POVM - Definition

Definition

In the simplest case, a POVM is a set of Hermitian positive semidefinite operators on a Hilbert space that sum to unity,

This formula is similar to the decomposition of a Hilbert space by a set of orthogonal projectors:

An important difference is that the elements of a POVM are not necessarily orthogonal, with the consequence that the number of elements in the POVM, n, can be larger than the dimension, N, of the Hilbert space they act in.

In general, POVMs can be defined in situations where outcomes can occur in a non-discrete space. The relevant fact is that measurements determine probability measures on the outcome space:

Definition. Let (X, M) be measurable space; that is M is a σ-algebra of subsets of X. A POVM is a function F defined on M whose values are bounded non-negative self-adjoint operators on a Hilbert space H such that F(X) = I_H and for every ξ H,

is a non-negative countably additive measure on the σ-algebra M.

This definition should be contrasted with that for the projection-valued measure, which is very similar, except that, in the projection-valued measure, the F are required to be projection operators.