Probability from Amplitude

Quantum mechanics does not say which outcome a measurement will produce — it says how likely each outcome is. The rule that converts amplitudes into probabilities is called the Born rule.

Amplitudes and their squares

A single-qubit state is written as

where $\alpha$ and $\beta$ are amplitudes — complex numbers that carry all the information about the state. On their own, amplitudes are not probabilities. To get a probability you must square the modulus of the amplitude:

The normalization condition $|\alpha|^2 + |\beta|^2 = 1$ guarantees that these probabilities always add up to 1.

A worked example

Consider the state

Both coefficients are real and positive, so $|\alpha| = \tfrac{3}{5}$ and $|\beta| = \tfrac{4}{5}$. Squaring gives

Check: $0.36 + 0.64 = 1.00$ ✓. The state is normalized because $3$, $4$, and $5$ form a Pythagorean triple ($3^2 + 4^2 = 5^2$), so dividing by $5$ makes both coefficients valid.

Try it

A qubit is prepared in $|\psi\rangle = \tfrac{3}{5}|0\rangle + \tfrac{4}{5}|1\rangle$. Apply the Born rule and return $P(0)$, the probability of measuring outcome $0$.

Sign in on the full site to ask questions and join the discussion.