Positive operator-valued measure

Definition

Let $(\Omega,\Sigma)$ be a measurable space and $H$ a complex Hilbert space. A positive operator-valued measure or POVM is a map

E:\Sigma\longrightarrow B(H)

such that:

$E(A)$ is a positive operator for every $A\in\Sigma$ ;
$E(\Omega)=I_H$ ;
for every pairwise disjoint sequence $(A_n)$ in $\Sigma$ , $\left\langle x,E\!\left(\bigcup_nA_n\right)y\right\rangle = \sum_n\langle x,E(A_n)y\rangle \qquad(x,y\in H).$

The third condition is countable additivity in the weak operator topology.

Born rule

For a density operator $\rho$ , the scalar measure

\mathbb P_\rho(A)=\operatorname{Tr}(\rho E(A))

is the probability distribution of the measurement outcome.

Discrete outcomes

For a finite or countable outcome set $I$ , a POVM is equivalently a family of positive operators $(E_i)_{i\in I}$ satisfying

\sum_{i\in I}E_i=I_H

in the weak operator topology. Then $\mathbb P_\rho(\{i\})=\operatorname{Tr}(\rho E_i)$ .

Projection-valued measures

A POVM is a projection-valued measure when every $E(A)$ is an orthogonal projection; equivalently,

E(A\cap B)=E(A)E(B).

General POVM effects need not be projections or mutually orthogonal.