Practical Quantum Metrology in Noisy Environments

Abstract

The problem of estimating an unknown phase $\phi$ using two-level probes in the presence of unital phase-covariant noise and using finite resources is investigated. We introduce a simple model in which the phase-imprinting operation on the probes is realized by a unitary transformation with a randomly sampled generator. We determine the optimal phase sensitivity in a sequential estimation protocol, and derive a general (tight-fitting) lower bound. The sensitivity grows quadratically with the number of applications $N$ of the phase-imprinting operation, then attains a maximum at some $N_{\text{opt}}$, and eventually decays to zero. We provide an estimate of $N_{\text{opt}}$ in terms of accessible geometric properties of the noise and illustrate its usefulness as a guideline for optimizing the estimation protocol. The use of passive ancillas and of entangled probes in parallel to improve the phase sensitivity is also considered. We find that multi-probe entanglement may offer no practical advantage over single-probe coherence if the interrogation at the output is restricted to measuring local observables.

10 pages, 5 figures; published version

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