The Bloch Sphere

A single qubit is not just “0 or 1”. Every pure state it can be in corresponds to exactly one point on the surface of this sphere, and every point on the surface is a legal state. That is the whole space — nothing more, nothing less.

The two poles are the only two things a measurement can ever hand you: the top is |0⟩ and the bottom is |1⟩. Everything else on the surface is a blend of the two, and pointing the arrow there does not mean the qubit is “secretly” 0 or 1 — it means that when you measure, the answer is genuinely decided at that moment, with odds set by how far the arrow leans toward each pole.

The polar angle θ is the only thing that sets the measurement odds: P(0) = cos²(θ/2) and P(1) = sin²(θ/2). At the equator (θ = 0.50π) both are exactly 50%.

The azimuthal angle φ is the relative phase between |0⟩ and |1⟩. Drag the φ slider and watch the two probability readouts: they do not move at all. Spinning around the vertical axis changes the state — |+⟩ and |−⟩ are different states — but it changes nothing you can see in a single measurement in the computational basis.

If one measurement cannot see φ, why track it? Because phase is what makes interference work. Take |+⟩ (φ = 0) and |−⟩ (φ = π): identical 50/50 outcomes, opposite sides of the equator. Apply one more Hadamard and they separate perfectly — |+⟩ becomes |0⟩ and |−⟩ becomes |1⟩, each with certainty. The phase you could not measure directly has been turned into an outcome you can. Every quantum algorithm worth running is built on that trick: arrange the phases so wrong answers cancel and right answers add.

One caveat about the picture: the surface only holds pure single-qubit states. If a qubit is entangled with another one, its own state is mixed and its arrow sits inside the sphere, shortened toward the centre. A maximally entangled qubit — half of a Bell pair — has an arrow of length zero, sitting exactly at the origin. It has no state of its own at all; all of the information lives in the correlation with its partner.