In 1981 the Northern Irish physicist John Bell opened a paper with a joke about a colleague. Reinhold Bertlmann, an Austrian physicist, always wore socks of different colors. If you saw that one of his feet wore a pink sock, you knew at once, without looking, that the other sock was not pink. The socks were simply chosen that way in the morning. Bell's point was that quantum entanglement looks at first like Bertlmann's socks, and that it is not like them at all. This course is about the difference.

A quick reminder about quantum states

Quantum mechanics describes a system, such as an electron or a particle of light called a photon, by its quantum state. The state does not usually fix the result of every measurement. Instead it gives probabilities. A photon can be in a state where a test of its polarization (the direction in which its electric field vibrates) has an even chance of passing or being blocked by a filter. This is often called superposition. Here we need only one fact about it: in the standard theory, the quantum state gives the probabilities of measurement results, not a hidden list of answers.

When two systems share one state

Things get stranger when two systems interact and then fly apart. Quantum mechanics often cannot give each one a state of its own. It gives only a single state for the pair. This is entanglement: a pair, or a larger group, is entangled when the whole has a definite quantum state but its parts do not have separate states of their own.

A standard example is two photons produced together so that their polarizations match. Measure either photon with a filter at any angle and the result is random: pass or block, fifty fifty. Yet if both photons are measured with filters at the same angle, the two results always agree, whatever angle is chosen. Each result is a coin toss, but the two coins always land the same way.

The word and its author

Erwin Schrödinger, the Austrian physicist, gave the idea its name in 1935. Writing in German he used the word Verschränkung, and in an English paper that same year he chose "entanglement". Schrödinger argued that entanglement was not one odd feature among many but the defining feature of quantum mechanics, the one that forced a complete break with classical ways of thinking.

Why it is not just socks

Correlations are everywhere in daily life, and most have a dull explanation. Split a pair of gloves between two boxes, send one to Lisbon and one to Tokyo, and the person in Tokyo who finds a left glove knows Lisbon has the right. The information was fixed when the boxes were packed. Physicists call this a common cause: a classical correlation comes from properties the objects carried from the start.

Entangled pairs show correlations that seem, at first, to work the same way. The puzzle is that they hold for any choice of measurement, including choices made after the particles have separated. With gloves there is only one question to ask: left or right? With photons you can ask about polarization at any angle you like, and the answers still line up in the pattern quantum mechanics predicts. As later levels show, Bell proved in 1964 that no set of properties carried from the start can reproduce all the quantum correlations. The experiments that followed made entanglement one of the best tested facts in physics.

What entanglement is not

Popular articles sometimes say that measuring one particle instantly changes its partner, as if a signal raced between them. Entanglement does not let anyone send a message, and no experiment has detected anything traveling between the partners. What it shows is that the results at two places are linked in a way that neither local signals nor shared instructions can explain. Level 6 returns to this point.

Worked example: a pair of photons

Imagine a lab where a laser shines into a special crystal. Now and then, one laser photon splits into two lower energy photons that leave in different directions, entangled so their polarizations match. One photon goes to a detector on the left side of the room, run by a student named Ana; the other goes to a detector on the right, run by Ben.

Ana sets her filter upright. Over a thousand photons, about half pass and half are blocked, in an order that looks completely random. Ben, with his filter also upright, sees the same thing: a random string of passes and blocks. Only when they meet and line up their records, photon by photon, do they find that every pass on Ana's side matches a pass on Ben's, and every block matches a block.

Now they both tilt their filters to forty five degrees. Again each sees randomness, and again the records match perfectly. A glove story can explain matching for one setting. The question that drives the rest of this course is whether any such story can explain the full pattern across all settings, including the runs where Ana and Ben choose different angles. Einstein suspected the answer was yes. Bell showed how to find out.