Entanglement

Two qubits, one prepared here and one carried light-years away. Measure yours and get a random 0 or 1 — and in that instant the distant qubit is guaranteed to give the matching answer, even though nobody has touched it. Einstein called it "spooky action at a distance" and hated it; it turned out to be real, and it is the engine of nearly everything quantum computers do that classical ones can't. The name for this deep, non-classical link is entanglement.

Separable states: two qubits that don't really interact

A two-qubit state is called separable (or a product state) when it factors into a state for the first qubit times a state for the second:

Each qubit has its own honest description; knowing one tells you nothing new about the other. For example |0\rangle \otimes |1\rangle = |01\rangle is separable (both qubits are definite), and so is

|{+}\rangle \otimes |{+}\rangle = \tfrac12\big(|00\rangle + |01\rangle + |10\rangle + |11\rangle\big),

which looks tangled but simply multiplies out from two independent |{+}\rangle qubits. A state that cannot be written as any such product — no matter how you choose a, b, c, d — is entangled. The two qubits then share a description that lives in neither qubit alone.

The Bell state — and why it isn't a product

The most famous entangled state is the Bell state |\Phi^+\rangle:

|\Phi^+\rangle = \frac{1}{\sqrt2}\big(|00\rangle + |11\rangle\big).

Worked example — prove it is not separable. Suppose it were a product. Then for some amplitudes we could write

Match this term by term against |\Phi^+\rangle = \tfrac{1}{\sqrt2}|00\rangle + 0\,|01\rangle + 0\,|10\rangle + \tfrac{1}{\sqrt2}|11\rangle:

ac = \tfrac{1}{\sqrt2}, \qquad bd = \tfrac{1}{\sqrt2}, \qquad ad = 0, \qquad bc = 0.

The last two force ad = 0 and bc = 0, so at least one of each pair vanishes. But if a = 0 then ac = 0 \ne \tfrac{1}{\sqrt2}, and if d = 0 then bd = 0 \ne \tfrac{1}{\sqrt2} — every branch contradicts the requirement that ac and bd are non-zero. No choice of a, b, c, d works, so |\Phi^+\rangle is entangled. \blacksquare

Worked example: the correlation table

Measuring both qubits of |\Phi^+\rangle in the computational basis, the Born rule gives probability |\tfrac{1}{\sqrt2}|^2 = \tfrac12 for 00 and \tfrac12 for 11 — and zero for 01 or 10. The two bits are perfectly correlated: they always agree. The grid below fills in only the matching diagonal.

Read it as a story about timing. Measure the first qubit alone: you get 0 or 1 at a fair 50/50. But the moment you see (say) 0, the whole state collapses to |00\rangle, so the second qubit is now determined to read 0 as well — no matter how far away it sits, and no matter who measures it first.

Contrast: a product state has independent qubits

Compare |{+}\rangle \otimes |{+}\rangle. Its four outcomes 00, 01, 10, 11 each have probability |\tfrac12|^2 = \tfrac14. Now measure the first qubit and get 0: the state collapses to |0\rangle \otimes |{+}\rangle, and the second qubit is still a fresh |{+}\rangle — a 50/50 coin, exactly as before you looked. The first result told you nothing about the second.

That is the whole distinction. In a product state, measuring one qubit leaves the other's odds unchanged (independent outcomes). In an entangled state, one measurement rewrites the other's odds (correlated outcomes) — here from 50/50 to a certainty.

a two-qubit state is separable if it factors as |\psi\rangle \otimes |\varphi\rangle, and entangled if it does not;
the Bell state |\Phi^+\rangle = \tfrac{1}{\sqrt2}(|00\rangle + |11\rangle) is entangled — no product of single-qubit states equals it;
measuring one qubit of |\Phi^+\rangle gives 0/1 at 50/50, and the other qubit then matches with certainty, however far apart they are;
each side's own statistics look completely random, so entanglement cannot send information faster than light.

In 1935, Einstein, Podolsky and Rosen (EPR) argued that the instantaneous correlations of an entangled pair meant quantum mechanics had to be incomplete — surely each qubit secretly carried its answer all along (a "hidden variable"), like two gloves sealed in boxes: open one, and you instantly "know" the other is the right hand. Three decades later John Bell found a way to settle it. He derived an inequality that any local hidden-variable theory must obey, and showed entangled qubits violate it. Experiments have confirmed the violation many times over (earning the 2022 Nobel Prize in Physics). The correlations are not pre-packed answers in boxes — they are a genuinely non-classical link, exactly the "spooky action" Einstein could not accept.

The single most common misconception is that entanglement lets you signal faster than light. It does not. Yes, the outcomes are perfectly correlated — but the person holding the distant qubit sees only a random string of 0s and 1s, with no way to tell whether you have measured your qubit yet, or what you got. The correlation only becomes visible after the two parties compare notes over an ordinary (light-speed-limited) channel. This is the no-signalling principle: local statistics on each side alone are indistinguishable from pure noise, so entanglement transmits no information by itself. It is a shared resource, not a telephone. Building on it, quantum teleportation moves a qubit's state using entanglement plus classical bits — never entanglement alone.