Interplay between time and energy in bosonic noisy quantum metrology

TL;DR

This paper investigates how finite energy and time resources influence quantum metrology with bosonic modes under noise, revealing optimal strategies and quantum advantages in parameter estimation.

Contribution

It introduces a comprehensive analysis of the interplay between energy and time resources in noisy bosonic quantum metrology, including bounds and optimal measurement strategies.

Findings

Optimal time-energy partitioning enhances precision.

Fast-prepare-and-measure with Fock states outperforms classical methods for temperature estimation.

Most optimal strategies do not require entanglement or adaptivity.

Abstract

Quantum entanglement and coherence often allow for protocols that outperform classical ones in estimating a system's parameter. When using infinite-dimensional probes (such as a bosonic mode), one could in principle obtain infinite precision in a finite time for both classical and quantum protocols, which makes it hard to quantify potential quantum advantage. However, such a situation is unphysical, as it would require infinite resources, so one needs to impose some additional constraint: typically the average energy employed by the probe is finite. Here we treat both energy and time as a resource, showing that, in the presence of noise, there is a nontrivial interplay between the average energy and the time devoted to the estimation. Our results are valid for the most general metrological schemes (e.g. adaptive schemes which may involve entanglement with external ancillae or any kind of continuous measurement). We apply recently derived precision bounds for all parameters characterizing the paradigmatic case of a bosonic mode, subject to Lindbladian noise. We show how the time employed in the estimation should be partitioned in order to achieve the best possible precision. In most cases, the optimal performance may be obtained without the necessity of adaptivity or entanglement with ancilla. We compare results with classical strategies. Interestingly, for temperature estimation, applying a fast-prepare-and-measure protocol with Fock states provides better scaling with the number of photons than any classical strategy.