Editing Quantum channel (section)

=== Observable ===

An observable associates a numerical value <math>f_i \in \mathbb{C}</math> to a quantum mechanical ''effect'' <math>F_i</math>. <math>F_i</math>'s are assumed to be positive operators acting on appropriate state space and <math display="inline">\sum_i F_i = I</math>. (Such a collection is called a [[POVM]].<ref>{{cite book|first=Asher |last=Peres |author-link=Asher Peres |title=Quantum Theory: Concepts and Methods |title-link=Quantum Theory: Concepts and Methods |year=1993 |publisher=[[Kluwer]] |isbn=0-7923-2549-4 |page=283}}</ref>{{sfn|Bengtsson|Życzkowski|2017|p=271}}) In the Heisenberg picture, the corresponding ''observable map'' <math>\Psi</math> maps a classical observable

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

to the quantum mechanical one

:<math>\; \Psi (f) = \sum_i f_i F_i.</math>

In other words, one [[Naimark's dilation theorem|integrates ''f'' against the POVM]] to obtain the quantum mechanical observable. It can be easily checked that <math>\Psi</math> is CP and unital.

The corresponding Schrödinger map <math>\Psi^*</math> takes density matrices to classical states:{{sfn|Wilde|2017|at=§4.6.6}}

:<math>
\Psi (\rho) = \begin{bmatrix} \langle F_1, \rho  \rangle \\ \vdots \\ \langle F_n, \rho \rangle \end{bmatrix}, 
</math>

where the inner product is the Hilbert–Schmidt inner product. Furthermore, viewing states as normalized [[density matrix#C*-algebraic formulation of states|functionals]], and invoking the [[Riesz representation theorem]], we can put

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