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: 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.
In the case of a unital C*-algebra, the result is as follows: Informally, one can say that every completely positive map
So Stinespring's result classifies completely positive maps.
, define and extend by semi-linearity to all of K. This is a Hermitian sesquilinear form because
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.
, acting on K, becomes the projection onto H. Symbolically, we can write In the language of dilation theory, this is to say that
It is therefore a corollary of Stinespring's theorem that every unital completely positive map is the compression of some *-homomorphism.
The triple (π, 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.
Define a partial isometry W : K1 → K2 by On V1H ⊂ K1, 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.
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 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).
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 (π, V, K) be a minimal Stinespring representation of Φ.
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.
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 Vi explicitly, unless one makes the various identification of spaces explicit.
On the other hand, Choi's original proof involves direct calculation of those operators.
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.
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.
The expression is sometimes called the operator sum representation of Φ.