Optimal tunnel-rate readout time (${\tau }_{\ast }=\log ({\Gamma }_f/{\Gamma }_s)/({\Gamma }_f-{\Gamma }_s)$)

For independent exponential escape, equal prior probabilities and no state relaxation, the mean classification error is $P_e(\tau )=[e^{-{\Gamma }_f\tau }+1-e^{-{\Gamma }_s\tau }]/2$. Setting its derivative to zero gives the displayed interval. The error increases both when the fast state is not given enough time to escape and when the slow state is given too much. Detector bandwidth, relaxation and reloading require an expanded measurement model.