of Fig. \[PicD\], where Alice and Bob share an output state $\rho_{AB}$, where only part of the purification is accessible to Eve (system $E$), while the inaccessible part $P$ accounts for all possible forms of extra noise in Alice’s and Bob’s apparata, including preparation noise and detection noise (quantum inefficiencies, etc...)

![Output state from a device-dependent QKD protocol. Alice and Bob extract a secret-key by applying rank-1 POVMs on their local systems $A$ and $B$. Eve steals information from system $E$, while the extra system $P$ is inaccessible and completes the purification of the global state $\Psi_{ABEP}$.[]{data-label="PicD"}](picD.eps){width="50.00000%"}

In direct reconciliation, Alice’s variable $X$ is the encoding to guess. The key rate is then given by $K(Y|X)=I(X,Y)-I(E,X)$, where $I(E,X)=S(E)-S(E|X)$ is the Holevo bound quantifying the maximal information that Eve can steal on Alice’s variable [@Notation1]. We can write an achievable upper bound if we allow Bob to use a quantum memory and a coherent detector. In this case, $I(X,Y)$ must be replaced by the Holevo quantity $I(B,X)=S(B)-S(B|X)$ and we get the forward Devetak-Winter (DW) rate [@DW1; @DW2; @DWnote]$$K(Y|X)\leq K(B|X):=I(B,X)-I(E,X).   \label{DWdr}$$ The optimal forward-rate is defined by optimizing on Alice’s individual detections $K(\blacktriangleright):=\sup_{\{M_{x}\}}K(B|X)$.

We can write similar quantities in reverse reconciliation, where 