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A self-adjoint operator whose quadratic form is nonnegative on every vector.
An operator on a complex Hilbert space is positive semidefinite, written , if
for every vector . Positivity implies that is self-adjoint. In finite dimension it is equivalent to all eigenvalues of being real and nonnegative, and also equivalent to a factorization .
Positive operators are the effects in a POVM and, after trace normalization, the density operators of quantum theory.
On a finite-dimensional complex Hilbert space , a positive operator-valued measure or POVM with finite outcome set is a family of positive semidefinite operators satisfying
For a density operator , outcome has probability . 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.
Let be a complex Hilbert space (typically finite-dimensional in basic quantum theory). A density operator (also called a density matrix) is an operator such that:
In finite dimension, these conditions are equivalent to being a positive semidefinite matrix with trace .