Describing quantum metrology with erasure errors using weight distributions of classical codes

Quantum sensors are expected to be a prominent use-case of quantum technologies, but, in practice, noise easily degrades their performance. Quantum sensors can for instance be afflicted with erasure errors. Here, we consider using quantum probe states with a structure that corresponds to classical [n,k,d] binary block codes of minimum distance d≥t+1. We obtain bounds on the ultimate precision that these probe states can give for estimating the unknown magnitude of a classical field after at most t qubits of the quantum probe state are erased. We show that the quantum Fisher information is proportional to the variances of the weight distributions of the corresponding 2t shortened codes. If the shortened codes of a fixed code with d≥t+1 have a nontrivial weight distribution, then the probe states obtained by concatenating this code with repetition codes of increasing length enable asymptotically optimal field sensing that passively tolerates up to t erasure errors.