Lindblad estimation with fast and precise quantum control

TL;DR

This work establishes the ultimate precision limit for Lindblad estimation under fast and precise quantum control, showing that the optimal strategy is to rapidly projectively measure and reinitialize the state, effectively separating signal from noise within the measurement basis. By deriving general QFI bounds and proving their saturation via measure-and-reset protocols, the authors unify a wide range of applications—from stochastic waveform estimation to multi-qubit Pauli Lindblad estimation—under a single framework. They reveal how the advantage of ancilla assistance, entanglement, and strategy choice depends on the Hermiticity and geometry of the jump operators, with concrete results for single-qubit and many-qubit scenarios, including correlated noise and Pauli structures. The findings highlight both fundamental limits and practical pathways for enhancing sensitivity in stochastic sensing tasks such as gravitational-wave searches and axion detection, while also outlining the gap between idealized fast-control limits and current experimental capabilities. Overall, the paper clarifies when and how rapid, projective control can beat conventional sensing limits in noisy quantum environments.

Abstract

Enhancing precision sensors for stochastic signals using quantum techniques is a promising emerging field of physics. Estimating a weak stochastic waveform is the core task of many fundamental physics experiments including searches for stochastic gravitational waves, quantum gravity, and axionic dark matter. Simultaneously, noise spectroscopy and characterisation, e.g. estimation of various decay mechanisms in quantum devices, is relevant to a broad range of fundamental and technological applications. We consider the ultimate limit on the sensitivity of these devices for Lindblad estimation given any quantum state, fast and precise control sequence, and measurement scheme. We show that it is optimal to rapidly projectively measure and re-initialise the quantum state. We develop optimal protocols for a wide range of applications including stochastic waveform estimation, spectroscopy with qubits, and Lindblad estimation.

Figures (6)

• Figure 1: Block diagrams of different quantum metrological strategies. (a) Sequential strategy with fast and precise control where the encoding channel $\Lambda$ is queried $M$ times on a short timescale $t$ and between each query a control operation $C$ is performed. The final state after a total time of $T = M t$ is then measured. (b) Measure-and-reset strategy where the encoding channel is queried once, a measurement is performed, and then the state is re-initialised and the procedure is repeated a total of $M$ times. (c) Parallel strategy where $M$ devices are simultaneously queried and measured. (d--f) Unextended (i.e. ancilla-free) cases of the extended (i.e. noiseless ancilla--assisted) cases shown in panels (a--c), respectively.
• Figure 2: Visualisation of the Gram-Schmidt process for orthogonalising the images of the initial state $\lvert\psi\rangle$ under $N$ noise operators $\text{span}\{\hat{L}_{j,(n)}\lvert\psi\rangle\}_{j=2}^{N+1}$ and one signal operator $\hat{L}_{1,(s)}\lvert\psi\rangle$. The component of the weak signal within the noisy subspace $\text{span}\{\lvert0\rangle,\lvert j\rangle\}_{j=2}^{N+1}$ is lost leaving only the noise-free component $\lvert1\rangle$.
• Figure 3: In the commuting Hermitian case, the jump operators $\hat{L}_j$ become real vectors $l_j$ and the state $\lvert\psi\rangle$ becomes a probability distribution $p$. (a) The signal vector $l_{1,(s)}$ and (b--d) noise vectors $l_{j,(n)}$ are randomly sampled from a uniform distribution. (f) Optimal probability distribution in the noiseless case. (g--j) Optimal distribution given the signal $l_{1,(s)}$ and first 1--4 noises, respectively. For example, panel (i) shows the distribution given the signal $l_{1,(s)}$ and three noises: $l_{2,(n)}$, $l_{3,(n)}$, and $l_{4,(n)}$. The support of the optimal distribution is length $N+2$ given $N$ noise operators.
• Figure 4: (a) QFI versus $\theta_1$ for the case of $\hat{L}_{1,(s)},\hat{L}_{2,(n)}$ both non-Hermitian given in Eq. \ref{['eq:outofplane operators']} where $\phi_1=\phi_2=0$ and $\theta_2=0$ for the $\hat{L}_{2,(n)}$ in-plane case and $\theta_2=\pi/4$ in the $\hat{L}_{2,(n)}$ out-of-plane case. In the in-plane case, an unextended state is optimal and a gap exists between the noisy and noiseless QFIs. In the out-of-plane case, there is a transition as $\theta_1$ increases below which an unextended state is optimal and above which a gap emerges between the optimal unextended and extended QFIs. (b) For the unextended parallel strategy, the optimal state allowing for entanglement discontinuously changes from being separable to highly entangled when the QFI per qubit equals $2T$. (c) Entanglement entropy versus $\theta_1$ normalised to the entanglement entropy of a maximally entangled state.
• Figure 5: Bloch ball representation of the reduced density matrix of the optimal state for the case of $\hat{L}_{1,(s)},\hat{L}_{2,(n)}$ both non-Hermitian and $\hat{L}_{2,(n)}$ out-of-plane. The families of optimal extended states and unextended states are shown for different values of $\theta_1\in(0,\pi/2)$ where $\theta_2=\pi/4$ and $\varphi_1=\varphi_2=0$. States on the surface of the Bloch ball are unextended, states inside the Bloch ball are extended, and states at the origin are maximally entangled such as the Bell state. Starting at $\theta_1=0$, the optimal state is $\lvert\uparrow\rangle$ which is shown at the North pole of the Bloch ball. The optimal state remains unextended for increasing values of $\theta_1$ and rotates down the $\hat{\sigma}_x,\hat{\sigma}_z$ plane until a transition point determined by $\theta_2=\pi/4$ is reached after which the optimal state becomes extended and increasingly entangled until it reaches a maximally entangled state when $\theta_1=\pi/4$. The behaviour is then symmetric about $\theta_1=\pi/4$.

Theorems & Definitions (6)

• Theorem 1
• Lemma 1
• Claim 1
• Example 1
• Example 2
• Claim 2