Both readings were written on the same line

Two qubits need two wires, and the amplitudes sit on pairs of readings, of which there are four, written 00, 01, 10 and 11. A product state splits into a line for the first qubit and a line for the second, each with amplitudes of its own. Entanglement is the name for a joint state that cannot be split into two such lines.

Start both wires certain to read 0, so the only amplitude is the one on 00. H on the first wire rewrites that wire as an equal superposition with matching signs and leaves the second wire certain to read 0, which is an equal superposition of 00 and 10, with nothing on 01 or 11. A controlled NOT then flips the second wire in any term where the first wire would read 1, and leaves the second wire alone where the first would read 0. The 00 term stays 00 and the 10 term becomes 11, so the state that remains has equal amplitudes on 00 and 11 and zero amplitude on the other two pairs.

John Bell’s 1964 paper showed that some correlations in this family cannot be copied by answers fixed in advance, if each wire’s answer depends only on how that wire is read. The equal superposition of 00 and 11 is the member of that family which fits on one sheet of graph paper. Square either amplitude and the chance of that pair is a half, while the chance of 01 is zero and the chance of 10 is zero.

Read the first wire in the computational basis, the arrangement that distinguishes 0 from 1, and the click is 0 or 1 with equal chance, with no way to ask the button for a particular one. A reading of 0 leaves the pair in 00, so the second wire is then certain to read 0, and a reading of 1 leaves 11, so the second wire is then certain to read 1. The bench also prepares the earlier superposition of 00 and 10, and there the second wire stays certain to read 0 whichever bit comes back from the first.

Taken alone, before the two bits are laid side by side, each wire of the shared pair is still a fair toss. Keith can press what he likes on his own wire after the pair has been prepared, and Malcolm’s click, seen without Keith’s bit next to it, remains an even chance of 0 or 1. The agreement shows up only once the two bits are on the same table, and bringing them there takes a letter or a phone call. Einstein, Podolsky and Rosen wrote about correlations of this family in 1935, and a slip saying that both wires will read 0, or that both will read 1, can still imitate the matches when both wires are read in this same arrangement. Bell’s argument needs further arrangements than that 0-and-1 reading, and those sums are taught on Quantum Computing Courses.