Fig. 4: Asymptotic key rate vs. detection efficiency (η) in the 2322-scenario.

We compare lower bounds on the asymptotic key rate for devices that have some fixed detection efficiency η ∈ [0.84, 1]. The curves based on the entropies \({H}_{(8/7)}^{\uparrow }(A| E)\), \({H}_{(4/3)}^{\uparrow }(A| E)\), \({H}_{(2)}^{\uparrow }(A| E)\), and \({H}_{\min }(A| E)\) were computed numerically using the NPA hierarchy, the red curve representing \(\inf H(A| E)-H(A| B)\) was computed using the analytical expression from ref. ¹⁰. The curves computed in the NPA hierarchy constrained the devices to satisfy some conditional probability distribution. We optimized the choice of distribution for each detection efficiency and entropy separately using a parametrization of two-qubit systems together with the method of ref. ⁵².