= Solution
A <positive operator-valued measure> consists of <positive semidefinite> effects $E_r$ with $\sum_rE_r=I$. Its outcome probabilities are $p_r=\operatorname{Tr}(\rho E_r)$, which are nonnegative and sum to one. Unlike a <projective measurement>, its effects need not be mutually <orthogonal projections>, and there can be more outcomes than the system's Hilbert-space dimension.
For example, choose three equatorial <Bloch vectors> $\mathbf n_r$ separated by $120^\circ$ and set $E_r=(I+\mathbf n_r\cdot\boldsymbol\sigma)/3$. Each effect has <eigenvalues> zero and $2/3$, and the vectors sum to zero, so the three effects sum to $I$. This <trine qubit POVM> gives $p_r=(1+\mathbf s\cdot\mathbf n_r)/3$ for input <Bloch vector> $\mathbf s$. It cannot be a three-outcome orthogonal <projective measurement> on that <qubit>.
A physical implementation can couple the system to an <ancilla> and projectively measure the enlarged system. More generally, operators $M_{r\alpha}$ implement an outcome with $E_r=\sum_\alpha M_{r\alpha}^\dagger M_{r\alpha}$; the normalized conditional state is $\sum_\alpha M_{r\alpha}\rho M_{r\alpha}^\dagger/p_r$. Thus a <POVM does not determine the post-measurement state>: different <measurement instruments> can have the same outcome probabilities and different disturbance. <Generalized measurements> are useful for state discrimination and realistic detection, including explicitly modeled inconclusive outcomes.
Back to article page
