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 as a composition of two completely positive maps each of which has a special form: 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: Informally , one can say that every completely positive map formula_6 can be "lifted" up to a map of the form formula_7. 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 formula_8. For formula_9, define and extend by semi-linearity to all of "K". This is a Hermitian sesquilinear form because formula_6 is compatible with the * operation. Complete positivity of formula_6 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 formula_14. The completion of this quotient space is then a Hilbert space, also denoted by formula_15. Next define formula_16 and formula_17. One can check that formula_18 and formula_19 have the desired properties. Notice that formula_19 is just the natural algebraic embedding of "H" into "K". One can verify that formula_21 holds. In particular formula_22 holds so that formula_19 is an isometry if and only if formula_24. In this case "H" can be embedded, in the Hilbert space sense, into "K" and formula_25, acting on "K , becomes the projection onto H". Symbolically, we can write In the language of dilation theory, this is to say that formula_27 is a "compression" of formula_28. It is therefore a corollary of Stinespring's theorem that every unital completely positive map is the compression of some *-homomorphism. Minimality. The triple (, "V", "K") is called a Stinespring representation of Φ. A natural question is now whether one can reduce a given Stinespring representation in some sense. Let "K" be the closed linear span of ("A") "VH". By property of *-representations in general, "K" is an invariant subspace of ("a") for all  a. Also, "K" contains "VH". Define We can compute directly and if "k" and "ℓ" lie in "K" So (, "V", "K") is also a Stinespring representation of Φ and has the additional property that "K" is the closed linear span of ("A") "V H". Such a representation is called a minimal Stinespring representation. Uniqueness. Let (, "V", "K") and (, "V", "K") be two Stinespring representations of a given Φ. Define a partial isometry "W" : "K" → "K" by On "V""H" ⊂ "K", this gives the intertwining relation In particular, if both Stinespring representations are minimal, "W" is unitary. Thus minimal Stinespring  representations are unique up to a unitary transformation. Some consequences. 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. GNS construction. 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 formula_34 is determined by for some formula_36 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). Choi's theorem. It was shown by Choi that if formula_38 is completely positive, where "G  and H" are finite-dimensional Hilbert spaces of dimensions "n" and "m" respectively, then Φ takes the form: This is called Choi's theorem on completely positive maps. Choi proved this using linear algebra techniques, but his result can also be viewed as a special case of Stinespring's theorem: Let (, "V", "K") be a minimal Stinespring representation of Φ. By minimality, "K" has dimension less than that of formula_40. So without loss of generality, "K" can be identified with Each formula_42 is a copy of the "n"-dimensional Hilbert space. From formula_16, we see that the above identification of "K  can be arranged so formula_44, where P" is the projection from "K" to formula_42. Let formula_46. We have and Choi's result is proved. Choi's result is a particular case of noncommutative Radon–Nikodym theorem for completely positive (CP) maps corresponding to a tracial completely positive reference map on the matrix algebras. In strong operator form this general theorem was proven by Belavkin in 1985 who showed the existence of the positive density operator representing a CP map which is completely absolutely continuous with respect to a reference CP map. The uniqueness of this density operator in the reference  Steinspring representation simply follows from the minimality of this representation. Thus, Choi's operator is the Radon–Nikodym derivative of a finite-dimensional CP map with respect to the standard trace. Notice that, in proving Choi's theorem, as well as Belavkin's theorem from Stinespring's formulation, the argument does not give the Kraus operators "V" explicitly, unless one makes the various identification of spaces explicit. On the other hand, Choi's original proof involves direct calculation of those operators. Naimark's dilation 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. Sz.-Nagy's dilation theorem. This result states that every contraction on a Hilbert space has a unitary dilation with the minimality property. Application. 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 formula_48 from the expression are called the Kraus operators of Φ. The expression is sometimes called the operator sum representation of Φ.