Precision limits for time-dependent quantum metrology under Markovian noise

TL;DR

This paper establishes fundamental precision bounds for time-dependent quantum parameter estimation under Markovian noise, extending existing frameworks and demonstrating tight bounds with error correction strategies.

Contribution

It extends the minimization-over-purifications framework to time-varying channels and derives universal scaling laws for quantum Fisher information under Markovian noise.

Findings

QFI scales as T^{2k} in noiseless dynamics, limited to T^{2k} or T^{2k-1} under Markovian noise depending on regime.

Demonstrates T^{4} and T^{3} scaling in qubit sensors with dephasing and spontaneous emission.

Provides explicit quantum error correction schemes that asymptotically saturate the bounds.

Abstract

We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields $Q_{\mathrm{coh}}(T)\sim T^{2k}$, then under Markovian noise the QFI scales at most as $Q(T)\sim T^{2k}$ in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to $Q(T)\sim T^{2k-1}$. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting $T^{4}$ and $T^{3}$ scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.