The definition of quantum mutual entropy is motivated by the classical case.
In other words, if we assume the two variables x and y to be uncorrelated, mutual information is the discrepancy in uncertainty resulting from this (possibly erroneous) assumption.
The quantum mechanical counterpart of classical probability distributions are modeled with density matrices.
So one can assign to ρ a state on the subsystem A by where TrB is partial trace with respect to system B.
The reduced von Neumann entropy of ρAB with respect to system A is S(ρB) is defined in the same way.
Quantum mutual information can be interpreted the same way as in the classical case: it can be shown that where
Note that there is an alternative generalization of mutual information to the quantum case.