Editing Quantum channel (section)

=== 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 <math>\{ F_1, \dots, F_n \}</math> be the effects (POVM) associated to an observable. In the Schrödinger picture, an instrument is a map <math>\Phi</math> with pure quantum input <math>\rho \in L(H)</math> and with output space <math>C(X) \otimes L(H)</math>:

:<math>
\Phi (\rho) =  \begin{bmatrix} \rho(F_1) \cdot F_1 \\ \vdots \\ \rho(F_n) \cdot F_n \end{bmatrix}.
</math>

Let

:<math>
f =  \begin{bmatrix} f_1 \\ \vdots \\ f_n \end{bmatrix} \in C(X).
</math>

The dual map in the Heisenberg picture is

:<math>
\Psi (f \otimes A) =  \begin{bmatrix} f_1 \Psi_1(A) \\ \vdots \\ f_n \Psi_n(A)\end{bmatrix}
</math>

where <math>\Psi_i</math> is defined in the following way: Factor <math>F_i = M_i ^2</math> (this can always be done since elements of a POVM are positive) then <math>\; \Psi_i (A) = M_i A M_i</math>.
We see that <math>\Psi</math> is CP and unital.

Notice that <math>\Psi (f \otimes I)</math> gives precisely the observable map. The map

:<math>{\tilde \Psi}(A)= \sum_i \Psi_i (A) = \sum _i M_i A M_i</math>

describes the overall state change.