Bob’s variable $Y$ is the encoding to infer. The secret key rate is given by $%
K(X|Y)=I(X,Y)-I(E,Y)$, where $I(E,Y)=S(E)-S(E|Y)$ is Eve’s Holevo information on $Y$. Assuming a coherent detector for Alice, this rate is bounded by the backward DW rate$$K(X|Y)\leq K(A|Y):=I(A,Y)-I(E,Y),   \label{DWrevEQ}$$ which gives the optimal backward-rate $K(\blacktriangleleft)$ by maximizing on Bob’s individual detections $\{M_{y}\}$.

Playing with system $P$, we can easily derive upper and lower bounds for the two optimal rates. Clearly, we get lower bounds $K_{\ast}\leq K$ if we assume $P$ to be accessible to Eve, which means to extend $E$ to the joint system $\mathbf{E}=EP$ in previous Eqs. (\[DWdr\]) and (\[DWrevEQ\]). By exploiting the purity of the global state $\Psi_{AB\mathbf{E}}$ and the fact that the encoding detections are rank-1 POVMs (therefore collapsing pure states into pure states), we can write the entropic equalities $S(AB)=S(%
\mathbf{E})$, $S(B|X)=S(\mathbf{E}|X)$ and $S(A|Y)=S(\mathbf{E}|Y)$. Then we easily derive$$K_{\ast}(\blacktriangleright)=I_{c}(A\rangle B),~K_{\ast}(\blacktriangleleft
)=I_{c}(B\rangle A),$$ where the coherent information $I_{c}(A\rangle B)$ and its reverse counterpart [@RevCOH; @RMP] $I_{c}(B\rangle A)$ quantify the maximal entanglement which is distillable by local operations and one-way CC, forward and backward, respectively.

It is also clear that we get upperbounds $K^{\ast}\geq K$ by assuming $P$ to be accessible to the decoding party, Alice 