Quantum Channel - Examples - Observable

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 Ψ 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 Ψ is CP and unital.

The corresponding Schrödinger map Ψ* takes density matrices to classical states:

\Psi (\rho) = \begin{bmatrix} \langle F_1, \rho \rangle \\ \vdots \\ \langle F_n, \rho \rangle \end{bmatrix}

,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

\Psi (\rho) = \begin{bmatrix} \rho (F_1) \\ \vdots \\ \rho (F_n) \end{bmatrix}.

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