Geometric Underpinning of Pure-Gaussian Metrology

• Geometric Underpinning of Pure-Gaussian Metrology is a framework that employs manifold geometry, symplectic invariance, and Riemannian metrics to set precision limits in quantum parameter estimation.
• It utilizes the Fubini–Study metric and Siegel upper half-space to translate quantum Fisher information into measurable distances on the Gaussian manifold.
• The approach provides practical design rules for continuous-variable sensor optimization by linking group symplectic actions and geometric invariants to probe performance.

The geometric underpinning of pure-Gaussian metrology refers to the foundational role played by manifold geometry, symplectic invariance, and Riemannian metrics—specifically those arising from the Fubini–Study distance and the Siegel upper half-space—in quantifying precision limits and optimization strategies for quantum parameter estimation with pure Gaussian states. The interplay between covariance matrix geometry, the structure of the quantum Fisher information (QFI), and the symplectic group action provides both conceptual clarity and practical design rules for continuous-variable (CV) quantum metrology.

1. Structure of the Pure-Gaussian Manifold

Pure $n$-mode Gaussian states form a manifold $\mathcal{M}$ parameterized explicitly by phase-space displacements and symplectic transformations acting on the vacuum. Any such state can be expressed (up to a global phase) as

$$|\psi(\lambda, S)\rangle = D(\lambda) S |0^n\rangle$$

where $D(\lambda)$ denotes displacement by real vector $\lambda \in \mathbb{R}^{2n}$ and $S \in \mathrm{Sp}(2n,\mathbb{R})$ is a real symplectic matrix effecting all Gaussian operations (squeezing, rotations, mode-mixing) (Sidhu et al., 2019). The covariance matrix $\sigma = \frac{1}{2} S S^T$ fully characterizes the state for zero mean.

The dimension of $\mathcal{M}$ is $2n$ (displacements) $+$ $\dim\mathrm{Sp}(2n) = n(2n+1)$ (squeeze/rotate parameters), minus one for a global $U(1)$ phase. Tangent vectors to $\mathcal{M}$ correspond to infinitesimal variations in $(\lambda, \sigma)$, constrained by the underlying symplectic structure.

2. Riemannian Geometry, Fubini–Study Metric, and QFI

On the projective Hilbert space of pure states, the Fubini–Study metric

$$ds_\mathrm{FS}^2 = 4\left(1 - |\langle\psi|\psi + d\psi\rangle|\right)$$

reduces, on the pure-Gaussian submanifold, to an explicit metric on covariance matrices: $$ds^2_{\mathrm{FS}} = \frac{1}{4}\mathrm{Tr}\left[(d\sigma\,\sigma^{-1})^2\right]$$ This is equivalently expressible in terms of the complex structure $J = \Omega \sigma^{-1}$ (for bosons) as $$ds^2_{\mathrm{FS}} = \frac{1}{8}\mathrm{Tr}[dJ\,dJ]$$ (Windt et al., 2020, Chatterjee et al., 10 Jan 2026). The resulting Riemannian metric provides a local measure of distinguishability for pure Gaussians and induces a Kähler geometry on the manifold (Sidhu et al., 2019).

Strikingly, the QFI metric for a family $|\psi(\theta)\rangle$ coincides—up to a scale factor—with the Fubini–Study metric: $$\text{QFI}(\theta) = 4\,ds^2_{\mathrm{FS}}\left(\frac{d}{d\theta}, \frac{d}{d\theta}\right) = \mathrm{Tr}\left[(\partial_\theta \sigma) \sigma^{-1} (\partial_\theta \sigma) \sigma^{-1}\right]$$ In the multi-parameter case, the QFI matrix is given by $$F_{ij} = \mathrm{Tr}\left[(\partial_i \sigma) \sigma^{-1} (\partial_j \sigma) \sigma^{-1}\right]$$ (Windt et al., 2020, Sidhu et al., 2019). Thus, the precision achievable in pure-Gaussian metrology is determined by distances on the Gaussian manifold.

