what is the mutual information between a input state and output state of a quantum channel?

Question

Suppose we have a quantum channel $\mathcal{N}$, the input ensemble is $A:\{p_i,\rho_i\}$, and the output is $B$. Then what is the mutual information between $A$ and $B$? If I want to get $I(A;B)$, I need to know $\rho_{AB}$, but I am having trouble to understand what is $\rho_{AB}$ in this problem? Or I can ask: what is the mutual information between $A$ and $B$?

Answer 1

Usually, what you are looking for, is written in terms of a so-called "classical-quantum state"

$$ \rho_{AB} := \sum_{i} p_i |i\rangle\langle{i}|_A \otimes \mathcal{N}(\rho_i),$$

where the $A$-System is a "dummy" quantum system introduced to "make the classical input system quantum".

The quantum mutual information then matches the Holevo quantity of the ensemble $\{p_i, \mathcal{N}(\rho_i)\}$.

More information can be found in Mark Wilde's book, especially Exercise 11.6.9 therein.