Overcoming Quantum Metrology Singularity through Sequential Measurements

TL;DR

The work tackles the problem of singularities in multi-parameter quantum metrology, where the classical Fisher information can be non-invertible even when the quantum Fisher information is invertible. It establishes a link that the singularity of the QFI implies CFI singularity for any measurement, and identifies a fundamental bound requiring at least $m \ge k+1$ measurement outcomes; sequential measurements provide $m=2^{n_{\mathrm{seq}}}$ outcomes, enabling fixed local measurements to overcome this limit with exponential growth of information. The authors validate the approach on a spin-chain probe and a Jaynes–Cummings light-matter system, showing that once the sequential-length condition is met, Bayesian post-processing yields convergent estimates and reduced uncertainties; they also demonstrate protocol optimizations—both in measurement basis and timing—that further enhance precision. Overall, the sequential-measurement protocol offers a minimal-control, scalable route to reliable simultaneous estimation of multiple parameters in quantum sensing, with explicit demonstrations and practical optimization guidance.

Abstract

The simultaneous estimation of multiple unknown parameters is the most general scenario in quantum sensing. Quantum multi-parameter estimation theory provides fundamental bounds on the achievable precision of simultaneous estimation. However, these bounds can become singular (no finite bound exists) in multi-parameter sensing due to parameter interdependencies, limited probe accessibility, and insufficient measurement outcomes. Here, we address the singularity issue in quantum sensing through a simple mechanism based on a sequential measurement strategy. This sensing scheme overcomes the singularity constraint and enables the simultaneous estimation of multiple parameters with a local and fixed measurement throughout the sensing protocol. This is because sequential measurements, involving consecutive steps of local measurements followed by probe evolution, inherently produce correlated measurement data that grows exponentially with the number of sequential measurements. Finally, through two different examples, namely a strongly correlated probe and a light-matter system, we demonstrate how such singularities are reflected when inferring the unknown parameters through Bayesian estimation.

Figures (8)

• Figure 1: Heisenberg probe: Trace of the inverse of the CFI matrix $\mathrm{Tr}[\mathcal{F}(\bm{B})^{-1}]$ as a function of the number of sequential measurements $n_\mathrm{seq}$ for different $k$ unknown magnetic fields $\bm{B}{=}(B_x^{(1)}{,}B_x^{(2)}{,}\ldots{,}B_x^{(k)})$. Blue bars (red cross-hatched) represent non-singular (singular) CFI matrix. We consider $N{=}10$ and $B_x^{(k)}{=}0.5J$$\forall k$.
• Figure 2: The light-matter probe: Trace of the inverse of the CFI matrix $\mathrm{Tr}[\mathcal{F}(\bm{\lambda})^{-1}]$ as a function of the number of sequential measurements $n_\mathrm{seq}$ for different $k$ unknown parameters $\bm{\lambda}{=}(\omega_1{,}\omega_2{,}J_1{,}J_2)$. Blue bars (red cross-hatched) represent non-singular (singular) CFI matrix. We consider $\bm{\lambda}{=}(0.9{,}1.1{,}0.1{,}0.2)\omega_a$ and $|\psi(0)\rangle{=}|\downarrow{,}\downarrow{,}\alpha\rangle$ with $\alpha{=}2$.
• Figure 3: Posterior distributions $P(\bm{\lambda}|\mathrm{data})$ as functions of $B_x^{(1)}$ and $B_x^{(2)}$ for different $n_\mathrm{seq}$ and $M$. Left (right) column corresponds to $n_\mathrm{seq}{=}1$ ($n_\mathrm{seq}{=}2$). The true field is $(B_x^{(1)}{,}B_x^{(2)}){=}(0.1{,}0.35)J$. The system size is $N=6$.
• Figure 4: Trace of covariance matrix $\text{Tr}(\text{Cov}[\hat{\boldsymbol{B}}])$ using a Bayesian estimator as a function of $M$. We encode $k{=}4$ unknown magnetic fields $\bm{B}{=}(B_x^{(1)}{,}B_x^{(2)}{,}B_x^{(3)}{,}B_x^{(4})$ where $B_x^{(k)}{=}0.4J$ for $1{\leq}k{\leq}4$. (a) $n_\text{seq}{=}1$, (b) $n_\text{seq}{=}2$, and (c) $n_\text{seq}{=} 3$. The system size is set to $N=6$ and $\mu = 10^4$.
• Figure S1: A schematic of the sequential measurement sensing protocol montenegro2022sequential. A quantum trajectory $\xi_j$ is constructed by starting with an initial state $\ket{\psi(0)}$, which undergoes a sequence of evolution steps governed by the operator $U(\bm{\lambda})$ to encode the unknown parameters $\bm{\lambda}$. Each evolution step is followed by a measurement, with outcomes $\uparrow_i$ or $\downarrow_i$, resulting in a sequence of $n_\mathrm{seq}$ outcomes, for instance $\xi_j = \{\uparrow_1, \downarrow_2, \dots, \downarrow_{n_\mathrm{seq}}\}$. While the evolution is assumed to be unitary, it can be generalized to non-unitary dynamics yang2024sequential. After completing $n_\mathrm{seq}$ measurements for a single trajectory, the quantum probe is reset to its initial state, and the process is repeated to generate a new trajectory.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2