Optimal phase estimation in the presence of correlated dephasing

TL;DR

This work addresses the problem of optimal phase estimation when sensing probes experience correlated dephasing, introducing a Gaussian noise model with covariance $\Sigma_{ij}=\sigma^2 c^{|i-j|}$ that spans anti-correlated to fully correlated regimes. It develops a refined classical-simulation bound on the quantum Fisher information and demonstrates, via tensor-network optimization, that finite-bond-dimension matrix-product-state probes in parallel protocols can outperform spin-squeezed states, especially for negative correlations, while adaptive strategies offer no clear advantage under comparable resources. The approach combines discretization of noise with the Rouwenhorst process, quantum comb formalism, and QCE-based bounds to saturate fundamental limits up to $N\approx 30$ channels, revealing a regime where tensor-network methods yield practical, near-optimal metrology in realistic noisy environments. These results advance understanding of metrological optimization under correlated noise and inform design of robust quantum sensors in correlated environments.

Abstract

We investigate optimal metrological protocols for phase estimation in the presence of correlated dephasing noise, including spin-squeezed states sensing strategies as well as parallel and adaptive protocols optimized using tensor-network based numerical methods. The results are benchmarked against fundamental bounds obtained either via a latest quantum comb extension method or an optimized classical simulation method. We find that the spin-squeezed offer practically optimal performance in the regime where phase fluctuations are positively correlated, but can be outperformed by tensor-network optimized strategies for negatively correlated fluctuations.

Figures (5)

• Figure 1: Schematic diagram of a metrological channel estimation protocol with continuous temporal correlations. Continuous-time frequency estimation problem (top) is mapped to a general discrete-time phase sensing problem involving $N$ steps (middle), where each step represents sensing for a time $\delta t$ followed by a quantum control operation $V_{n}$, with $n= 1,2,\ldots, N-1$. Both the probe sensing dynamics $\Lambda_\theta^{(N)}$ as well as the entire control protocol $C$ can be represented mathematically as a quantum combs (bottom).
• Figure 2: Comparison of standard CS bound \ref{['eq:upper']}---solid lines---and refined CS bound \ref{['eq:new CS bound']}---dashed lines---to precision of SS probe state strategy with $J_y$ measurement \ref{['eq:lower']}---dotted lines---for Markovian Gaussian noise model \ref{['eq:gaussian']}. The actual QFI lies inside the shaded regions. We see that the refined bound is tight at $c=0$ and significantly tighter for weakly correlated noise, for strongly correlated noise the bounds coincide.
• Figure 3: The dashed lines indicate the asymptotic upper bounds of the quantum Fisher information per channel, $\lim_{N\rightarrow \infty}\mathcal{F}_{Q}^{(N)}/N$, as a function of the correlation parameter $c$. (a) The left panel contains the binary phase, i.e., $M = 2$ and (b) the right pannel demonstrates $M=4$. The red curve represents the upper bound obtained from the QCE method with Rouwenhorst approximation, and the blue curve is the tighter upper bound obtained by classical simulation bound \ref{['eq:new CS bound']} for the actual Gaussian noise model. The dotted green line denotes the lower bound for the approximate model with an SS state as the initial probe state at $N = 30$. The scattered points correspond to the computed values of $\mathcal{F}_{Q}^{(N)}/N$ for $N = 30$, obtained via tensor network optimization for both the parallel (MPS bond dimensions $4$ (entangled state), $1$ (product state) and adaptive (ancilla dimension $3$) strategies for the discretized model. We also plot results for $M=2$ on the right panel with grayed lines to better appreciate the effect of more refined phase discretization. Here $\sigma^2 = 1$.
• Figure 4: Fisher information per channel $F_{Q}^{(N)}/N$ with the number of channel uses $N$ with SS probe state.
• Figure 5: Asymptotic estimator variance calculated using Eq. \ref{['eq:classical estimator']} for Rouwenhorst process with $M=3$ phases state space with either $J_y$ (dashed lines) or higher moment $\vec{X}$ (solid lines) measurement. Here we use only $\vec{X}=(J_y,X_{1,2})^T$, as including more observables has negligible effect.