Fig. 3: Global randomness vs. detection efficiency (η) in the 2222-scenario.

We compare lower bounds on different measures of the global randomness produced by 2-input 2-output devices that have some fixed detection efficiency η ∈ [0.7, 1]. The curves for \({H}_{(4/3)}^{\uparrow }(AB| E)\), \({H}_{(2)}^{\uparrow }(AB| E)\), and \({H}_{\min }(AB| E)\) were computed numerically, the red curve representing \(\inf H(A| E)\) was computed using the analytical expression from ref. ¹⁰ and the TSGPL bound uses data from the authors of ref. ¹⁵. The red curve (analytic) was computed by maximizing the CHSH score over two-qubit systems with a fixed η. All other curves constrained the devices to satisfy some fixed probability distribution. For the TSGPL bound this distribution was chosen by maximizing the CHSH score for a fixed η. For the remainder of the curves we optimized our choice of distribution using the method of ref. ⁵². Note that this optimization is important when in the presence of inefficient detectors. For example, if we always use the two-qubit system which achieves Tsirelson’s bound for the CHSH game in the noiseless case then we could not certify any entropy for detection efficiencies lower than η ≈ 0.83. However, by allowing ourselves to optimize over partially entangled states we can certify entropy down to detection efficiencies of η ≈ 0.67.