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.