Finite POVMs and their outcome law

Let $H$ be a nonzero finite-dimensional complex inner-product space; the pairing is linear in its first argument (Finite-dimensional vector space, and its dimension ${\dim }_{F}V$; infinite-dimensional means having no finite basis, Real and complex inner-product spaces and their induced length). Write $\mathcal{L}(H)$ for its complex-linear endomorphisms (The space $\mathcal{L}(V,W)$ of linear maps with pointwise addition and scalar multiplication) and let $\mathrm{\Omega }$ be a finite nonempty outcome set. A finite POVM on $H$ with outcomes $\mathrm{\Omega }$ is a family $E=(E_a{)}_{a\in \mathrm{\Omega }}$ with each $E_a$ nonnegative and ${\sum }_{a\in \mathrm{\Omega }}{E}_a=I_H$; after any enumeration of $\mathrm{\Omega }$, these operators form the finite complete positive family of Finite positive effect family. Reordering does not alter the finite sum. The complete-family lemma A complete positive family consists of effects shows that every $E_a$ also has nonnegative complement $I_H-E_a$, so each is an effect.

For a density operator $\rho$ on $H$ (a nonnegative operator of trace one, Density operator), its POVM outcome weights are

$p_{\rho }(a):=\mathrm{tr}(\rho E_a),a\in \mathrm{\Omega }\mathrm{.}$

Here $\mathrm{tr}$ is the finite-dimensional endomorphism trace (The basis-independent trace of an endomorphism of a finite-dimensional vector space).

The positivity and normalization of these weights are proved in POVM outcome weights are probabilities ↗. A POVM specifies this outcome law; it does not by itself prescribe a conditional output state.