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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
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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 $\partial_\lambda S$ 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.

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Francis J. Headley, Mahdi RouhbakhshNabati, Henry Harper-Gardner, Daniel Braun, Henning Schomerus, Emre Köse. 2026-04-14. Path Integral Approach to Quantum Fisher Information. https://arxiv.org/abs/2604.12763

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