between the density matrices is specified, a standard linearity argument, together with the finite-dimensional assumption, allow us to extend uniquely to the full space of operators. This leads to the adjoint map , which describes the action of in the Heisenberg picture: The spaces of operators L(HA) and L(HB) are Hilbert spaces with the Hilbert–Schmidt inner product. Therefore, viewing as a map between Hilbert spaces, we obtain its adjoint * given by While takes states on A to those on B, maps observables on system B to observables on A. This relationship is same as that between the Schrödinger and Heisenberg descriptions of dynamics. The measurement statistics remain unchanged whether the observables are considered fixed while the states undergo operation or vice versa. It can be directly checked that if is assumed to be trace preserving, is unital, that is,. Physically speaking, this means that, in the Heisenberg picture, the trivial observable remains trivial after applying the channel. Classical information So far we have only defined quantum channel that transmits only quantum information. As stated in the introduction, the input and output of a channel can include classical information as well. To describe this, the formulation given so far needs to be generalized somewhat. A purely quantum channel, in the Heisenberg picture, is a linear map Ψ between spaces of operators: that is unital and completely positive (CP). The operator spaces can be viewed as finite-dimensional C*-algebras. Therefore, we can say a channel is a unital CP map between C*-algebras: Classical information can then be included in this formulation. The observables of a classical system can be assumed to be a commutative C*-algebra, i.e. the space of continuous functions on some set . We assume is finite so can be identified with the n-dimensional Euclidean space with entry-wise multiplication. Therefore, in the Heisenberg picture, if the classical information is part of, say, the input, we would define to include the relevant classical observables. An example of this would be a channel Notice is still a C*-algebra. An element of a C*-algebra is called positive if for some . Positivity of a map is defined accordingly. This characterization is not universally accepted; the quantum instrument is sometimes given as the generalized mathematical framework for conveying both quantum and classical information. In axiomatizations of quantum mechanics, the classical information is carried in a Frobenius algebra or Frobenius category. Examples States A state,