= Born rule
{c}
{title2=$p_j=|\langle j|\psi\rangle|^2$}
{wiki}
The Born rule connects a <quantum state> to measurement <probabilities>. Let $\{|j\rangle\}$ be a complete <orthonormal basis>, finite or countably infinite, for a <Hilbert space>, and let
$$
|\psi\rangle=\sum_j a_j|j\rangle,
\qquad
a_j=\langle j|\psi\rangle,
\qquad
\langle\psi|\psi\rangle=1.
$$
An ideal measurement distinguishing these basis vectors returns label $j$ with <probability>
$$
p_j=|a_j|^2=|\langle j|\psi\rangle|^2.
$$
This assignment is a physical measurement rule. Its compatibility with total <probability> one follows from <orthonormality> and normalization:
$$
\sum_jp_j=\sum_j|a_j|^2
=\sum_{j,k}a_j^*a_k\langle j|k\rangle
=\langle\psi|\psi\rangle=1.
$$
The <probabilities> are the squared <moduli>, not the amplitudes themselves or the squares $a_j^2$. https://quantum.cloud.ibm.com/learning/en/courses/basics-of-quantum-information/single-systems/quantum-information[Watrous, IBM Quantum Learning, Quantum information, Measuring quantum states]{ref}.
If the basis is an <eigenbasis> of an <observable>, the reported numerical value is its <eigenvalue>. Several vectors can share one <eigenvalue>; the <probability> of that value is then the sum over the corresponding basis labels. More generally, a <projective measurement> with <orthogonal projections> $P_r$ assigns
$$
p_r=\langle\psi|P_r|\psi\rangle=\|P_r|\psi\rangle\|^2,
\qquad
p_r=\operatorname{Tr}(\rho P_r)
$$
for a <pure state> and a general <density matrix> $\rho$, respectively. This also covers <mixed states> and degenerate <eigenspaces>. The <Born rule> determines outcome statistics; the ideal conditional state change is specified separately by the <Lüders rule>. See <projective measurement> for that formulation and its assumptions. https://quantum.cloud.ibm.com/learning/en/courses/general-formulation-of-quantum-information/general-measurements/formulations-of-measurements[Watrous, IBM Quantum Learning, Mathematical formulations of measurements, Projective measurements]{ref}.
For a worked example, take the normalized <qubit> <quantum state>
$$
|\psi_\phi\rangle=\frac{\sqrt3}{2}|0\rangle+\frac{e^{i\phi}}2|1\rangle.
$$
The squared <moduli> add to $3/4+1/4=1$. A <computational-basis measurement>, equivalently a measurement of the $Z$ <Pauli matrix>, has
$$
p_0=\left|\frac{\sqrt3}{2}\right|^2=\frac34,
\qquad
p_1=\left|\frac{e^{i\phi}}2\right|^2=\frac14.
$$
These <probabilities> do not depend on the relative <quantum phase> $\phi$. Now choose the <Hadamard basis>
$$
|\pm\rangle=\frac{|0\rangle\pm|1\rangle}{\sqrt2},
$$
whose vectors are <eigenstates> of the $X$ <Pauli matrix>, also used by the <Pauli X gate>. The new <probability amplitudes> and <probabilities> are
$$
b_\pm=\langle\pm|\psi_\phi\rangle
=\frac{\sqrt3\pm e^{i\phi}}{2\sqrt2},
\qquad
p_\pm=|b_\pm|^2
=\frac12\pm\frac{\sqrt3}{4}\cos\phi.
$$
Both <probabilities> are nonnegative and their sum is one. The relative <quantum phase> becomes visible because the two <probability amplitudes> add before their squared <modulus> is taken:
* At $\phi=0$, $p_+=(2+\sqrt3)/4\simeq0.9330127$ and $p_-\simeq0.0669873$.
* At $\phi=\pi/2$, $p_+=p_-=1/2$.
* At $\phi=\pi$, $p_+\simeq0.0669873$ and $p_-\simeq0.9330127$.
\Image[/born-rule-basis-probabilities.png]
{title=The same qubit preparation has phase-independent Z probabilities and phase-sensitive X probabilities}
A <Hadamard gate> followed by a <computational-basis measurement> realizes these $X$ statistics, since $H|\pm\rangle=|0\rangle,|1\rangle$, respectively. Compare the two bases using fresh, identically prepared copies: a preceding measurement may change the state. The <quantum superposition> section shows why replacing the preparation by the <mixed state> $\operatorname{diag}(3/4,1/4)$ keeps the $Z$ statistics but gives $p_+=p_-=1/2$ for every $\phi$.
The usefulness of the <Born rule> is this conversion from <linear algebra> to experimental predictions: calculate overlaps in the basis actually measured, then take squared <moduli>. Its appealing geometry is that changing basis redistributes probability while preserving total squared <norm>. The same rule relates an <energy eigenstate> expansion to an <energy measurement> and a position <wavefunction> to a <probability density>.
Born's 1926 paper on <quantum scattering> interpreted the outgoing-wave coefficients in terms of the <probabilities> of collision outcomes; a corrective footnote specified proportionality to the square of the amplitude. This is a historical scattering formulation, rather than the general <density matrix> formula above. <Zur Quantenmechanik der Stoßvorgänge by Max Born>[Born, Zur Quantenmechanik der Stoßvorgänge, Zeitschrift für Physik 37, 863–867, 1926]{ref}; https://www.neo-classical-physics.info/uploads/3/0/6/5/3065888/born_-_qm_for_collisions_i.pdf[English translation of Born's paper, p. 3 and footnote 1]{ref}.
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