# How do I prove the following maps are completely positive?

I am trying to prove that the following superoperators are quantum channels, that is completely positive and trace-perserving linear maps

1 $$\Psi[M]=WMW^\dagger$$ where $$W$$ is an isometry

2 $$\Psi[M_A]=M_A\otimes\sigma_B$$ where $$\sigma_B$$ is a state

3 partial trace: $$\Psi[M_{AB}]=Tr_B[M_{AB}]$$

4 $$\Psi[M]=\sum_{x,y}p(y|x)\langle x|M|x\rangle|y\rangle\langle y|$$ with $$p(y|x)$$ a conditional probability distribution

I have already verified the trace-preserving property but I have no idea how to deal with the completely positive definition . How does one do that?

My defintion is:

Definition: A superoperator $$\Phi$$ in $$L(L(H_A),L(H_B))$$is completely positive if $$(\Phi \otimes \mathcal{I}_R)(M_A) \geq 0$$ is true for all $$\mathcal{H}_R$$, given that $$M_A\geq0\,.$$

• Check the relevant properties of the Choi matrix or show it has a Kraus form, then you know it is a quantum channel. Commented Mar 8 at 14:07

The standard method is the apply your superoperator to one half of a maximally entangled state on $$\mathbb{C}^d\otimes\mathbb{C}^d$$ (as a density matrix) where your operator takes inputs from the space $$\mathbb{C}^d$$. The outcome is a state called the Choi map. Its eigenvalues are non-negative if and only if the superoperator is completely positive.
Take case (2) as an example. if $$|B\rangle$$ is your Bell state, then $$I\otimes\Psi(|B\rangle\langle B|)=|B\rangle\langle B| \otimes \sigma_B.$$ This is clearly a valid density matrix, so the eigenvalues are all between 0 and 1. Hence, the operation is completely positive.