# Probability amplitude
**Probability amplitude** is a complex number associated with a possible outcome of a quantum measurement. The probability of that outcome is the squared modulus of the amplitude, a rule known as the [[born-rule|Born rule]].

Classical probability uses real numbers in $[0, 1]$. Quantum mechanics uses complex numbers instead. The reason is interference: amplitudes can add or cancel before squaring, producing effects that no real-valued probability theory can describe.

For a [[qubit]] $\lvert\psi\rangle = a\lvert 0\rangle + b\lvert 1\rangle$, $a$ is the amplitude for outcome 0 and $b$ is the amplitude for outcome 1.  

$$P_0 = |a|^2 \qquad P_1 = |b|^2$$

## Normalization

Since probabilities must sum to 1, the amplitudes of any valid quantum state satisfy:

$$|a|^2 + |b|^2 = 1$$

For an $n$-qubit system with $2^n$ basis states this generalises to $\sum_x |c_x|^2 = 1$. Geometrically, the condition means the state vector has unit length in $\mathbb{C}^{2^n}$. Quantum gates are unitary matrices, which preserve inner products and therefore automatically preserve normalization.

## Interference

The key feature that distinguishes amplitudes from probabilities is that amplitudes can cancel. Two paths leading to the same outcome contribute their amplitudes before squaring, not their probabilities. If path 1 contributes amplitude $c_1$ and path 2 contributes $c_2$ to the same outcome, the total probability is $|c_1 + c_2|^2$, not $|c_1|^2 + |c_2|^2$.

A concrete example: start with $\lvert 0\rangle$ and apply the Hadamard $H$ twice. After the first $H$:

$$H\lvert 0\rangle = \frac{1}{\sqrt{2}}\lvert 0\rangle + \frac{1}{\sqrt{2}}\lvert 1\rangle$$

After the second $H$, each basis state routes through both branches. The amplitude for $\lvert 1\rangle$ receives $+\tfrac{1}{2}$ from the $\lvert 0\rangle$ branch and $-\tfrac{1}{2}$ from the $\lvert 1\rangle$ branch — they cancel. The amplitude for $\lvert 0\rangle$ receives $+\tfrac{1}{2}$ from each branch — they reinforce. The result is $\lvert 0\rangle$ with certainty. If probabilities rather than amplitudes were summed at each step, this cancellation would be impossible.

## Phase

A complex amplitude $c = r e^{i\theta}$ has a magnitude $r \geq 0$ and a phase $\theta \in [0, 2\pi)$. Only the magnitude contributes to probability directly; the phase only matters relative to other terms in the same superposition.

**Global phase** is unobservable: the states $\lvert\psi\rangle$ and $e^{i\theta}\lvert\psi\rangle$ give identical probabilities for every possible measurement.

**Relative phase** between terms is physically meaningful. The states $\lvert +\rangle = \tfrac{1}{\sqrt{2}}(\lvert 0\rangle + \lvert 1\rangle)$ and $\lvert -\rangle = \tfrac{1}{\sqrt{2}}(\lvert 0\rangle - \lvert 1\rangle)$ have identical Z-basis probabilities ($P(0) = P(1) = \tfrac{1}{2}$ for both), but they are physically distinct states. The $-1$ relative phase on $\lvert 1\rangle$ is detectable by measuring in the X basis, where $\lvert +\rangle$ always gives $+1$ and $\lvert -\rangle$ always gives $-1$.