---
tag: X8K0Z7484R
kind: statement
role: theorem
name: Stinespring dilation theorem
msc: [46L07, 47A20, 81P47]
uses: [CGY299QMC3, HM2HHE3J51]
about: [CGY299QMC3]
sources:
  - url: https://en.wikipedia.org/wiki/Stinespring_dilation_theorem
same_as:
  wikidata: Q7617407
  wikipedia: Stinespring dilation theorem
---

## Statement

Let $A$ be a unital C*-algebra, let $H$ be a [[Hilbert space|HM2HHE3J51]], and let $B(H)$ be the C\*-algebra of bounded operators on $H$.
Let $\varphi \colon A \to B(H)$ be a [[completely positive map|CGY299QMC3]].
Then there exist a Hilbert space $K$, a unital $*$-homomorphism $\pi \colon A \to B(K)$, and a bounded operator $V \colon H \to K$ such that
$$\varphi(a) = V^* \pi(a) V \quad \text{for all } a \in A.$$
Moreover $\norm{\varphi} = \norm{V}^2 = \norm{\varphi(1)}$.

## 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.

## Explanations

### L1 human

A $*$-homomorphism into operators is the most rigid kind of map between operator algebras.
Stinespring's theorem says that every completely positive map is obtained from such a homomorphism, by enlarging the Hilbert space and then cutting back down to the original one.
So completely positive maps are "shadows" of representations.

### L2 human

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|EQG2K8ZFM4]].
It is a basic tool in the theory of operator systems and operator spaces.