Amplitudes vs Probabilities

A quantum state is described by numbers called amplitudes — not probabilities. Every possible measurement outcome is assigned an amplitude, and those amplitudes can be negative or even complex. Probabilities only appear at the moment you measure.

The state vector

A single qubit in an arbitrary state is written

where $\alpha$ and $\beta$ are complex numbers. The pair $(\alpha, \beta)$ is the state vector. The two coefficients must satisfy the normalisation condition

which simply says "the total probability of all outcomes is 1."

The Born rule

When you measure $|\psi\rangle$ in the computational basis, the probability of each outcome is the squared magnitude of the corresponding amplitude:

This is the Born rule. It is one of the fundamental postulates of quantum mechanics — it cannot be derived from anything more basic; it is how theory connects to experiment.

For example, if $\alpha = \tfrac{3}{5}$ and $\beta = \tfrac{4}{5}$ (both real and positive), then

Note that $0.36 + 0.64 = 1$, as required by normalisation. You can check: $\alpha^2 + \beta^2 = \tfrac{9}{25} + \tfrac{16}{25} = \tfrac{25}{25} = 1$, so the state is indeed normalised.

Why amplitudes, not just probabilities?

A classical probability distribution over {0, 1} is just two non-negative numbers that sum to 1. The extra freedom in amplitudes — the ability to be negative or complex — is what allows quantum states to interfere. Two paths to the same outcome can have amplitudes that cancel (destructive interference) or reinforce (constructive interference). This interference is the engine behind quantum algorithms like Grover search and the quantum Fourier transform; it has no classical analogue.

Try it

Apply the Born rule to find the probability of measuring $|1\rangle$ for the state $\tfrac{3}{5}|0\rangle + \tfrac{4}{5}|1\rangle$.

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