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arXiv:2604.12763

Path Integral Approach to Quantum Fisher Information

Francis J. Headley, Mahdi RouhbakhshNabati, Henry Harper-Gardner, Daniel Braun, Henning Schomerus, Emre Köse

quant-phcond-mat.stat-mechhep-thSubmitted Apr 14, 2026• Announced Apr 15, 2026• 0 citations
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Abstract

We present a real-time path-integral formulation of the quantum Fisher information for dynamical parameter estimation. For pure states undergoing unitary evolution, we show that the quantum Fisher information can be expressed as a connected symmetrized covariance of a time-integrated action deformation, equivalently as an integrated insertion of $\partial_λS$ in the propagator. This reformulation avoids explicit state reconstruction by rewriting the quantum Fisher information in terms of real-time correlators that are natural targets for many-body methods. We further embed the construction into the Schwinger-Keldysh closed-time-path formalism, identifying the quantum Fisher information with the Keldysh component of an appropriate contour-ordered correlator generated by forward and backward propagating sources. Finally, using the Van Vleck-Gutzwiller approximation we re-derive the compact semiclassical quantum Fisher information expression, clarifying how classical trajectory data control leading-order metrological sensitivity.

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This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.

•Dec 16, 2025•1 citation
We present a real-time path-integral formulation of the quantum Fisher information for dynamical parameter estimation. For pure states undergoing unitary evolution, we show that the quantum Fisher information can be expressed as a connected symmetrized covariance of a time-integrated action deformation, equivalently as an integrated insertion of ∂λS\partial_λS∂λ​S in the propagator. This reformulation avoids explicit state reconstruction by rewriting the quantum Fisher information in terms of real-time correlators that are natural targets for many-body methods. We further embed the construction into the Schwinger-Keldysh closed-time-path formalism, identifying the quantum Fisher information with the Keldysh component of an appropriate contour-ordered correlator generated by forward and backward propagating sources. Finally, using the Van Vleck-Gutzwiller approximation we re-derive the compact semiclassical quantum Fisher information expression, clarifying how classical trajectory data control leading-order metrological sensitivity.
Driven-dissipative many-body quantum systems can be exploited in quantum enhanced sensing protocols. Here, non-classical correlations may allow one to reach a measurement precision that surpasses the standard quantum limit. The relevant figure of merit for assessing such quantum advantage is the quantum Fisher information (QFI). In a many-body setting its computation is challenging as it requires knowledge of the full system-environment state. Here we make use of a field-theoretical approach, which is particularly well suited for systems with many degrees of freedom. In particular, we exploit that the QFI can be linked to a Keldysh path-integral. This route leads to a semiclassical expression of the QFI, which is obtained through a controlled expansion in the so-called quantum fields. We benchmark the framework on three systems of increasing complexity: the driven-dissipative harmonic oscillator, the boundary time crystal - which becomes semiclassical at large system sizes - and a two-dimensional array of collectively emitting atoms. We assess in which parameter regimes the semiclassical description holds and gives access to system sizes beyond the reach of exact methods.
In this work we develop a real-time Schwinger-Keldysh formulation of Krylov dynamics that treats Krylov complexity as an in-in observable generated by a closed time contour path integral. The resulting generating functional exposes an emergent phase-space description in which the Lanczos coefficients define an effective Hamiltonian governing operator motion along the Krylov chain. In the semiclassical limit, exponential complexity growth arises from hyperbolic trajectories, and asymptotically linear Lanczos growth appears as a universal chaotic fixed point, with sub-leading deformations classified as irrelevant, marginal or relevant. Going beyond the saddle, the Schwinger-Keldysh framework provides controlled access to fluctuations and large deviations of Krylov complexity, revealing sharp signatures of integrability-chaos crossovers that are invisible at the level of the mean. This formulation reorganises Krylov complexity into a dynamical field-theoretic framework and identifies new fluctuation diagnostics of operator growth in closed quantum systems.
