|ψ⟩ looks like something you need a degree to read. It is a column of numbers wearing a funny hat.
That is genuinely all. |0⟩ is the column (1, 0). |1⟩ is (0, 1). And a qubit holding a mixture of both is (α, β) — one number for each.
P(0) = |0.6|² = 0.36 · P(1) = |0.8|² = 0.64 · they sum to 1
The one rule they have to follow
Not any two numbers will do. Square them both, add them up, and you must get exactly 1:
|α|² + |β|² = 1
That is just saying "something definitely happens when you look". Draw every allowed pair and you get the surface of a ball — which is exactly the globe from the Fundamentals track.
Gates spin you around that surface. They never take you off it.
Four worth memorising
| Name | The numbers | Where it is |
| --- | --- | --- |
| \|0⟩ | (1, 0) | north pole |
| \|1⟩ | (0, 1) | south pole |
| \|+⟩ | (1, 1)/√2 | on the equator |
| \|−⟩ | (1, −1)/√2 | equator, opposite side |
Look at the last two. |+⟩ and |−⟩ differ only by a minus sign. Measure either and you get 50/50 — identical, indistinguishable.
But one H gate turns that invisible difference into a dead certainty. |+⟩ becomes a definite 0; |−⟩ becomes a definite 1. The minus sign was there all along, waiting to be cashed in.
Why a ball, specifically?
Two complex numbers means four ordinary numbers. The "must add to 1" rule uses one up. And multiplying everything by the same phase changes nothing you could ever detect, so that uses up another.
Four minus two is two. Two numbers is exactly what you need to pin down a spot on a globe: latitude and longitude.
Worth remembering
- A ket is a column of numbers. |0⟩ = (1,0), |1⟩ = (0,1).
- The squares must add to 1, which is why states live on a globe.
- |+⟩ and |−⟩ look identical when measured, but one H tells them apart perfectly.
- The globe picture works for one qubit only.