PROBABILITY, NOT MAGIC: HOW MEASUREMENT CONNECTS TO THE SPHERE
The Born Rule, in its geometric form, with fully worked numeric examples.
Every previous lesson in this module has used words like "leans toward," "roughly," and "50/50." That's been fine for building intuition, but it won't survive contact with actually programming a quantum computer (Module 11) or predicting the output of a real circuit.
We need a precise, quantitative rule that converts "where the arrow points on the Bloch sphere" into "exact probability of each measurement outcome." This rule is called the Born Rule.
The rule is named after physicist Max Born, who proposed in 1926 that the quantity describing a quantum state should be squared to get a probability. This single idea — "square it to get a probability" — earned him the 1954 Nobel Prize in Physics, and it's the exact reason you'll see squared terms everywhere in quantum mechanics.
The Formula (Geometric Form)
Recall from Lesson 3: θ (theta) is the angle measured from the North Pole (|0⟩), ranging from 0° to 180°.
The Born Rule states:
P(measure 0) = cos²(θ / 2)
P(measure 1) = sin²(θ / 2)
That's it — two formulas, both depending only on θ (not on φ — we'll explain why below). Note the "half-angle": we divide θ by 2 before taking the cosine or sine.
Worked Examples
Example 1: North Pole (θ = 0°)
- θ/2 = 0°
- P(0) = cos²(0°) = (1)² = 1 → 100%
- P(1) = sin²(0°) = (0)² = 0 → 0%
- ✓ Matches Lesson 4's claim about
|0⟩
Example 2: South Pole (θ = 180°)
- θ/2 = 90°
- P(0) = cos²(90°) = (0)² = 0 → 0%
- P(1) = sin²(90°) = (1)² = 1 → 100%
- ✓ Matches Lesson 4's claim about
|1⟩
Example 3: Equator (θ = 90°), e.g., |+⟩
- θ/2 = 45°
- cos(45°) = √2/2 ≈ 0.7071
- P(0) = cos²(45°) = (0.7071)² = 0.5 → 50%
- P(1) = sin²(45°) = (0.7071)² = 0.5 → 50%
- ✓ Matches Lesson 5's claim that all equator states give 50/50
More Examples: Non-Pole, Non-Equator States
| Θ | POSITION | P(0) | P(1) |
|---|---|---|---|
| 30° | Close to North Pole | 93.3% | 6.7% |
| 60° | Leaning toward |0⟩ | 75% | 25% |
| 90° | Equator | 50% | 50% |
| 120° | Leaning toward |1⟩ | 25% | 75% |
| 150° | Close to South Pole | 6.7% | 93.3% |
Notice the symmetry: θ = 120° is the "mirror image" of θ = 60° around the equator (60° from the North Pole vs 60° from the South Pole), and the probabilities are exactly swapped. This symmetry will always hold — a useful sanity check.
Why φ (Phase) Doesn't Appear at All
The Z-basis measurement probability genuinely does not depend on phase at all.
This is exactly the mathematical reason behind Lesson 5's surprising fact: |+⟩, |i⟩, |−⟩, |−i⟩ all sit at θ = 90° (they only differ in φ), so they all get the exact same 50%/50% answer.
Phase becomes measurable once you change which basis you measure in (X or Y instead of Z) — which effectively asks a question aligned with a different axis. The precise mechanics of this basis-change are covered in Module 8.
Practice Questions
Test your understanding
KEY TAKEAWAYS
Remember these points
The Born Rule (geometric form) converts a Bloch sphere angle into an exact measurement probability: P(0) = cos²(θ/2), P(1) = sin²(θ/2)
This formula is symmetric: an angle θ away from the North Pole gives the same probability magnitude as (180° − θ) away from the South Pole, just swapped
This Z-basis probability depends only on θ, never on φ — exactly why all four equator states from Lesson 5 share identical 50/50 statistics
Phase becomes measurable when you change measurement basis (Module 8) or use interference (Module 4, 9) — it's just invisible to this particular Z-basis formula
This formula is exactly what you'll compute programmatically once you reach Qiskit in Module 11 — every simulated "shot" of a circuit is a random draw governed by probabilities like these