Bloch Sphere — Qubit State & Robertson Uncertainty

|ψ⟩ = cos(θ/2)|0⟩ + eiφsin(θ/2)|1⟩ — orthographic projection (AZ=30°, EL=25°)

Bloch Sphere
Polar angle θ (0° = |0⟩, 180° = |1⟩) 90.0°
Azimuthal angle φ 0.0°
—

Born Rule Probabilities

P(|0⟩)
0.500
P(|1⟩)
0.500

Bloch Vector ⟨σ⟩

⟨σ_x⟩ = sin θ cos φ —
⟨σ_y⟩ = sin θ sin φ —
⟨σ_z⟩ = cos θ —
|r|² (purity) —

Robertson Uncertainty: Δσ_x · Δσ_z ≥ |⟨σ_y⟩|

Δσ_x = √(1 − ⟨σ_x⟩²) —
Δσ_z = √(1 − ⟨σ_z⟩²) —
Product Δσ_x · Δσ_z —
Bound |⟨σ_y⟩| —
Ratio (product / bound) —

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