The phenomenon 01 · Established physics
Spooky Action at a Distance
Perfectly correlated outcomes, and not a single bit of signal.
You can make two particles together so that they share a single linked state, then send them to two people, Alice and Bob, as far apart as you like. When each of them measures their particle, the results always line up. Say that whenever Alice gets one answer, Bob gets the opposite. Every time, no matter the distance between them. It looks for all the world as if the two had agreed on their answers in advance.
But you cannot send a message this way
This is the part pop science usually gets wrong. On her own, Alice just sees a random string of results, and so does Bob. The striking pattern only shows up after the two of them get together and compare notes, over an ordinary phone line, no faster than light. Nothing actually crosses the gap between them.
The tempting picture is a pair of gloves. Split a pair into two boxes, ship them apart, and the moment you open yours and find the left glove you know the other box holds the right. But entanglement is stranger than that. With gloves the answer was already fixed when they were packed. Here the answer isn't decided until the measurement actually happens. The next figure, Bell's test, is exactly how we know that.
The two particles occupy one shared entangled state. For the spins of two electrons that state is the singlet, a state with zero total spin in which the two are always found pointing opposite ways. Measure both along the same direction and the outcomes are perfectly anti-correlated. Up here means down there, every single trial. Yet each side on its own behaves like a fair coin: 50/50 up or down, with no pattern at all.
Now let Alice and Bob measure along different angles. The correlation doesn't simply switch off. It weakens smoothly as the angle between their settings grows, and that smooth angle-dependence is the whole crux. Einstein, Podolsky and Rosen (EPR, 1935) argued that the only sensible explanation was that each particle secretly carried pre-set answers for every possible angle, the so-called "hidden variables", and that quantum mechanics was just an incomplete description of them.
Could such pre-set answers reproduce the angle-dependence? That is Bell's question, and the next figure settles it. But notice already why no signal is possible: because each side alone is perfectly random, nothing Alice chooses to do shows up in Bob's results. There is correlation to find later, never a message to read now.
The singlet state of two spin-½ particles is
\[ |\psi^-\rangle = \tfrac{1}{\sqrt{2}}\big(|01\rangle - |10\rangle\big), \]
and for spin measurements along unit directions \(\mathbf a, \mathbf b\) the correlation is
\[ E(\mathbf a,\mathbf b) = \langle \psi^-|\,\sigma_{\mathbf a}\!\otimes\!\sigma_{\mathbf b}\,|\psi^-\rangle = -\,\mathbf a\cdot\mathbf b = -\cos\theta_{\mathbf a\mathbf b}. \]
This state is non-separable: it cannot be written as any product \(\rho_A\otimes\rho_B\), nor as a classical mixture of products, so the correlations are not carried by shared local properties. Crucially, the reduced state on either side is maximally mixed, \(\rho_A = \operatorname{Tr}_B|\psi^-\rangle\langle\psi^-| = \tfrac12 \mathbb{I}\). Every local measurement is therefore uniformly random and, crucially, completely independent of the distant setting.
That independence is the no-communication theorem: because Alice's marginal statistics do not depend on Bob's choice of measurement, no information can be transmitted by entanglement alone. The correlations are revealed only by bringing the two records together through a classical, light-speed-limited channel. Entanglement is a resource for correlation, never for signalling.
Pairs born at the centre fly to two distant stations; the outcomes always come out opposite. Yet each column alone is random, so no message ever crosses. The singlet: same setting → perfectly opposite, each side 50/50. The link is correlation, not a signal. \(|\psi^-\rangle\) with \(E=-\cos\theta\); each reduced state is \(\tfrac12\mathbb{I}\). Locally random ⇒ nothing is signalled.
Bell's test 02 · Settled · Clauser · Aspect · Zeilinger
Bell's Theorem
A number Einstein's world can't exceed, and reality does.
Albert Einstein thought quantum particles must have secret rules hidden inside them, deciding their behaviour in advance. John Bell created a test to see if Einstein was right. He showed that if particles were just following pre-programmed, ordinary rules, their matching behaviour could only go so far. But when we test real quantum particles, they match up way more often than ordinary rules allow. It proves that the quantum world breaks normal rules of reality. Watch the meter sail over the dashed line.
Bell's idea, in the practical CHSH form: each side chooses between two measurement settings, and from many trials you build four correlations and combine them into a single number,
\[ S = E(a,b) - E(a,b') + E(a',b) + E(a',b'). \]
Any theory in which the particles carry pre-set local values, what we call a local hidden-variable theory, must obey Bell's inequality \(|S| \le 2\). Quantum mechanics does not. With the singlet and the right choice of angles it reaches
\[ |S| = 2\sqrt{2} \approx 2.83, \]
the maximum quantum mechanics itself allows (Tsirelson's bound). The meter cycles the settings: \(S\) climbs above the classical line at 2 and tops out at \(2\sqrt2\) when the angles are optimal, then eases back. Anything above 2 is a region no local-hidden-variable theory can enter.
Why \(S\le 2\) classically
If each particle carries pre-set outcomes \(a,a',b,b' \in \{+1,-1\}\) (local realism, no faster-than-light influence), then \(ab - ab' + a'b + a'b' = a(b-b') + a'(b+b')\); one of \((b-b'),(b+b')\) is \(0\) and the other \(\pm2\), so the bracket is exactly \(\pm2\) for every assignment, and averaging gives \(|S|\le2\). Quantum mechanics, with \(E(a,b) = -\cos\theta\), gives \(|S| = |{-}3\cos\delta + \cos 3\delta|\), maximised at the \(45^\circ\) spacing to \(2\sqrt2\).
The experiments, and the loopholes
A clean refutation has to close three gaps: the locality loophole (settings chosen so slowly a light-speed influence could coordinate them), the detection / fair-sampling loophole (only a biased subset of pairs detected), and the freedom-of-choice loophole (the settings correlated with the source). Clauser & Freedman ran the first Bell test (1972); Aspect (1981–82) switched the settings fast enough to close locality; and in 2015 three groups (Delft, NIST, Vienna) closed the major loopholes together in single experiments.
The 2022 Nobel Prize in Physics went to John Clauser, Alain Aspect and Anton Zeilinger for these Bell-inequality experiments and for pioneering quantum information science, the foundation of quantum cryptography, quantum computing and quantum teleportation.
Stated precisely. The experiments rule out local hidden variables, that is local realism. They do not establish faster-than-light signalling, and nothing here does: entanglement still cannot send information. What fell was Einstein's assumption that the world is local and pre-determined, not his cosmic speed limit.
Two players, random questions, answers compared after the fact. No pre-agreed strategy beats the dashed line, yet quantum pairs sail past it. The CHSH number \(S\): classical theories stay \(\le 2\); quantum reaches \(2\sqrt2 \approx 2.83\). Both bounds are marked: watch \(S\) cross 2. \(|S|\le2\) for local hidden variables; \(2\sqrt2\) for quantum. Loopholes closed by 2015: local realism is out, FTL signalling still not in.