viewed as a mapping from observables to their expectation values, is an immediate example of a channel. Time evolution For a purely quantum system, the time evolution, at certain time t, is given by where and H is the Hamiltonian and t is the time. Clearly this gives a CPTP map in the Schrödinger picture and is therefore a channel. The dual map in the Heisenberg picture is Restriction Consider a composite quantum system with state space For a state the reduced state of ρ on system A, ρA, is obtained by taking the partial trace of ρ with respect to the B system: The partial trace operation is a CPTP map, therefore a quantum channel in the Schrödinger picture. In the Heisenberg picture, the dual map of this channel is where A is an observable of system A. Observable An observable associates a numerical value to a quantum mechanical effect . 's are assumed to be positive operators acting on appropriate state space and . (Such a collection is called a POVM.) In the Heisenberg picture, the corresponding observable map maps a classical observable to the quantum mechanical one In other words, one integrate f against the POVM to obtain the quantum mechanical observable. It can be easily checked that is CP and unital. The corresponding Schrödinger map takes density matrices to classical states: where the inner product is the Hilbert–Schmidt inner product. Furthermore, viewing states as normalized functionals, and invoking the Riesz representation theorem, we can put Instrument The observable map, in the Schrödinger picture, has a purely classical output algebra and therefore only describes measurement statistics. To take the state change into account as well, we define what is called a quantum instrument. Let be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map with pure quantum input and with output space : Let The dual map in the Heisenberg picture is where is defined in the following way: Factor (this can always be done since elements of a POVM are positive) then . We see that is CP and unital. Notice that gives precisely the observable map. The map describes the overall state change. Measure-and-prepare channel Suppose two parties A and B wish to communicate in the following manner: A performs the measurement of an observable and communicates the measurement outcome to B classically. According to the message he