in the joint state $\rho_{AE|y}=\sum_{k}p_{k|y}\rho_{A}(k)\otimes|k\rangle_{E}%
\langle k|$, where $p_{k|y}:=\langle k|M_{y}|k\rangle$. Then, Eve retrieves $%
K=\{k,p_{k|y}\}$ by a projective POVM, while Alice decodes a variable $X$ with distribution $$p_{x|y}=\mathrm{Tr}(M_{x}\rho_{A|y})=\sum_{k}p_{x|k}p_{k|y},~p_{x|k}:=%
\mathrm{Tr}[M_{x}\rho_{A}(k)].$$ This equation defines a Markov chain $Y\rightarrow K\rightarrow X$, so that $%
I(Y,K)\geq I(Y,X)$ by data processing inequality, i.e., Eve gets more information than Alice [@cohe].

As expected, system $a$ sent through the channel must be quantum $D(A|a)>0$ in order to have a secure QKD. Indeed, this is equivalent to sending an ensemble of non-orthogonal states. By contrast, the classicality of the private system $A$ is still acceptable, i.e., we can have $D(a|A)=0$. In fact, we may build QKD protocols with preparation noise using classical-quantum states $$\rho_{Aa}=\sum_{x}p_{x}|x\rangle_{A}\langle x|\otimes\rho_{a}(x),
\label{cqSTATE}$$ whose local detection (on system $A$) prepares any desired ensemble of non-orthogonal signal states $\{\rho_{a}(x),p_{x}\}$. For instance, the classical-quantum state of two qubits $\rho_{Aa}=(|0,0\rangle_{Aa}%
\langle0,0|+|1,\varphi\rangle_{Aa}\langle1,\varphi|)/2$, with $\{|0\rangle
,|1\rangle\}$ orthonormal and $\langle0|\varphi\rangle\neq0$, realizes the B92 protocol [@B92].

*Secret-key rates*.– Once we have clarified that non-zero input discord $D(A|a)>0$ is a necessary condition for QKD, we now study the secret-key rates which can be achieved by device-dependent protocols. Our next derivation refers to the protocol of Fig. \[picU\] and, more generally, to the scheme 