3. Even–Odd Splitting of the Gaussian QFI

The QFI for centered multimode Gaussian states admits a canonical even–odd decomposition (Chatterjee et al., 10 Jan 2026):

• Even QFI: Encodes changes in the symplectic spectrum (e.g., thermal fluctuations, changes in purity).
• Odd QFI: Associated with correlation-generating dynamics (e.g., squeezing, entanglement generation). For pure Gaussian states, all symplectic eigenvalues are fixed at $1/2$, so the even contribution vanishes identically: $\text{Pure Gaussian:}\quad QFI_e = 0$ Only the odd part survives, which precisely coincides with the Siegel metric on the Siegel upper half-space $$\mathfrak{h}_n = \{Z = X + iY: X^\top = X, Y^\top = Y > 0\}$$: $$ds^2_{\mathrm{Siegel}} = \mathrm{Tr}\left[Y^{-1} dZ\, Y^{-1} dZ^*\right]$$

$$QFI_{\rm pure} = 4\, \mathrm{Tr} [ Y^{-1} \dot{Z}\, Y^{-1} \dot{Z}^* ]$$

This geometric structure allows one to translate metrological resource quantification into questions about geodesics and distances on $\mathfrak{h}_n$. The QFI for any pure-Gaussian 1-parameter family is given by the length squared of the tangent vector in the Siegel metric, simplifying both analysis and optimization.

4. Symplectic Group Action and Tangent Vector Structure

The tangent bundle of the pure-Gaussian manifold is generated by the symplectic group (bosonic case) or orthogonal group (fermionic case). Every state is an orbit of the vacuum under Sp($2n$), i.e., $\sigma = M \sigma_0 M^T$. Infinitesimal symplectic transformations $K \in \mathfrak{sp}(2n)$ induce tangent directions: $d\sigma = K \sigma + \sigma K^T, \quad dJ = [K, J]$ A basis for the tangent space at a given state is constructed from a basis $\{K_\mu\}$ of $\mathfrak{sp}(2n)$ modulo stabilizer directions (the $\mathfrak{u}(n)$ subalgebra fixing $\sigma$) (Windt et al., 2020). The action of the group enables the push-forward of metric tensors and gradient fields efficiently across the manifold, avoiding repeated inversion at every point.

5. Geometric Optimization of Quantum Metrological Objectives

The aforementioned geometric structure naturally supports efficient optimization of metrological figures of merit, notably the maximization of QFI for probe design. The typical workflow is as follows (Windt et al., 2020):

1. Fix a reference pure-Gaussian state (typically the vacuum).
2. Parametrize pure Gaussian states via group elements $M \in \mathrm{Sp}(2n)$ acting on the reference.
3. Define the objective $f(M) = \text{QFI}(\theta, \sigma(M))$.
4. Compute derivatives $\partial_\mu f$, using the explicit group action and the chain rule for $d\sigma$.
5. Construct the gradient vector field on the group manifold, making use of the precomputed metric at the reference point.
6. Iterate a local retraction (e.g., the Cayley approximation) along the gradient direction to increase QFI, subject to experimental or resource constraints. This setup accommodates constraints such as fixed energy, restricted unitaries, or multi-parameter trade-offs, facilitating flexible probe design.

6. Geometric Invariants and Metrological Resource Classification

Geometric invariants—quantities unchanged under the group action—play a central role in classifying metrological resources. For a set of generators closing under commutation (a Lie algebra $\mathfrak{g}$), the QFI matrix $F^g$ and the Uhlmann curvature tensor $\Omega$ retain their spectra under group evolutions. This leads to a metrological analogue of Liouville's theorem: statistical distances, volumes (as $\sqrt{\mathrm{pdet} F}$), and curvatures are preserved under unitary group flows (Wilson et al., 8 Jul 2025). In the Gaussian setting, this implies that resources such as squeezing or entanglement—quantified by the QFI eigenvalues—cannot be increased by symmetry-preserving Gaussian unitaries, establishing a “metrological budget” fixed by the initial state.

The spectrum of $F^g$ and its companion invariants partition state space into orbits of equivalence for parameter estimation: two states within the same orbit have equal metrological capabilities with respect to the algebra $\mathfrak{g}$. This establishes a rigorous classification framework for CV quantum sensing and guides the selection of probe states for optimal estimation tasks.

7. Practical Implications and Sensor Design

The geometric approach underpins practical design rules for continuous-variable sensors. Because QFI and optimal probe structure are geometric, experimentally relevant resource trade-offs—such as the combination of squeezing, mode-mixing, and photon number—can be read off as distances and trajectories on the Siegel upper half-space. Optimal sensing strategies correspond to geodesics of the Siegel metric, and multi-parameter estimation invokes the full metric tensor to analyze precision bounds and compatibility (Chatterjee et al., 10 Jan 2026, Sidhu et al., 2019). No increase in metrological power is possible via Gaussian mixing beyond what is set by the initial squeezing/entanglement content.

Typical scenarios include:

• Single/multi-mode squeezing and displacement estimation: QFI derived directly from the metric structure.
• Analysis of resource interconversion under Gaussian channels.
• Explicit graphical calculus for realizing and evaluating measurement strategies. This geometric framework supplies both deep theoretical insight and concrete prescriptions for the benchmarking and design of quantum-enhanced sensors using pure-Gaussian probes.