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--- probability-amplitude [May 08, 2026 at 22:04] – yanevskiv
+++ probability-amplitude [June 13, 2026 at 03:49] (current) – Ivan Janevski
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+# 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 $-1$.