Even–Odd Splitting of Gaussian QFI

• The paper decomposes the QFI of centered multimode Gaussian states into an even sector capturing spectral sensitivity and an odd sector reflecting correlation dynamics.
• It rigorously shows that passive operations do not generate odd-sector QFI, while thermometric parameters affect only the even sector, enabling clearer resource classification.
• The framework provides actionable design rules for optimizing quantum sensors by leveraging the geometric structure of the Gaussian state manifold.

The even–odd splitting of the Gaussian quantum Fisher information (QFI) establishes a canonical decomposition of the QFI for centered multimode Gaussian states into two additive contributions: an "even" part encoding the sensitivity of the symplectic spectrum and an "odd" part linked to correlation-generating dynamics. This structure provides a rigorous algebraic and geometric classification of metrological resources in continuous-variable quantum systems, revealing sharp distinctions between spectral (thermodynamic) and correlational (squeezing, entanglement) aspects of Gaussian sensing and estimation. The framework also yields concrete design rules for the optimization of quantum sensors and offers geometric insight into the structure of the quantum statistical manifold for Gaussian states (Chatterjee et al., 10 Jan 2026).

1. Canonical Even–Odd Decomposition of Gaussian QFI

Let $\rho(\theta)$ be an $n$-mode centered Gaussian state parameterized by $\theta$, with covariance matrix $V(\theta)$. The infinitesimal variation $dV$ induces the Bures metric

$$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$

where $L_A(X) = AXA$ and $\Omega$ is the $2n \times 2n$ symplectic form. For a one-parameter path $\theta \mapsto V(\theta)$, the QFI is

$n$0

with $n$1. To obtain the even–odd split, one works in the symplectic (Williamson) frame where $n$2 commutes with $n$3 and $n$4 with $n$5 diagonal. Parity projectors on symmetric $n$6 matrices $n$7 are defined as

$n$8

Pulling back the derivative, denote $n$9. The decomposition is then:

$\theta$0

such that $\theta$1. In block form, $\theta$2 with $\theta$3 and $\theta$4; the same definitions apply.

2. Geometric Structure on the Pure-State Manifold

For pure centered Gaussian states, the symplectic spectrum satisfies $\theta$5 for all $\theta$6. This enforces $\theta$7, leading to $\theta$8 vanishing on the even sector and $\theta$9. The odd part coincides exactly with the symmetric Riemannian metric on the Siegel upper half-space $V(\theta)$0 of complex adjacency matrices $V(\theta)$1 with $V(\theta)$2, yielding:

$V(\theta)$3

Thus, for pure states,

$V(\theta)$4

with $V(\theta)$5. All pure-Gaussian metrology is thus governed by the geometry of the Siegel metric.

3. Prototypical Cases and Lower Bounds

Distinct physical evolutions and parameterizations map cleanly onto the even–odd structure:

• Passive Gaussian unitaries (orthogonal symplectics): For $V(\theta)$6 (generator $V(\theta)$7 skew, symmetric part $V(\theta)$8), the $V(\theta)$9-frame formula $dV$0 yields $dV$1. Hence, passive operations never generate odd-sector QFI.
• Thermometric parameters (spectrum-only): When $dV$2 enters only through symplectic eigenvalues $dV$3, $dV$4 and

$dV$5

For a single-mode thermal state ($dV$6 and $dV$7):

$dV$8

• Lower bound via purity change: Let $dV$9 (purity). Then,

$$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$0

showing that the even-QFI sector bounds the rate of global purity change.

4. Multi-Parameter QFI Matrix Decomposition

Consider parameters $$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$1 with corresponding frame-pulled derivatives $$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$2 decomposed into even/odd blocks $$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$3, and analogously for $$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$4. The QFI matrix is then:

$$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$5

$$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$6

Cross-terms between parameters affecting purely the spectrum (even sector) and those affecting only the frame (odd sector) vanish. When families of parameters separately move only spectrum or only frame, the QFI matrix is block-diagonal.

5. Physical Applications

(a) Unitary Sensing:

• For a pure two-mode squeezed input with local squeezing $$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$7, followed by a beam-splitter of angle $$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$8, the QFI is

$$ds^2 = \frac{1}{2} \operatorname{Tr}\left[dV \, (4 L_V + L_\Omega)^{-1} dV\right],$$9

which is entirely odd-sector due to $L_A(X) = AXA$0. By contrast, in a thermal-contrast interferometer where two thermal states $L_A(X) = AXA$1 are locally squeezed and then mixed,

$L_A(X) = AXA$2

The even part witnesses temperature difference, while the odd part reflects correlation-based (squeezing) resources.

(b) Gaussian Channels—Loss and Amplification:

• For a loss channel with transmissivity $L_A(X) = AXA$3 on a squeezed vacuum (squeeze $L_A(X) = AXA$4 then loss),

$L_A(X) = AXA$5

gives

$L_A(X) = AXA$6

with $L_A(X) = AXA$7 and $L_A(X) = AXA$8, $L_A(X) = AXA$9. As $\Omega$0 or $\Omega$1, $\Omega$2 so $\Omega$3 diverges as $\Omega$4 or $\Omega$5 (reflecting purity-breaking), while $\Omega$6 remains finite.

• Amplification channels are analogous, with odd-QFI dominant when purity changes slowly.

(c) Joint Phase–Loss Estimation:

• For $\Omega$7, the QFI is block-diagonal:

$\Omega$8

$\Omega$9

Off-diagonal covariance $2n \times 2n$0 arises from parameter sector orthogonality.

6. Design Principles and Geometric Perspective

Metrological strategies are dictated by the sector structure:

• Sensing thermometric parameters (temperature, loss, gain) requires optimization within the even sector; correlation-generating operations are ineffectual for $2n \times 2n$1.
• Sensing phase-type parameters is governed by the odd sector, maximized by pure and squeezed states.
• Joint estimation is optimal when parameters are sector-separated, ensuring $2n \times 2n$2 and statistical independence.
• Monitoring global purity provides an experimental lower bound on $2n \times 2n$3, formalizing a "purity-speed-limit."
• Geometrically, the Gaussian-state manifold forms a fiber bundle: base coordinates are symplectic eigenvalues (even directions), fiber is the symplectic frame modulo orthogonal symplectics (odd directions). The QFI metric is block-diagonal with respect to this Cartan decomposition.

In summary, the even–odd splitting yields a transparent algebraic, metrological, and geometric classification of Gaussian resources: the "even" sector encodes spectral (thermodynamic) aspects, while the "odd" sector reflects correlations such as squeezing. This decomposition rigorously guides the analysis and design of continuous-variable sensing protocols, anchoring quantum metrology within symplectic geometric structure (Chatterjee et al., 10 Jan 2026).