Even Odd Splitting of the Gaussian Quantum Fisher Information: From Symplectic Geometry to Metrology

TL;DR

The paper introduces a canonical, parity-based decomposition of the quantum Fisher information (QFI) for centered multimode Gaussian states into an even part, tied to changes in the symplectic spectrum, and an odd part, tied to correlation-generating dynamics, leading to ds^2 = ds_e^2 + ds_o^2 with no cross term. Using a fiber-bundle geometry over Gaussian state space and a Cartan decomposition of the symplectic algebra, the authors derive closed-form expressions for the even and odd QFI in the Williamson frame and show that for pure states the even sector vanishes, leaving the QFI governed by the Siegel upper-half-space geometry. The framework extends to the full multi-parameter QFIM with sector decoupling and clarifies when cross-parameter information can arise; it is illustrated on unitary sensing and Gaussian channels where the even sector captures spectral/thermality resources and the odd sector captures correlation resources. These results provide design principles for continuous-variable metrology, enabling a geometric, resource-aware benchmarking of probes and channels, and point toward extensions beyond Gaussian states. The work thus offers a rigorous, geometrical lens to separate population/spectral changes from correlation-based enhancements in Gaussian quantum metrology.

Abstract

We introduce a canonical decomposition of the quantum Fisher information (QFI) for centered multimode Gaussian states into two additive pieces: an even part that captures changes in the symplectic spectrum and an odd part associated with correlation-generating dynamics. On the pure-state manifold, the even contribution vanishes identically, while the odd contribution coincides with the QFI derived from the natural metric on the Siegel upper half-space, revealing a direct geometric underpinning of pure-Gaussian metrology. This also provides a link between the graphical representation of pure Gaussian states and an explicit expression for the QFI in terms of graphical parameters. For evolutions completely generated by passive Gaussian unitaries (orthogonal symplectics), the odd QFI vanishes, while thermometric parameters contribute purely to the even sector with a simple spectral form; we also derive a state-dependent lower bound on the even QFI in terms of the purity-change rate. We extend the construction to the full QFI matrix, obtaining an additive even odd sector decomposition that clarifies when cross-parameter information vanishes. Applications to unitary sensing (beam splitter versus two-mode squeezing) and to Gaussian channels (loss and phase-insensitive amplification), including joint phase loss estimation, demonstrate how the decomposition cleanly separates resources associated with spectrum versus correlations. The framework supplies practical design rules for continuous-variable sensors and provides a geometric lens for benchmarking probes and channels in Gaussian quantum metrology.

Figures (10)

• Figure 1: Various spaces and their commutative diagram. Here, $\phi$ is the map that defines the trivialization. This is for the set of Gaussian states.
• Figure 2: Various spaces and their commutative diagram. Here, $\phi$ is the map that defines the trivialization. This is for the Symplectic group.
• Figure 3: Graphical representation of how the beam-splitter changes and distributes correlations. Here $c_\theta=cos(\theta)$ and $sh(r)=sinh(r)$.
• Figure 4: Plot of the even quantum Fisher information $QFI_e(v_1,v_2,r=0)$. It is zero when $v_1 = v_2$ and hence can witness a temperature difference between modes. For very asymmetric mode temperatures (even for small differences), the corners of the plot light up.
• Figure 5: . Plot of the odd quantum Fisher information $QFI_o(v_1,v_2,r=0.5)$. Unlike its even counterpart, this contribution never vanishes and increases more rapidly with growing asymmetry. A large odd component reflects that the parameter is probed along a direction that enhances inter-mode correlations, leading to an increased sensitivity of the state in this sector.

Theorems & Definitions (19)

• Definition 1: Fiber bundle
• Definition 2: Section
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• Corollary 3.1
• Corollary 3.2
• Definition 3: Even and Odd QFI:
• proposition 1: Invariance under even-frame changes