Positive operator-valued measure

On a finite-dimensional complex Hilbert space $H$, a positive operator-valued measure or POVM with finite outcome set $I$ is a family of positive semidefinite operators $(E_i)_{i\in I}$ satisfying

For a density operator $\rho$, outcome $i$ has probability $\operatorname{Tr}(\rho E_i)$. Positivity makes these probabilities nonnegative, and the normalization makes them sum to one.

Orthogonal projective measurements are a special case; general POVM effects need not be projections or mutually orthogonal.