Gaussian Fisher Information Is Superadditive

Abstract

Quantum Fisher information adds over independent probes, so a single parameter is best measured probe by probe. We show that Gaussian measurements, the linear optics and homodyne detection of optical and microwave experiments, break this rule: two independent modes are better measured together. The uncertainty principle leaves a linear detector half of phase space, and for two modes the choice of half is a resource. Bell homodyne, a balanced beam splitter followed by two homodyne detectors, beats the best separate readout when the modes differ in quadrature width and in how the width responds to the parameter. Heterodyne detection splits a mode on a beam splitter to read both quadratures and pays one unit of vacuum noise for the empty port. Bell homodyne fills that port with the second mode, so the noise turns into signal. We prove that the gain stays below $(\sqrt2-1)^2=17.157\%$ for every parameter carried by widths. For thermal modes at one temperature, the canonical case, a gain was conjectured impossible. We prove that any frequency difference opens a temperature window and that the gain peaks at $12.699\%$ at frequency ratio $3.318$, where Bell homodyne is the optimal Gaussian measurement. Standard hardware reaches the gain: two coupled resonators with two homodyne detectors, or a phase-preserving amplifier whose idler band is fed by the same thermal source.