Spin-½ Uncertainty Ensemble
Robertson bound σ
Sx
·σ
Sy
≥ ½|⟨S
z
⟩| — 200 spins on the Bloch sphere (ħ/2 = 1)
θ (polar angle)
90°
0° (|+z⟩)
90° (|+x⟩)
180° (|-z⟩)
φ (azimuthal angle)
0°
0°
90°
180°
270°
360°
Bloch Sphere State
Measurement Ensemble (100 × S
x
⊕ 100 × S
y
)
Resample Ensemble
⟨S_z⟩ = cos θ
0.000
(ħ/2 units)
σ_x (analytic) / σ_x (MC)
—
σ_x = √(1−sin²θcos²φ)
σ_y (analytic) / σ_y (MC)
—
σ_y = √(1−sin²θsin²φ)
Product / Bound (≥1)
—
σ_x·σ_y ≥ ½|⟨S_z⟩|
σ_x · σ_y (analytic)
—
product
Bound = ½|⟨S_z⟩|
—
Robertson lower bound
MC σ_x (empirical)
—
from 100 S_x samples
MC σ_y (empirical)
—
from 100 S_y samples