Nobel Prize.

Two particles are made in a joint singlet state. Measure Alice's along angle a and Bob's along b, each gets ±1, and quantum mechanics predicts their correlation depends only on the angle between the settings, θ = a − b: E(a, b) = −cos(a − b). At equal settings the outcomes are perfectly anti-correlated (E = −1); at right angles, uncorrelated (E = 0). Crucially, each side alone is still 50/50, so you cannot signal with entanglement, and relativity is safe.

To test reality itself, Clauser, Horne, Shimony & Holt (1969) combined four such correlations into one number, the CHSH parameter: S = E(a, b) − E(a, b′) + E(a′, b) + E(a′, b′).

Bell's theorem (1964): in any theory where the outcomes are set by local hidden variables carried by the particles, |S| ≤ 2. No exceptions, no loopholes in the maths, it is a theorem. Quantum mechanics, with the cosine correlation and the right angles, reaches |S| = 2√2 ≈ 2.83, the Tsirelson bound. (In this simulation the four terms collapse, by symmetry, to a single spacing φ, so the meter shows S = 3·E(φ) − E(3φ), maximal at φ = 45°.)

The simulation runs two sources. The quantum one samples each pair from the exact singlet probabilities. The local hidden-variable one gives each pair a shared random instruction λ and lets each particle answer from λ and its own local angle, a genuine local-realist model. Watch it: its correlation can only ever be the straight line, and its S sits at 2. It physically cannot reach the cosine. A noise slider mimics imperfect sources (real experiments land near S ≈ 2.4, not 2.83), and below about 71% visibility the violation vanishes entirely.

The toggle is the whole argument in one switch.

• Quantum (entangled singlet). The particles share one state with no separate properties of their own until measured. Correlations follow −cos θ; the CHSH meter climbs to ~2.83 and the verdict reads local realism ruled out. This is our universe. Verdict: violates Bell, nonlocal/non-real.
• Local hidden-variable (secret instructions). Einstein's preferred world: each particle leaves the source carrying a definite answer for every possible measurement. Reasonable, intuitive, local. But its correlations are forced onto the straight line, and its CHSH parameter cannot pass 2. Verdict: obeys Bell, and disagrees with experiment.

The point of running both is to feel the trap close: the local-realist universe is the sensible one, and it is the one nature rejects.

The honest line between what is exact and what is dialled for display:

In one line: the −cos θ correlation, the |S| ≤ 2 theorem, the 2√2 quantum ceiling, the experimental violation and the no-signaling rule are exact, established physics; only the event rate, the noise model and the analyzer cartoon are dialled for display, and what the violation means about reality is the part still argued over.

• Einstein, A., Podolsky, B., & Rosen, N. (1935). Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Phys. Rev. 47, 777. The EPR argument for hidden variables.
• Bohm, D. (1951). Quantum Theory. Prentice-Hall. The spin-½ ("Bohm") reformulation of EPR used here.
• Bell, J. S. (1964). On the Einstein Podolsky Rosen Paradox. Physics 1, 195. The inequality that made it testable.
• Clauser, J. F., Horne, M. A., Shimony, A., & Holt, R. A. (1969). Proposed Experiment to Test Local Hidden-Variable Theories. Phys. Rev. Lett. 23, 880. The CHSH form.
• Freedman, S. J., & Clauser, J. F. (1972). Experimental Test of Local Hidden-Variable Theories. Phys. Rev. Lett. 28, 938. First experimental violation.
• Aspect, A., Dalibard, J., & Roger, G. (1982). Experimental Test of Bell's Inequalities Using Time-Varying Analyzers. Phys. Rev. Lett. 49, 1804. Closing the locality loophole.
• Cirel'son (Tsirelson), B. S. (1980). Quantum Generalizations of Bell's Inequality. Lett. Math. Phys. 4, 93. The 2√2 bound.
• Hensen, B., et al. (2015). Loophole-free Bell inequality violation using electron spins separated by 1.3 km. Nature 526, 682. Plus Giustina et al. and Shalm et al., 2015.
• The Nobel Prize in Physics 2022, Aspect, Clauser & Zeilinger, "for experiments with entangled photons, establishing the violation of Bell inequalities."