In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra A as a composition of two completely positive maps each of which has a special form:
A *-representation of A on some auxiliary Hilbert space K followed by An operator map of the form T ↦ V*TV. Moreover, Stinespring's theorem is a structure theorem from a C*-algebra into the algebra of bounded operators on a Hilbert space. Completely positive maps are shown to be simple modifications of *-representations, or sometimes called *-homomorphisms.
Formulation In the case of a unital C*-algebra, the result is as follows:
Theorem. Let A be a unital C*-algebra, H be a Hilbert space, and B(H) be the bounded operators on H. For every completely positive
Φ : A → B ( H ) , {\displaystyle \Phi :A\to B(H),}
there exists a Hilbert space K and a unital *-homomorphism
π : A → B ( K ) {\displaystyle \pi :A\to B(K)}
such that
Φ ( a ) = V ∗ π ( a ) V , {\displaystyle \Phi (a)=V^{\ast }\pi (a)V,}
where V : H → K {\displaystyle V:H\to K} is a bounded operator. Furthermore, we have
‖ Φ ( 1 ) ‖ = ‖ V ‖ 2 . {\displaystyle \|\Phi (1)\|=\|V\|^{2}.}
Informally, one can say that every completely positive map Φ {\displaystyle \Phi } can be "lifted" up to a map of the form V ∗ ( ⋅ ) V {\displaystyle V^{*}(\cdot )V} . The converse of the theorem is true trivially. So Stinespring's result classifies completely positive maps.
Sketch of proof We now briefly sketch the proof. Let K = A ⊗ H {\displaystyle K=A\otimes H} . For a ⊗ h , b ⊗ g ∈ K {\displaystyle a\otimes h,\ b\otimes g\in K} , define
⟨ a ⊗ h , b ⊗ g ⟩ K := ⟨ Φ ( b ∗ a ) h , g ⟩ H = ⟨ h , Φ ( a ∗ b ) g ⟩ H {\displaystyle \langle a\otimes h,b\otimes g\rangle _{K}:=\langle \Phi (b^{*}a)h,g\rangle _{H}=\langle h,\Phi (a^{*}b)g\rangle _{H}}
and extend by semi-linearity to all of K. This is a Hermitian sesquilinear form because Φ {\displaystyle \Phi } is compatible with the * operation. Complete positivity of Φ {\displaystyle \Phi } is then used to show that this sesquilinear form is in fact positive semidefinite. Since positive semidefinite Hermitian sesquilinear forms satisfy the Cauchy–Schwarz inequality, the subset
K ′ = { x ∈ K ∣ ⟨ x , x ⟩ K = 0 } ⊂ K {\displaystyle K'=\{x\in K\mid \langle x,x\rangle _{K}=0\}\subset K}
is a subspace. We can remove degeneracy by considering the quotient space K / K ′ {\displaystyle K/K'} . The completion of this quotient space is then a Hilbert space, also denoted by K {\displaystyle K} . Next define π ( a ) ( b ⊗ g ) = a b ⊗ g {\displaystyle \pi (a)(b\otimes g)=ab\otimes g} and V h = 1 A ⊗ h {\displaystyle Vh=1_{A}\otimes h} . One can check that π {\displaystyle \pi } and V {\displaystyle V} have the desired properties. Notice that V {\displaystyle V} is just the natural algebraic embedding of H into K. One can verify that V ∗ ( a ⊗ h ) = Φ ( a ) h {\displaystyle V^{\ast }(a\otimes h)=\Phi (a)h} holds. In particular V ∗ V = Φ ( 1 ) {\displaystyle V^{\ast }V=\Phi (1)} holds so that V {\displaystyle V} is an isometry if and only if Φ ( 1 ) = 1 {\displaystyle \Phi (1)=1} . In this case H can be embedded, in the Hilbert space sense, into K and V ∗ {\displaystyle V^{\ast }} , acting on K, becomes the projection onto H. Symbolically, we can write
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