Optimal arbitrary superpositions for lossy phase estimation per absorbed photon

Characterize whether arbitrary superpositions of entangled optical probes that optimize lossy phase estimation can outperform a single recycled photon when performance is measured by quantum Fisher information per photon absorbed by the sample.

Background

The paper analyzes several entangled probe families, including N00N states and the two-mode states |m::m'⟩, under a resource measure based on absorbed photons rather than photons sent or channel uses. Within the studied families, the performance remains below that of a single recycled photon, although the best |m::m'⟩ states approach a supremum of approximately 0.868 of the single-photon benchmark as the photon number increases.

The unresolved case is the full class of arbitrary superpositions previously considered for lossy phase estimation. Their decoherence under loss is not captured by the simple survival-power model used for the analyzed families, and the paper does not determine their quantum Fisher information per absorbed photon relative to the single recycled-photon benchmark.