Phase estimation for thermal Gaussian states

TL;DR

This paper derives optimal bounds on phase estimation precision for mixed Gaussian states, analyzing how temperature and measurement strategies affect fidelity in quantum optical metrology.

Contribution

It provides the first comprehensive analysis of phase estimation bounds for displaced and squeezed thermal states, comparing measurement strategies and considering loss effects.

Findings

Temperature affects estimation fidelity differently for displaced and squeezed thermal states.

Adaptive homodyning attains bounds for displaced thermal states but not for squeezed states.

Heterodyne measurements perform well for mixed states, close to optimal bounds.

Abstract

We give the optimal bounds on the phase-estimation precision for mixed Gaussian states in the single-copy and many-copy regimes. Specifically, we focus on displaced thermal and squeezed thermal states. We find that while for displaced thermal states an increase in temperature reduces the estimation fidelity, for squeezed thermal states a larger temperature can enhance the estimation fidelity. The many-copy optimal bounds are compared with the minimum variance achieved by three important single-shot measurement strategies. We show that the single-copy canonical phase measurement does not always attain the optimal bounds in the many-copy scenario. Adaptive homodyning schemes do attain the bounds for displaced thermal states, but for squeezed states they yield fidelities that are insensitive to temperature variations and are, therefore, sub-optimal. Finally, we find that heterodyne measurements perform very poorly for pure states but can attain the optimal bounds for sufficiently mixed states. We apply our results to investigate the influence of losses in an optical metrology experiment. In the presence of losses squeezed states cease to provide Heisenberg limited precision and their performance is close to that of coherent states with the same mean photon number.