Fig. 5: Optimal randomness generation rates as a function of the loss of VOA with different values of \({\hat{p}}_{\alpha }\) and error estimation methods including random sampling (RS), entropy inequality (EI) and infinite length (IL).

RS, EI, and IL represent the models without the afterpulse and RSP, EIP, and ILP represent the models with the afterpulse \({\hat{p}}_{\alpha }=5 \%\). The experimental parameters are listed in Table 1, and we assume that the pulse distribution is the coherent state that contains ν = 50 photons initially. a The condition with low loss. b The condition with the loss of up to 50 dB.