Normalisation Constraint
The squared amplitudes of a qubit must sum to 1, ensuring total probability is 100 %.
Formula
Variables
Example
α = √(0.3), β = √(0.7) → 0.3 + 0.7 = 1 ✓
How it works
A qubit is described by two complex amplitudes, one for measuring |0⟩ and one for measuring |1⟩. The rule |α|^2 + |β|^2 = 1 comes from the fact that these squared magnitudes are probabilities, and all possible measurement outcomes must add up to 100%. If the sum were not 1, the state would not represent a physically valid qubit.
When to use it
Use this when checking whether a proposed qubit state is valid or when finding an unknown amplitude from a known one. It is also useful before doing quantum measurements, normalizing state vectors, or verifying answers in basic quantum computing problems.
Worked example, step by step
Suppose a qubit has amplitude α = 0.6 for |0⟩. Step 1: Square the known amplitude: |α|^2 = 0.6^2 = 0.36 Step 2: Use the normalisation constraint: |α|^2 + |β|^2 = 1 Step 3: Substitute: 0.36 + |β|^2 = 1 Step 4: Solve for |β|^2: |β|^2 = 1 - 0.36 = 0.64 Step 5: Take the square root: |β| = √0.64 = 0.8 Final check: 0.36 + 0.64 = 1 ✓
Common mistakes to avoid
- adding amplitudes directly instead of adding their squared magnitudes
- forgetting that complex amplitudes use absolute value, so the probability is |α|^2 not α^2
- using numbers that do not sum to 1 and treating the state as valid anyway
Did You Know?
This constraint is why qubits live on the surface of a sphere (the Bloch sphere) rather than inside it.
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