Path Integral Approach to Quantum Fisher Information

TL;DR

This paper introduces a real-time path-integral framework for quantum Fisher information, enabling parameter estimation without explicit state reconstruction, and connects it with many-body and semiclassical methods.

Contribution

It develops a novel path-integral and Schwinger-Keldysh formalism for quantum Fisher information, bridging quantum metrology with real-time correlators and semiclassical approximations.

Findings

Expresses quantum Fisher information as a symmetrized covariance of action deformations.

Rewrites quantum Fisher information in terms of real-time correlators suitable for many-body methods.

Derives a semiclassical expression linking classical trajectories to metrological sensitivity.

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.