For a bipartite system, the quantum mutual information is defined via the Von Neumann entropy as follows: $$I(A:B)=S(A)+S(B)-S(AB).$$
It's always positive. Is it an entanglement measurement?
Also how can you find out if a bipartite system is separable by the use of mutual information?
Coherent information is the measure of quantum correlations. Positivity of coherent information indicates that quantum correlations are present. But this is not an iff relation.
Coherent information is denoted as
$$I(A\rangle B) = H(B) - H(AB) = -H(A|B)\,.$$
If $I(A\rangle B)$ quantity is positive then there is quantum entanglement between system $A$ and $B$.
The relation between mutual information and coherent information is as follows:
$$I(A;B) = H(A) + I(A\rangle B) = H(B) + I(B\rangle A)\,.$$
For example, consider $$\rho_{AB} = | \Phi_+ \rangle \langle \Phi_+|\,.$$
Where $| \Phi_+ \rangle$ is a bell state. You can see that $$H(AB) =H(\rho_{AB})= 0$$ $$H(B)= H(\text{Tr}_A(\rho_{AB}) )= 1$$
Hence; $$H(A|B) = -1$$
Thus, entanglement is present between system $A$ and $B$.
TL;DR: Negative conditional entropy $H(A|B)$, i.e. Positive Coherent Information $I(A\rangle B)= -H(A|B)$ is a hallmark of quantum correlations - that is, of entanglement!
