= Quantum measurement
{title2=$p_m=\operatorname{Tr}(E_m\rho)$}
{wiki=Measurement_in_quantum_mechanics}
= Measurement in quantum mechanics
{synonym}
A <quantum measurement> turns a <quantum state preparation> into a classical outcome, such as a detector label. Two questions must be distinguished: how likely is each outcome, and what <quantum state> remains conditional on that outcome? A <positive operator-valued measure> answers the first; a <quantum instrument> answers both. This distinction is useful whenever a measured system participates in a later experiment, including <quantum state tomography> and <measurement-based quantum feedback>.
For a <density operator> $\rho$, the <Born rule> gives $p_m=\operatorname{Tr}(E_m\rho)$, where the $E_m$ are <positive operators> satisfying $\sum_m E_m=I$. Here $I$ is the <identity operator>, and $\operatorname{Tr}$ denotes the <trace>. Positivity makes every $p_m$ nonnegative; completeness makes their sum one. For a <pure state>, $\rho=|\psi\rangle\langle\psi|$ and $p_m=\langle\psi|E_m|\psi\rangle$. A <projective measurement> is the special case in which the $E_m$ are <orthogonal projections> onto mutually orthogonal subspaces.
For a <qubit>, choose the <computational basis> and write a normalized <pure state>, up to <global phase>, as
$$
|\psi(\theta,\phi)\rangle=\cos(\theta/2)|0\rangle+e^{i\phi}\sin(\theta/2)|1\rangle,
\qquad 0\leq\theta\leq\pi,\quad 0\leq\phi<2\pi.
$$
The angle $\phi$ is a relative <quantum phase>; at the poles it does not distinguish states. Measuring the <Pauli matrix> $Z$ reports $+1$ for label zero and $-1$ for label one, with
$$
p_0=\cos^2(\theta/2),\qquad p_1=\sin^2(\theta/2).
$$
These <probabilities> describe repeated identical preparations. They do not assign a hidden, pre-existing value to every possible measurement basis.
\Image[/qubit-measurement-probabilities.png]
{title=Qubit measurement probabilities: Z measures populations, while X detects relative phase}
The <Born rule> grew out of Max Born's 1926 work on <quantum scattering>: squared <probability amplitudes> determine the relative frequencies of scattering outcomes. Born explained this statistical interpretation again in his 1954 Nobel lecture. This probability law does not by itself select a <post-measurement state>. https://doi.org/10.1007/BF01397477[Born, 1926 collision paper]{ref}; https://www.nobelprize.org/uploads/2018/06/born-lecture.pdf[Born, Nobel lecture, 1954, pp. 261–262]{ref}.
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