Quantum metrology in the presence of correlated noise via Markovian embedding

TL;DR

Addresses quantum metrology with correlated noise by proposing a Markovian embedding using pseudomodes to convert non-Markovian bath dynamics into a finite Markovian enlarged system. It combines quantum comb formalism with Iterative See-Saw optimization to design optimal adaptive protocols and derive bounds for correlated noise, utilizing the channel QFI $F_Q$ and related constructs. Demonstrated on a damped Jaynes–Cummings model, environmental memory carried by pseudomodes can boost the achievable $F_Q$, particularly with an ancilla, and universal correlated bounds bound the QFI in these scenarios. The framework offers a scalable, unifying approach to quantum metrology under quantum correlated noise, with potential extensions to more intricate open-system dynamics.

Abstract

We analyze quantum metrological protocols, where the sensing system is linearly coupled to a bosonic environment, by performing a Markovian embedding of the problem based on pseudomode formalism. This allows us to effectively model the problem using low-dimensional environment and apply recently developed powerful tools that yield optimal metrological protocols and fundamental metrological bounds for correlated-noise models. We illustrate the method by investigating a frequency estimation protocol in the presence of noise modeled effectively as a damped Jaynes-Cummings dynamics.

Figures (4)

• Figure 1: Schematic illustration of two equivalent setup leading to same reduced dynamics of the system. (a) System is linearly coupled to a bath (infinite) with arbitrary coupling strength. The total system-bath dynamics is governed by the Unitary $U$, generated by the total Hamiltonian $H$ as in Eq. (\ref{['original_H']}). The corresponding metrological protocol (below) should in principle involve the whole bath-environment (inaccessible) to account for the correlation built up in course of the dynamics. (b) System space enlarged with a finite number of discrete pseudomodes together obeys a Markovian master equation as in Eq. (\ref{['sm-master']}) ($\Lambda$ is the corresponding dynamical map) such that the reduced system dynamics is the same as the original one. The metrological protocol (below) now involves finite pseudomode space $M$ (inaccessible) to effectively model all the correlations present in the model.
• Figure 2: Schematic of general adaptive metrological protocol involving $N$ independent ($\Lambda_\varphi^{\otimes N}$) or correlated quantum channels ($\Lambda_\varphi^{(N)}$) with arbitrary quantum controls ($C$).
• Figure 3: Comparison of QFI growth with time $t$ for three models, one with correlated environment (dynamics as given in Eq. (\ref{['eq:smmaster']})), one with fresh environment (dynamics as given in Eq. (\ref{['S-dynamical-map']})) and the third one is the no control scheme repeated over the course of time. (a) Schematic diagrams of the first two models. (b) QFI with qubit ancilla in the Markov limit; QFI with no control scheme repeated over the course of total sensing time and correlated bound for the dynamics with fresh/correlated environment (block sizes $=3$). Parameters for all the curves are $\gamma_0=2.457$, $\tau_B\sim\lambda^{-1}=0.01$ and $\Delta t=0.5$. (c) QFI for correlated, fresh and repeated no control schemes with qubit ancilla and without ancilla; bounds with correlated and fresh dynamics (block size $=3$). Parameter values for all the curves are $\gamma_0=2.457$, $\tau_B\sim\lambda^{-1}=0.4$ and $\Delta t=0.5$.
• Figure 4: Plots for QFI with time $t$, for different $\Delta t$ and qubit ancilla, with fixed $\lambda=2.5$ ($\tau_B=0.4$) and different $\gamma_0$. Markovian bound (constant decay rate $\gamma_0 \approx 1.14$ ) and the ultimate limit with unitary encoding are also illustrated in the figure.