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  "body": "\n## Statement\n\nLet $A$ be a unital C*-algebra, let $H$ be a [[Hilbert space|HM2HHE3J51]], and let $B(H)$ be the C\\*-algebra of bounded operators on $H$.\nLet $\\varphi \\colon A \\to B(H)$ be a [[completely positive map|CGY299QMC3]].\nThen 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\n$$\\varphi(a) = V^* \\pi(a) V \\quad \\text{for all } a \\in A.$$\nMoreover $\\norm{\\varphi} = \\norm{V}^2 = \\norm{\\varphi(1)}$.\n\n## Remarks\n\nProof idea: on the algebraic tensor product $A \\odot H$ define\n$$\\ip{a \\otimes \\xi}{b \\otimes \\eta} = \\ip{\\varphi(b^* a) \\xi}{\\eta}.$$\nComplete positivity of $\\varphi$ is exactly what makes this sesquilinear form positive semidefinite.\nLet $K$ 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 $1 \\otimes \\xi$.\n\nThe converse holds: every map of the form $a \\mapsto V^* \\pi(a) V$ is completely positive.\nIf $\\varphi(1) = 1$, then $V$ is an isometry and $H$ can be viewed as a subspace of $K$, so $\\varphi$ is the compression of $\\pi$ to $H$.\nOne 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.\n\nExample: 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.\n\n## Explanations\n\n### L1 human\n\nA $*$-homomorphism into operators is the most rigid kind of map between operator algebras.\nStinespring'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.\nSo completely positive maps are \"shadows\" of representations.\n\n### L2 human\n\nIn quantum information, the theorem says that every quantum channel comes from a unitary interaction with an environment followed by discarding the environment.\nFor finite-dimensional algebras it is equivalent to the Kraus form of [[Choi's theorem|EQG2K8ZFM4]].\nIt is a basic tool in the theory of operator systems and operator spaces.\n",
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