$A$ minimized over an arbitrary measurement on $B$. This local measurement is generally described by a positive operator valued measure (POVM) $\{M_{y}\}$, defining a random outcome variable $%
Y=\{y,p_{y}\} $ and collapsing system $A$ into conditional states $%
\rho_{A|y}~$[@Nota2]. Thus, we have$$S_{\min}(A|B):=\inf_{\{M_{y}\}}S(A|Y),~S(A|Y)=\sum_{y}p_{y}S(\rho_{A|y}),$$ where the minimization can be restricted to rank-1 POVMs [@VedralRMP].

The quantity $C(A|B)$ quantifies the classical correlations between the two systems, corresponding to the maximal common randomness achievable by local measurements and one-way classical communication (CC) [@Winter]. Thus, quantum discord is defined as the difference between total and classical correlations [@Qdiscord; @Qdiscord2; @VedralRMP]$$D(A|B):=I(A,B)-C(A|B)=S_{\min}(A|B)-S(A|B)\geq0.$$ An equivalent formula can be written by noticing that $I_{c}(A\rangle
B):=-S(A|B)$ is the coherent information [@Qcap; @Qcap2]. Then, introducing an ancillary system $E$ which purifies $\rho_{AB}$, we can apply the Koashi-Winter relation [@Koashi] and write $S_{\min}(A|B)=E_{f}(A,E)$, where the latter is the entanglement of formation between $A$ and $E$. Therefore$$D(A|B)=I_{c}(A\rangle B)+E_{f}(A,E)\geq\max\{0,I_{c}(A\rangle B)\}.
\label{KWvar}$$

It is important to note that $D(A|B)$ is different from $D(B|A)$, where system $A$ is measured. For instance, in classical-quantum states $%
\rho_{AB}=\sum _{x}p_{x}|x\rangle_{A}\langle x|\otimes\rho_{B}(x)$, where $A$ embeds a classical variable via the orthonormal set $\{|x\rangle\}$ and $B$ is prepared in non-orthogonal states $\{\rho_{B}(x)\}$, we have $D(B|A)=0$ while $D(A|B)>0$. By contrast, for quantum-classical 