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arXiv:2610.02625

Gaussian Fisher Information Is Superadditive

Jiaxin Liu, Zuoxian Wang, Danyue Ma

quant-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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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.

Comments
32 pages, 10 figures

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A weak force displaces a quantum oscillator in phase space, and the quantum Fisher information sets how precisely that displacement can be measured. The usual way to raise it is to squeeze the probe, but squeezing is fragile, has to be remade each cycle, and fades as the oscillator loses energy. We propose instead a quadratic Hamiltonian, varying in time, whose propagator traces a closed loop in the symplectic group Sp(2,R)\textrm{Sp}(2,\mathbb {R})Sp(2,R) , the probe passes through strong parametric gain and returns to the vacuum with no squeezing left. The gain enlarges the response to the signal instead of lowering the noise, and because the probe ends where it began, cycles follow one another with nothing to prepare in between, one quadrature being read at the end by ordinary homodyne detection. One smooth function carries the whole propagator, thus a loop is designed by choosing it and reading off the drive it implies, and the Fisher information is two integrals of the signal against the propagator. We give a family of such loops in closed form and three upper bounds, the sharpest reached exactly by the signal matched to the loop. When the oscillator also loses photons to a bath at rate κ\kappa κ , the Fisher information F\mathcal {F}F over a cycle from t0t_{0}t0​ to T obeys F≤4eκ(T−t0)∥s~∥2[(T−t0)+η/κ]\mathcal {F} \le 4\textrm{e}^{\kappa (T-t_{0})}\Vert \tilde{s}\Vert ^{2}[(T-t_{0}) + \eta /\kappa ]F≤4eκ(T−t0​)∥s~∥2[(T−t0​)+η/κ] , where ∥s~∥2\Vert \tilde{s}\Vert ^{2}∥s~∥2 is the signal energy and η\eta η the noise the bath injects while the gain is on. The inequality has the shape of a second law: the first term is what a loop that merely rotates already achieves, the second is the price of the gain.
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Abstract The central issue in quantum parameter estimation is to find out the optimal measurement setup that leads to the ultimate lower bound of an estimation error. We address here a question of whether a Gaussian measurement scheme can achieve the ultimate bound for phase estimation in single-mode Gaussian metrology that exploits single-mode Gaussian probe states in a Gaussian environment. We identify three types of optimal Gaussian measurement setups yielding the maximal Fisher information depending on displacement, squeezing, and thermalization of the probe state. We show that the homodyne measurement attains the ultimate bound for both displaced thermal probe states and squeezed vacuum probe states, whereas for the other single-mode Gaussian probe states, the optimized Gaussian measurement cannot be the optimal setup, although they are sometimes nearly optimal. We then demonstrate that the measurement on the basis of the product quadrature operators X^P^+P^X^\hat X\hat P + \hat P\hat XX^P^+P^X^ X ̂ P ̂ + P ̂ X ̂ , i.e., a non-Gaussian measurement, is required to be fully optimal.
We find the optimal initial states of a linearized cavity-optomechanical system for estimating the single-photon coupling by quantum Fisher information. In the red-detuned (beam-splitter) regime the interaction exchanges excitations between the two modes, and for pure inputs the Fisher information is four times the variance of the excitation-exchange operator, reducing the optimization to variance maximization with a closed-form solution. With the mechanical mode in a Fock state the probe is degenerate: any state saturating the photon-number constraint is optimal, and a Fock reference with many phonons enhances the per-photon sensitivity linearly. For an arbitrary fixed mechanical reference it becomes a phase-matched squeezed vacuum, from an exact bound on that coherence obtained by a Lagrange-dual argument. Jointly optimizing both modes at fixed total excitation number, the quantum Fisher information obeys a Heisenberg bound quadratic in the energy, saturated by equal superpositions of the two extremal eigenstates of the generator, equivalently a two-mode NOON state after a balanced beam splitter; balanced product Fock states reach only a linear enhancement. Exact diagonalization in truncated Fock spaces confirms all the results, the familiar single-photon input being the simplest saturating case. Three extensions are worked out: the blue sideband has an unbounded two-mode-squeezing spectrum, so the entangled optimum becomes impossible and the balanced product Fock state is exactly optimal at definite energy; under cavity loss the information accumulation turns linear on the cavity-lifetime timescale, at a renewal-optimal duration of about two and a half lifetimes; and for a state-of-the-art quantum-coherent coupling experiment the per-hour relative precision lies between one part in a million and ten parts in a million, number and thermal references being the most robust.
A simple but rigorous analysis of the important sources of noise in homodyne detection is presented. Output noise and signal-to-noise ratios are compared for direct detection, conventional (one-port) homodyning, and two-port homodyning, in which one monitors both output ports of a 50-50 beam splitter. It is shown that two-port homodyning is insensitive to local-oscillator quadrature-phase noise and hence provides (1) a means of detecting reduced quadrature-phase fluctuations (squeezing) that is perhaps more practical than one-port homodyning and (2) an output signal-to-noise ratio that can be a modest to significant improvement over that of one-port homodyning and direct detection.