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--- probability-amplitude [June 13, 2026 at 03:46] – external edit 127.0.0.1
+++ probability-amplitude [June 13, 2026 at 03:49] (current) – Ivan Janevski
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 **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]].
-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:
+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.
-$$P(0) = |a|^2 \qquad P(1) = |b|^2$$
+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.
-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.
+$$P_0 = |a|^2 \qquad P_1 = |b|^2$$
 ## Normalization
@@ Line 30: / Line 30: @@
 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 $-1$.
+**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 $-1$.