DW rate satisfies Eq. (\[DWdr\]) with $%
\geq$, the separation ($>$) being possible for higher-rank POVMs [DW1,DW2]{}. These POVMs are considered here by including their detection noise in system $P$ (e.g., as in Fig. \[PicD\]). What remain are rank-1 POVMs applied to $A $ and $B$.

R. García-Patrón, S. Pirandola, S. Lloyd, J.H. Shapiro, Phys. Rev. Lett. **102**, 210501 (2009).

[Supplementary Material]{}

Keys from separable Gaussian states
===================================

Here we provide a simple example of device-dependent QKD protocol which is based on the distribution of a bipartite Gaussian state which is mixed and separable (not in a tensor-product, therefore having non-zero discord). We show that the key rates can be positive despite no entanglement being present. The reader not familiar with the formalism of bosonic systems and Gaussian states can find these concepts in Ref. [@RMP2], whose notation is here adopted ($\hslash=2$ and unit vacuum noise).

Let us consider a continuous variable QKD protocol where Alice prepares two bosonic modes, $A$ and $a$, in a separable Gaussian state $\rho_{Aa}$, with zero mean and covariance matrix (CM)$$\mathbf{V}_{Aa}=\left(
\begin{array}{cc}
\mu\mathbf{I} & \mathbf{G} \\
\mathbf{G} & \mu\mathbf{I}\end{array}
\right) ,$$ where $\mu\geq1$ and $\mathbf{G}$ is a diagonal correlation block which can be in one 