Criteria for unbiased estimation: applications to noise-agnostic sensing and quantum channel estimation

We establish the necessary and sufficient conditions for local unbiased estimation in multi-parameter estimation tasks. More specifically, we first consider quantum state estimation, where multiple parameters are encoded in a quantum state, and derive simple and intuitive necessary and sufficient conditions for a local unbiased estimation based on the derivatives of the encoded state. To demonstrate the utility of our framework, we consider phase estimation under unknown Pauli noise. We show that while local unbiased phase estimation is infeasible with a naive scheme, employing an entangled probe with a noiseless ancilla enables local unbiased estimation. We then extend our analysis to quantum channel estimation and derive a necessary and sufficient condition for local unbiased estimability of channel parameters, allowing noiseless ancillae, general CPTP controls, and multiple uses of the channel. As a concrete application, we analyze unbiased estimation of noise parameters affecting non-Clifford gates via cycle benchmarking under SPAM errors.