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### Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

Distraction The matrix $P$ is a distraction, because the $dd'$ matrices $A_k$ are an orthonormal basis with respect to the inner product $\langle A,B\rangle_P:=\mathrm{tr}(PA^\dagger B)$ if and only ...
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### Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

This solution uses the spectral theorem and some elementary linear algebra computations. If you let $P = \sum_{i = 1}^{d'} \lambda_i \vert{\psi_i}\rangle\langle{\psi_i}\vert$ be the spectral ...

### What is the computational power of classically mixed states?

(The other answers and comments have already touched on the main points, but I have to take the opportunity to answer in my own words.) What you're describing is known as randomized or probabilistic ...
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Mixtures like your given $(|00101\rangle\langle00101|+|00111\rangle\langle00111| )/2$ are in fact describing random distributions of classical results, so you are asking here about a non-deterministic ...