Quantum-trajectory Monte Carlo estimators

For the GKSL convention used previously, the no-jump Hamiltonian subtracts i hbar over two times the sum of rate-weighted L-adjoint L operators. Its norm decreases according to the instantaneous jump hazard. In a small-time-step implementation, the probability of channel r is gamma-r times timestep times the corresponding expectation in the normalized state, to leading order. A large timestep can invalidate that linear probability approximation or miss multiple jumps.

Exact waiting-time or carefully controlled small-step schemes update both the channel and post-jump state with the right conditional distribution. The normalized no-jump state and the jump weights must be treated consistently; deterministic renormalization without stochastic compensation gives a different ensemble. Different valid unravelings can yield different-looking paths and variances while reproducing the same density evolution. Numerical convergence should target the ensemble observable, not the visual appearance of an individual random path.

The density estimator is the average of normalized pure-state projectors. It is positive and has unit trace for every finite sample, while its expectation equals the master-equation density when the paths are correctly sampled. An observable estimator averages each path's expectation value. Averaging state amplitudes instead is not equivalent: independent global phases can change that average without changing any projector or physical member of the ensemble.

Finite sampling noise and algorithmic bias are distinct. Increasing the number of paths reduces random fluctuations around the distribution produced by the implemented algorithm, but does not fix an incorrect jump rule, finite timestep or too-small basis. Compare timestep and Hilbert dimension at sufficiently large sample size, and report random seeds or reproducibility information as appropriate. Common-random-number comparisons can reduce differences but introduce correlations that must be included in uncertainty interpretation.

For independent identically distributed paths at one fixed evaluation time, an observable sample mean has variance equal to the single-path variance divided by N. For a basis-state indicator this reduces to the familiar binomial probability variance. If its true probability is unknown, estimating that variance from data adds its own finite-sample uncertainty. A standard error is a statistical scale, not a deterministic bound that every run must satisfy.

Trajectory samples at different times on the same path are correlated, as can be different observables or deliberately paired runs. Time averaging those samples does not automatically create the same number of independent paths. Rare jumps can also require many samples before a normal approximation is persuasive. Exact binomial examples offer a transparent benchmark for the simplest indicator, while general coherent paths require the variance of their actual per-path expectation values.