State
Measure
Limits
State Parameters
θ polar angle
0°
moves |0⟩ (north) ↔ |1⟩ (south)
φ relative phase
0°
rotates around equator · changes X/Y basis outcomes
γ global phase
0°
changes the numbers — the point does not move
Notable States
State Vector
Parametric form
|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩
Current values
|ψ⟩ = 1|0⟩ + 0|1⟩
With global phase γ
|ψ⟩ = eiγ[cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩]
    = α|0⟩ + β|1⟩
α = eiγcos(θ/2) = 1.000
β = ei(γ+φ)sin(θ/2) = 0
|α|² = 1.000 → P(|0⟩)
|β|² = 0.000 → P(|1⟩)
eiγ multiplies the entire state equally.
It cancels out of every |·|² calculation.
→ global phase is physically unobservable — drag γ to see
what to discover Drag γ (global phase) — the numbers change, the dot doesn't move. Then drag φ (relative phase) with θ = 90° and watch the dot orbit the equator. That orbit is real, measurable physics.
Measurement Basis
Z basis · measures |0⟩ and |1⟩ · the poles
Run Measurements
—
run a measurement
Total: 0
|0⟩
|1⟩

—

Reflect

The second measurement gave the same outcome with certainty. Before the first measurement, the state was |+⟩ — genuinely 50/50.

Did measurement reveal a pre-existing value, or create a new one?

What the Bloch Sphere Can Represent
✓ Single qubit ✓ Pure states ✗ Entangled states ✗ Multi-qubit
single-qubit restriction The Bloch sphere represents all pure states of a single qubit — every point on the surface is a valid quantum state. Mixed states (like a qubit affected by noise) live inside the sphere, not on the surface.
Bell State Challenge
The Bell state |Φ⁺⟩ = (1/√2)(|00⟩ + |11⟩) involves two entangled qubits.

Can each qubit be placed on its own Bloch sphere surface?

Both qubits drifted to the center — the maximally mixed state.

Entangled qubits have no individual pure state. The information is in the correlations between them — a 4-component vector. Two Bloch spheres cannot represent this.

Reflect

Each qubit individually is maximally mixed — center of the sphere, not on the surface.

Where did the information go? What would you need instead of two Bloch spheres to fully describe |Φ⁺⟩?