Quantum Fisher information (QFI) quantifies the sensitivity of a quantum state to a parameter change and plays a key role in quantum metrology. Meanwhile, quantum coherence is a crucial resource for quantum information processing. However, despite extensive studies on various aspects of quantum metrology, the interplay between the dynamics of the QFI and quantum coherence remains unexplored. Here, we explore the factorization relationship between the QFI and quantum coherence, both theoretically and experimentally. We prepare pure states in qubit and qutrit systems and apply unitary evolution to investigate this relationship. Our results confirm the factorization law, offering strong evidence for this relationship. Furthermore, we establish a lower bound on the variance of the estimated parameter that does not require the final state information. This approach reveals a deep connection between the QFI and quantum coherence, providing insights into quantum parameter estimation. These findings have potential applications in quantum sensing and quantum metrology.
Non-Hermitian systems have attracted considerable interest over the last few decades due to their unique spectral and dynamical properties not encountered in Hermitian counterparts. An intensely debated question is whether non-Hermitian systems, described by pseudo-Hermitian Hamiltonians with real spectra, can offer enhanced sensitivity for parameter estimation when they are operated at or close to exceptional points. However, much of the current analysis and conclusions are based on mathematical formalism developed for Hermitian quantum systems, which is questionable when applied to pseudo-Hermitian Hamiltonians, whose Hilbert space is intrinsically nonflat. Here, we develop a covariant formulation of quantum Fisher information (QFI) defined on the deformed Hilbert space of pseudo-Hermitian Hamiltonians. This covariant framework ensures the preservation of the state norm and enables a consistent treatment of parameter sensitivity. We further show that the covariant QFI of pseudo-Hermitian systems is dual to the ordinary QFI of corresponding Hermitian systems. Importantly, this correspondence naturally imposes an upper bound on the covariant QFI and allows one to identify optimal projections which saturate the corresponding classical Fisher information to this ultimate limit. The developed framework also enables to set the criteria under which pseudo-Hermitian sensors can exhibit an advantage over their Hermitian counterparts of the same dimensionality.
The quantum Fisher information (QFI) is a geometric measure of state deformation calculated along the trajectory parametrizing an ensemble of quantum states. It serves as a key concept in quantum metrology, where it is linked to the fundamental limit on the precision of the parameter that we estimate. However, the QFI is notoriously difficult to calculate due to its nonlinear mathematical form. For mixed states, standard numerical procedures based on eigendecomposition quickly become impractical with increasing system size. To overcome this limitation, we introduce a numerical approach based on Lyapunov integrals that combines the concept of symmetric logarithmic derivative and tensor networks. Importantly, this approach requires only the elementary matrix product states algorithm for time evolution, opening a perspective for broad usage and application to many-body systems. We discuss the advantages and limitations of our methodology through an illustrative example in quantum metrology, where the thermal state of the transverse-field Ising model is used to measure magnetic field amplitude.
This work develops a Lorentz-invariant variational framework in which Fisher-information geometry appears as an intrinsic structural contribution to quantum dynamics. Motivated by longstanding attempts to connect quantum mechanics with information-theoretic principles, we introduce an action functional depending on the density and phase fields in the Madelung representation. Variation of this action yields a modified Klein–Gordon equation containing a single nonlinear term proportional to the four-dimensional Fisher-information curvature of the probability density. The standard Klein–Gordon equation is recovered when the structural parameter vanishes, ensuring full compatibility with established relativistic dynamics. Taking the nonrelativistic limit, we obtain a uniquely determined nonlinear Schrödinger equation in which the correction term is the functional derivative of the Fisher information. The resulting dynamics preserve probability, maintain the Hamilton–Jacobi correspondence, and contain the linear Schrödinger equation as a special case. Analytical expressions for Gaussian and superposed states demonstrate how the structural modification scales with spatial localization and interference structure, providing clear qualitative signatures that distinguish the model from previous nonlinear extensions and offer a theoretical basis for future experimental verification. The results establish a mathematically transparent link between information geometry and quantum dynamics and provide a foundation for future extensions to fermionic, gauge, and many-body systems.