Accuracy demonstration of the Bayesian rule against the quantum trajectory equation.
(a,d) Stochastic currents in the homodyne detection for dispersive coupling χ = 0.1 and 0.5. The black curves denote the coarse-grained results for visual purpose, while the original currents (blue ones) are actually used for state estimate (inference). (b,c) State (density matrix) evolution under continuous measurement for a relatively weak qubit-cavity coupling, χ = 0.1. (e,f) State evolution under continuous measurement for a strong qubit-cavity coupling, χ = 0.5. (b,c,e,f) The curves “E” (red), “G” (green) and “K” (blue) denote, respectively, our exact Bayesian rule, Eqs. (18) and (19) together with (17), the approximate one involving instead the usual Gaussian distribution of Eq. (21) and that proposed by Korotkov16 under the bad-cavity and weak-response limits. In each figure, the lower panel plots also the difference from the quantum trajectory equation result, indicating that the BR proposed in this work is indeed exact. In all these numerical simulations, we chose the LO’s phase φ = π/4 and adopted a system of reduced units with parameters Δr = 0, εm = 1.0 and κ = 2.0.