variance $\protect\omega$. $K$ is positive in the white area, while it is zero in the black area. *Right panel*. Rate $K(X|Y)$ in reverse reconciliation, as function of $\protect\tau$ and $\protect\omega$. White area ($K>0$) is wider at low $\protect\omega$.[]{data-label="soglie"}](Soglie.eps){width="45.00000%"}

In particular, for a pure loss channel ($\omega=1$), the previous asymptotical rates simplify to the following$$K(Y|X)=\log_{2}\frac{\tau}{e(1-\tau^{2})}+h(3-2\tau),$$ which is positive for any $\tau>0.693$, and$$K(X|Y)=\log_{2}\frac{\tau}{e(1-\tau^{2})}+h\left( \frac{2}{\tau}-1\right) ,$$ which is positive for any $\tau>0.532$.

Discord bound can be tight
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Here we discuss a typical scenario where the optimal backward rate $K(\blacktriangleleft)$ of an ideal QKD protocol is exactly equal to the output discord $D(B|A)$ shared by Alice and Bob. This happens in continuous variable QKD, where reverse reconciliation is important for its ability to beat the 3dB loss-limit affecting direct reconciliation [@RMP2].

Consider an ideal QKD protocol which is based on the distribution of an EPRstate $\rho_{Aa}$, with CM $\mathbf{V}_{Aa}=\mathbf{V}(\mu)$ defined according to Eq. (\[Vomega\]) with $\mu\geq1$. By performing a rank-1 Gaussian POVM on mode $A$, Alice remotely prepares an ensemble of Gaussianly-modulated pure Gaussian states on the other mode $a$. For instance, heterodyne prepares coherent states, while homodyne prepares squeezed states. On average, mode $a$ is described by a thermal state with CM 