Fig. 2.

Lower bound on the conditional entropy $H{(A_i|B_i{X}_i{Y}_i{T}_i{E}_i^{\prime })}_{{\rho }_{|T_i=0}}$ for any state generated as in Fig. 1 and such that on test rounds the obtained winning probability for the CHSH game is $\omega$. This lower bound was obtained by using the method from [22]. For each input $y\in \mathcal{Y}$, the channel ${\mathcal{B}}_y$ is modelled as ${\mathcal{B}}_y(\omega )={\sum }_b{\mathrm{\Pi }}_y^{(b)}\omega {\mathrm{\Pi }}_y^{(b)}$, where ${\{{\mathrm{\Pi }}_y^{(b)}\}}_{b\in \mathcal{B}}$ are orthogonal projectors summing to the identity, and similarly for the map $\mathcal{A}$. It is simple to see that this is without loss of generality