Information limits and measurement-assisted learning for fidelity estimation of composed quantum channels

Abstract Reset-bath marginal channels leave fidelity ambiguity in quantum processes with memory. Fixed-input counterfactual audits give mean fidelity spans of 0.229 for dephasing and 0.100 for exchange dynamics over the target retention interval. Numerical optimisation confirms these spans along fixed-parameter retention trajectories, and a two-qubit bath also leaves nonzero ambiguity. We compare ancilla-free Direct Fidelity Estimation, or DFE, with constant, ridge and neural priors calibrated from identical finite-shot labels. Fusion weights use calibration-derived variance proxies independent of final query outcomes. Validation selects each method's budget before a fresh test. In dephasing, ridge and neural shrinkage both use 20 rather than 32 shots at average MAE target 0.065, and 36 rather than 48 at 0.053. The two priors share the selected budgets under this grid and selection rule, with lower neural mean errors. Constant shrinkage saves four shots per query relative to DFE, while ridge and neural save eight more. Both recover their 4096-shot calibration measurement cost from query 342, compared with 1025 for constant shrinkage. This accounting excludes input characterisation. In exchange, calibrated ridge meets both targets without query measurements.