Stinespring dilation theorem

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.

We now briefly sketch the proof. Let For , define

and extend by semi linearity to all of K. This is a Hermitian sesquilinear form because is compatible with the * operation. Complete positivity of 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

is a subspace. We can remove degeneracy by considering the quotient space The completion of this quotient space is then a Hilbert space, also denoted by Next define and One can check that and have the desired properties. Notice that is just the natural algebraic embedding of H into K. One can verify that holds. In particular holds so that is an isometry if and only if In this case H can be embedded, in the Hilbert space sense, into K and , acting on K, becomes the projection onto H. Symbolically, we can write

In the language of dilation theory, this is to say that is a compression of It is therefore a corollary of Stinespring's theorem that every unital completely positive map is the compression of some * homomorphism.

We mention a few of the results which can be viewed as consequences of Stinespring's theorem. Historically, some of the results below preceded Stinespring's theorem.

The Gelfand–Naimark–Segal (GNS) construction is as follows. Let H in Stinespring's theorem be 1 dimensional, i. e. the complex numbers. So Φ now is a positive linear functional on A. If we assume Φ is a state, that is, Φ has norm 1, then the isometry is determined by

for some of unit norm. So

and we have recovered the GNS representation of states. This is one way to see that completely positive maps, rather than merely positive ones, are the true generalizations of positive functionals. A linear positive functional on a C* algebra is absolutely continuous with respect to another such functional (called a reference functional) if it is zero on any positive element on which the reference positive functional is zero. This leads to a noncommutative generalization of the Radon–Nikodym theorem. The usual density operator of states on the matrix algebras with respect to the standard trace is nothing but the Radon–Nikodym derivative when the reference functional is chosen to be trace. Belavkin introduced the notion of complete absolute continuity of one completely positive map with respect to another (reference) map and proved an operator variant of the noncommutative Radon–Nikodym theorem for completely positive maps. A particular case of this theorem corresponding to a tracial completely positive reference map on the matrix algebras leads to the Choi operator as a Radon–Nikodym derivative of a CP map with respect to the standard trace (see Choi's Theorem).

Naimark's theorem says that every B(H) valued, weakly countably additive measure on some compact Hausdorff space X can be "lifted" so that the measure becomes a spectral measure. It can be proved by combining the fact that C(X) is a commutative C* algebra and Stinespring's theorem.

This result states that every contraction on a Hilbert space has a unitary dilation with the minimality property.

In quantum information theory, quantum channels, or quantum operations, are defined to be completely positive maps between C* algebras. Being a classification for all such maps, Stinespring's theorem is important in that context. For example, the uniqueness part of the theorem has been used to classify certain classes of quantum channels. For the comparison of different channels and computation of their mutual fidelities and information another representation of the channels by their "Radon–Nikodym" derivatives introduced by Belavkin is useful. In the finite dimensional case, Choi's theorem as the tracial variant of the Belavkin's Radon–Nikodym theorem for completely positive maps is also relevant. The operators from the expression

are called the Kraus operators of Φ. The expression

is sometimes called the operator sum representation of Φ.