Two qubits: independence and entanglement

Two dials set the qubits as if they were independent (a product state). The entanglement dial mixes in correlation. Watch the joint amplitudes stop factorizing.

qubit A
|0> |1>
qubit B
|0> |1>
0° (independent)
Joint amplitudes (sign matters)
Joint probabilities = amplitude²
entanglement (concurrence) = 0.00 — the two circles tell the whole story (independent).

The solid arrow on each circle is u0 (the direction you drag); the dashed arrow is u1 (90° rotated). Both arrows' lengths are Schmidt coefficients: solid = cos η (weight of "both at u0"), dashed = sin η (weight of "both at u1"). At η = 0 the solid is full and the dashed is gone: the state is purely "both at u0," a product. Turn up η and weight flows from solid to dashed: the state becomes a mix of "both at u0" and "both at u1." At η = 45° both arrows are equally long (0.71 each): maximally entangled. The formula above shows the same numbers; the arrows are its visual.