Imprinting Complete Information about a Quantum Channel on its Output State

We introduce a novel property of bipartite quantum states, which we call faithfulness, and we say that a state is faithful when acting with a channel on one of the two quantum systems; the output state carries complete information about the channel. The concept of faithfulness can also be extended to sets of states, when the output states altogether carry a complete imprinting of the channel. Measures of degrees of faithfulness are proposed.

Figure 1

The input state $R$ is called faithful when the correspondence between the output state $R_{\mathcal{E}}\doteq \mathcal{E}\otimes \mathcal{I}(R)$ and the quantum channel $\mathcal{E}$ is one to one, namely, the output state $R_{\mathcal{E}}$ carries a complete imprinting of the quantum channel $\mathcal{E}$.

Figure 2

A generally mixed state $R=\sum _l|A_l》《A_l|$ is faithful when the map $\mathcal{R}(M)=\sum _l{A}_l^{\tau }M{A}_l^{*}$ is invertible, in order to guarantee the one-to-one correspondence between $R_{\mathcal{E}}$ and $\mathcal{E}$.