= Probability amplitude
{title2=$a_j=\langle j|\psi\rangle$}
{wiki}
A probability amplitude is a <complex number> that describes the contribution of a <pure state> to a specified measurement outcome. For a normalized <quantum state> $|\psi\rangle$ and a rank-one outcome associated with a normalized vector $|j\rangle$, it is the <inner product> $a_j=\langle j|\psi\rangle$. The <Born rule> gives the corresponding <probability> $p_j=|a_j|^2=a_j^*a_j$, using the <complex conjugate>. Thus an amplitude can be negative or complex, while a <probability> is real and nonnegative. A <wavefunction> is the analogous position amplitude: its squared <modulus> gives a <probability density>, which must be integrated over a region to obtain a <probability>. https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2016/d87c768c4e1181fde8d4366112169719_MIT8_04S16_LecNotes6.pdf[Zwiebach, MIT 8.04, lecture 6, section 2, pp. 3–4]{ref}.
The extra information is the <quantum phase>. If $a=re^{i\theta}$, its <modulus> $r$ fixes $|a|^2=r^2$, but its angle $\theta$ controls how it adds to another contribution. In the <complex plane>, picture amplitudes as arrows: add the arrows, then square the resulting length. For coherent contributions $A$ and $B$ to the same outcome,
$$
|A+B|^2=|A|^2+|B|^2+2\operatorname{Re}(A^*B).
$$
The last term is <interference>. Adding <probabilities> first loses it. Contributions with different relative <quantum phases> can reinforce or cancel even when their individual squared <moduli> are fixed.
For the <qubit> <quantum superposition> $|\psi_\phi\rangle=(\sqrt3/2)|0\rangle+(e^{i\phi}/2)|1\rangle$, project onto the plus vector of the <Hadamard basis>. By linearity, the two contributions combine as
$$
\langle+|\psi_\phi\rangle
=\underbrace{\frac{\sqrt3}{2\sqrt2}}_{A}
+\underbrace{\frac{e^{i\phi}}{2\sqrt2}}_{B}.
$$
Here $|A|^2=3/8$ and $|B|^2=1/8$, while the <interference> term is $(\sqrt3/4)\cos\phi$. At $\phi=0$, the arrows align and the plus <probability> is $1/2+\sqrt3/4\simeq0.9330127$. At $\phi=\pi$, they oppose and that <probability> is $1/2-\sqrt3/4\simeq0.0669873$; cancellation is incomplete because the lengths differ. At $\phi=\pi/2$, the arrows are perpendicular and the cross term vanishes. The minus amplitude uses $A-B$, reversing the cross term. These are alternate <projective measurements> on identically prepared copies, rather than a sequence of measurements on one copy.
\Image[/quantum-relative-phase-interference.png]
{title=Relative quantum phase redistributes Hadamard-basis probabilities; a diagonal mixture stays at one half}
A <global phase> multiplies every contribution by the same factor $e^{i\gamma}$, giving
$$
|\langle j|e^{i\gamma}\psi\rangle|^2
=|e^{i\gamma}\langle j|\psi\rangle|^2
=|\langle j|\psi\rangle|^2.
$$
It therefore changes no outcome <probability>. Changing a relative <quantum phase> can change the angle between contributions and hence their <interference>. Keeping amplitudes until the last step makes <quantum circuits> and <matter-wave interference> calculable; discarding their angles prematurely discards physically testable information. The <Born rule> example compares two measurement bases explicitly.
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