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

TL;DR

This paper links classical code structures to quantum probe states, deriving bounds on precision in quantum sensing under erasure errors and showing how certain code-based states can asymptotically optimize field estimation despite erasures.

Contribution

It introduces a novel approach connecting classical code weight distributions to quantum Fisher information, enabling passive error tolerance in quantum metrology.

Findings

Quantum Fisher information relates to code weight variances.

Code-based probe states can asymptotically achieve optimal sensing.

Passive erasure error tolerance is possible with concatenated code states.

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 $d \geq t+1$. 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 $d \geq t+1$ have a non-trivial 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.