 Solution

A positive operator-valued measure consists of positive semidefinite effects $E_r$ with ${\sum }_r{E}_r=I$. Its outcome probabilities are $p_r=\mathrm{Tr}(\rho E_r)$, 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 $E_r=(I+{\mathbf{n}}_r\cdot \mathbf{\sigma })/3$. 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 $E_r={\sum }_{\alpha }{M}_{r\alpha }^{†}{M}_{r\alpha }$; the normalized conditional state is ${\sum }_{\alpha }{M}_{r\alpha }\rho M_{r\alpha }^{†}/p_r$. 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.