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 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 . So without loss of generality, K can be identified with Each is a copy of the n-dimensional Hilbert space. From , we see that the above identification of K can be arranged so , where Pi is the projection from K to . Let . 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 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. 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