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1

Let's say $|a\rangle$ is the pure state qubit and $|b\rangle$ is the bell state. Putting $|a\rangle$ through the Pauli channel depends on which of the Pauli gates (which of the $\sigma_j$) you want acting on it. Once you choose this (call it $\sigma_a$), apply $2\times2$ Identity operators to it on either side until it's the same size as $\rho$, then just ...


5

Mathematically it is a relationship between a bipartite linear operator vector space $L(X\otimes Y)$ and a superoperator vector space $C(X): L(X)\to L(Y)$ (maps of linear operators to linear operators). Bipartite density matrices are contained in the former, and quantum channels in the latter. The real "physical" meaning of the isomorphism for quantum ...


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In my understanding, the key part for entanglement to increase capacity is to have a suboptimal channel. Suppose the input of you channel can take value in the set $X$, and note $G(X)$ the graph where nodes are possible inputs and there is an edge between to inputs if the range of their corresponding output is overlapping, i.e. When going through the channel ...


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