01 · The state
Superposition
A qubit is not a bit that is 0 and 1. It is two complex amplitudes whose squares sum to one — a point on a sphere.
In English
A coin in the air is a bad metaphor. A qubit is more like an arrow on a globe: two numbers (amplitudes) whose sizes-squared are the chances of 0 and 1. Until you look, you have the arrow, not a secret bit hiding inside.
- Try this
- Drag the blob on the sphere. Read the two amplitudes and the two chances. Try the |0⟩, |1⟩, |+⟩ presets.
- Keep this
- A qubit is a direction. The chances are the square of the amplitudes.
Lab · drag the sphere
Drag the ball · or tap |0⟩ |1⟩ |+⟩ |−⟩ on the sphere
State vector
0.71|0⟩ + 0.71|1⟩
|0⟩0.7150%|1⟩0.7150%
P(|0⟩) = cos²(θ/2) = 50.0%·P(|1⟩) = 50.0%
Write a qubit as α|0⟩ + β|1⟩ with |α|² + |β|² = 1. The Bloch sphere is just a map of those two complex numbers (up to a global phase you can ignore) onto a point: θ sets the odds, φ sets the relative phase.
“Both 0 and 1 at once” is a slogan. The state is one vector. Superposition means that vector is not a computational-basis vector — until you measure, which is the next chapter.