= Positive operator-valued measure
{title2=POVM}
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= POVM
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{synonym}
A <positive operator-valued measure> is a family of <positive operators> $E_i$ satisfying $\sum_i E_i=I$. Each $E_i$ is called an effect and determines a <probability> $p_i=\operatorname{Tr}(E_i\rho)$ for input <density operator> $\rho$. The effects need not be <orthogonal projections> or commute. This permits measurements with more outcomes than the dimension of the <Hilbert space>, such as the <trine qubit POVM>. Coupling to a <quantum ancilla> and measuring the larger system supplies a physical route to generalized measurements. https://www.preskill.caltech.edu/ph229/notes/chap3.pdf[Preskill, Quantum Information, §§3.1.2–3.1.4]{ref}.
For an elementary unsharp $Z$ example on a <qubit>, take
$$
E_\pm=\frac12(I\pm\eta Z),\qquad 0\leq\eta\leq1.
$$
On the <eigenstate> $|0\rangle$, $p_\pm=(1\pm\eta)/2$; at $\eta=1/2$ they are $3/4$ and $1/4$. At $\eta=0$ the outcome is a fair coin independent of the input, and at $\eta=1$ these effects specify a sharp $Z$ <projective measurement>. Choosing $K_\pm=\sqrt{E_\pm}$ specifies one <quantum instrument>, but $U_\pm\sqrt{E_\pm}$ for outcome-dependent <unitary operators> has the same effects and different state updates. Thus <POVM does not determine the post-measurement state>.
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