Quantum mutual information

Remarks

Example: let $\rho_{AB} = \lvert \Phi \rangle\langle \Phi \rvert$ with $\lvert \Phi \rangle = (\lvert 00 \rangle + \lvert 11 \rangle)/\sqrt{2}$, a Bell state. The state is pure, so $S(\rho_{AB}) = 0$. Each marginal is $I/2$, so $S(\rho_A) = S(\rho_B) = \log 2$. Hence $I(A:B) = 2 \log 2$.

Non-example of correlation: for a product state $\rho_{AB} = \rho_A \otimes \rho_B$, the entropy is additive, $S(\rho_A \otimes \rho_B) = S(\rho_A) + S(\rho_B)$, so $I(A:B) = 0$.

An equivalent form uses the quantum relative entropy: $I(A:B)_\rho = D(\rho_{AB} \,\|\, \rho_A \otimes \rho_B)$. By Klein's inequality this gives $I(A:B)_\rho \ge 0$, with equality exactly for product states. It also satisfies $I(A:B)_\rho \le 2 \min(\log d_A, \log d_B)$, where $d_A$ and $d_B$ are the dimensions. With $\log$ to base $2$, the unit is the bit.