Stinespring dilation theorem

Remarks

Proof idea: on the algebraic tensor product $A \odot H$ define $$\ip{a \otimes \xi}{b \otimes \eta} = \ip{\varphi(b^* a) \xi}{\eta}.$$ Complete positivity of $\varphi$ is exactly what makes this sesquilinear form positive semidefinite. Let $K$ be the completion of the quotient by the null vectors, let $\pi(a)$ act by left multiplication, and let $V \xi$ be the class of $1 \otimes \xi$.

The converse holds: every map of the form $a \mapsto V^* \pi(a) V$ is completely positive. If $\varphi(1) = 1$, then $V$ is an isometry and $H$ can be viewed as a subspace of $K$, so $\varphi$ is the compression of $\pi$ to $H$. One can choose $K$ minimal, meaning that the vectors $\pi(a) V \xi$ span a dense subspace; the minimal dilation is unique up to unitary equivalence.

Example: for $H = \C$, a completely positive map $A \to \C$ with $\varphi(1) = 1$ is a state, and the theorem gives $\varphi(a) = \ip{\pi(a) \xi}{\xi}$ for a unit vector $\xi = V 1$; this is the Gelfand–Naimark–Segal construction.

Going deeper

For a student

In quantum information, the theorem says that every quantum channel comes from a unitary interaction with an environment followed by discarding the environment. For finite-dimensional algebras it is equivalent to the Kraus form of Choi's theorem. It is a basic tool in the theory of operator systems and operator spaces.