Fisher information of a squeezed-state interferometer with a finite photon-number resolution

Squeezed-state interferometry plays an important role in quantum-enhanced optical phase estimation, as it allows the estimation precision to be improved up to the Heisenberg limit by using ideal photon-number-resolving detectors at the output ports. Here we show that for each individual $N$-photon component of the phase-matched coherent $\otimes$ squeezed vacuum input state, the classical Fisher information always saturates the quantum Fisher information. Moreover, the total Fisher information is the sum of the contributions from each individual $N$-photon components, where the largest $N$ is limited by the finite number resolution of available photon counters. Based on this observation, we provide an approximate analytical formula that quantifies the amount of lost information due to the finite photon number resolution, e.g., given the mean photon number $\bar{n}$ in the input state, over $96$ percent of the Heisenberg limit can be achieved with the number resolution larger than $5\bar{n}$.

Comments: A new analytical result of the Fisher information (Eq.(18), or more generally, Eq.(B9)) has been derived to quantify how much the phase information is kept (or lost) due to finite number resolution of the photon-counting detectors

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