The fundamental limits of precision measurement with quantum states are currently being redefined through new understandings of the Fisher information. Kaustav Chatterjee (Technical University of Denmark), Tanmoy Pandit (QMill Oy), and Varinder Singh (Korea Institute for Advanced Study), alongside Pritam Chattopadhyay and Ulrik Lund Andersen, have presented a novel decomposition of the Gaussian Quantum Fisher Information (QFI) into even and odd components. Their research reveals a surprising connection between the symplectic geometry of quantum states and the practical limits of metrology, offering a deeper insight into how correlations and spectral properties contribute to measurement precision. This work demonstrates that the established framework can be used to design more effective continuous-variable quantum sensors and benchmark the performance of Gaussian quantum channels. By cleanly separating resources associated with spectrum and correlations, the team provides practical design rules for optimising quantum metrology protocols.
The research reveals that the even part of the QFI captures changes in the symplectic spectrum, while the odd part is associated with correlation-generating dynamics within the quantum system. Experiments on pure states demonstrate the even contribution vanishes, with the odd contribution coinciding with the QFI derived from the natural metric on the Siegel upper half-space, establishing a direct geometric underpinning for pure-Gaussian metrology. The team measured that for evolutions governed entirely by passive Gaussian unitaries, the odd QFI vanishes completely, with thermometric parameters contributing solely to the even sector in a simple spectral form.
Researchers derived a state-dependent lower bound on the even QFI, determined by the rate of change of the state’s purity, demonstrating that this contribution controls how quickly the state deviates from or approaches purity. Measurements confirm this bound diverges when approaching the pure-state manifold along directions that alter the symplectic eigenvalues, mirroring behaviour observed in transmissivity estimation. Tests prove that passive Gaussian unitaries, represented by orthogonal symplectic matrices, generate no odd contribution, resulting in a vanishing odd QFI. Conversely, genuinely active Gaussian operations, such as two-mode squeezing, contribute non-trivially to the odd QFI by creating or redistributing correlations while maintaining a fixed symplectic spectrum.
This establishes the odd sector as a quantifier of correlation-generating resources, providing a clear distinction between sensing protocols relying on population changes and those exploiting active Gaussian dynamics. Specifically, parameters with generators lying purely in the even sector do not mix, at the level of the QFI, with those whose generators lie purely in the odd sector. This clarifies when cross-parameter information, and potential incompatibilities, can arise between thermometric and correlation-generating parameters. Demonstrations in joint estimation scenarios, such as simultaneous sensing of phase and loss, show how this framework cleanly separates the roles of population changes versus correlations in attainable precision and the structure of the QFI matrix.
Gaussian State Decomposition Reveals Metrological Geometry
This work introduces a decomposition of the Fisher information for centered multimode Gaussian states, dividing it into even and odd sectors. This decomposition is rooted in the symplectic Lie algebra and implemented through analysis of tangent vectors on the Gaussian state manifold, allowing any change to a covariance matrix to be understood as a combination of spectral deformation and correlation-generating dynamics. The research demonstrates that on pure Gaussian states, the even sector vanishes, revealing a direct connection between the odd component of the QFI and the natural metric on the Siegel upper half-space, thus establishing a geometric basis for pure-Gaussian metrology. The authors further extended this framework to mixed Gaussian states, identifying a fiber-bundle structure where the even sector relates to changes in the symplectic spectrum and the odd sector quantifies deformations of the symplectic frame. This separation clarifies the roles of spectral and correlation-based resources in metrological protocols, offering practical design rules for continuous-variable systems and a means to benchmark probes and channels.
👉 More information
🗞 Even Odd Splitting of the Gaussian Quantum Fisher Information: From Symplectic Geometry to Metrology
🧠ArXiv: https://arxiv.org/abs/2601.06513