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

Abstract

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 $d\ensuremath{\ge}t+1$. 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 ${2}^{t}$ shortened codes. If the shortened codes of a fixed code with $d\ensuremath{\ge}t+1$ 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.