Naimark's dilation theorem

Remarks

For a POVM with finitely many outcomes $F_1, \dots, F_n$ the dilation is explicit. Take $\mathcal{K} = \mathcal{H} \otimes \C^n$ with standard basis $e_1, \dots, e_n$ of $\C^n$, put $V\psi = \sum_k F_k^{1/2}\psi \otimes e_k$, and let $P_k$ be the projection $I \otimes e_k e_k^*$. Then $\norm{V\psi}^2 = \sum_k \langle \psi, F_k\psi \rangle = \norm{\psi}^2$, so $V$ is an isometry, and $V^* P_k V = F_k^{1/2} F_k^{1/2} = F_k$. In general, the main step is the Stinespring dilation theorem. One applies it to the positive map $f \mapsto \int f \, dF$ on bounded measurable functions, which is completely positive because its domain is commutative. Example: for the trine POVM on $\C^2$ the construction embeds $\C^2$ in $\C^2 \otimes \C^3$, and a smaller dilation into $\C^3$ also exists, since the three effects have rank $1$. In quantum mechanics this means that every generalized measurement is a projective measurement of the system together with an ancilla.

Going deeper

For a student

Mark Naimark proved the theorem in the 1940s. It reduces questions about POVMs to the spectral theory of projection-valued measures. The dilation can be chosen minimal, meaning that the vectors $P(E)V\psi$ span a dense subspace of $\mathcal{K}$, and then it is unique up to unitary equivalence. In quantum information, the finite-dimensional form is used to implement a POVM by coupling the system to an auxiliary system and doing a projective measurement on both.