Unified and computable approach to optimal strategies for multiparameter estimation

Abstract

Precise estimation of physical parameters underpins both scientific discovery and technological development. A central goal of quantum metrology and sensing is to exploit quantum resources like entanglement to devise optimal strategies for estimating physical parameters as precisely as possible. While substantial progress has been made in single-parameter quantum metrology, the multiparameter scenario remains significantly more challenging due to the issue of parameter incompatibility. In this work, we present a unified and computable approach for the simultaneous estimation of multiple parameters that attains the ultimate precision permitted by quantum mechanics. The core of our approach is to integrate the quantum tester formalism into the recently proposed tight Cramér-Rao type bound. This formulation enables us to figure out the highest achievable precision via upper and lower bounds that are computable via semidefinite programs. More importantly, within this formulation, diverse quantum resources, including entanglement, coherence, quantum control, and indefinite causal order, are treated on equal footing and systematically optimized for the purpose of achieving the ultimate precision in multiparameter estimation. As a result, our approach is applicable to various metrological strategies both in the presence and absence of noise. To demonstrate its utility, we revisit three-dimensional magnetic-field estimation, uncovering the strengths and limitations of existing analytical results and further establishing a strict hierarchy among different types of strategies.

Figures (7)

• Figure 1: Schematic of quantum testers with $N=2$. The dashed boxes in the left panels are concrete realizations of estimation strategies: (a) parallel strategies, (b) sequential strategies, (c) causal superposition strategies, and (d) general indefinite-causal-order strategies. Here $\rho$ is the input state, $\mathcal{E}_{\bm{\theta}}$ represents the parameter-encoding channel, and $U_i$ denotes the unitary control operations. In each of these strategies, a final measurement $\{\Pi_x\}_x$ is performed on the output state. The black and red curves in (c) represent two different causal orders. The right panels display the associated quantum testers.
• Figure 2: Estimation error as a function of $t$ for $\theta_1=\theta_2=1/2$ and $\theta_3=\sqrt{2}/2$ with $N=2$. The blue curve depicts the estimation error given by \ref{['eq_ana_par']}. The yellow diamonds represent the estimation error associated with the heuristic states proposed in Ref. HZX20. The red circles are the upper bounds on the optimal estimation error, obtained from our approach by randomly generating $m=125$ real unit vectors $\left\{ |w_x\rangle \right\} _{x=1}^{m}$.
• Figure 3: Illustration of the coincidence between the analytical bound in \ref{['eq_ana_lower_par']} and the upper/lower bounds obtained from our approach. The yellow diamonds represent the precision associated with the heuristic states proposed in Ref. HZX20. The blue line corresponds to the analytical bound in \ref{['eq_ana_lower_par']}. The red circles and green crosses denote the upper and lower bounds obtained from our approach with $m=125$ and $n=2$, respectively. Here, we set $\theta_1=\theta_2=1/2$ and $\theta_3=\sqrt{2}/2$ with $N=2$.
• Figure 4: Differences between the analytical lower bound in \ref{['eq_ana_lower_par']} and the upper bound $B_+^{(i)}$ obtained from our method for different parameter configurations satisfying $\left\| \boldsymbol{\theta } \right\| =1$. Each point corresponds to a specific choice of $\theta_1$ and $\theta_2$, with $\theta_3$ determined through the relation $\theta_3 = \sqrt{1 - \theta_1^2 - \theta_2^2}$. The color denotes the magnitude of the difference. The white region represents parameter configurations that do not satisfy the constraint $\left\| \boldsymbol{\theta } \right\| =1$. Here, we set $t=3,m=700$, and $N=2$.
• Figure 5: Lower bound $B_-^{(i)}$ for parallel strategies and upper bound $B_+^{(ii)}$ for sequential strategies as functions of the noise strength $\gamma$. Here $B_-^{(i)}$ and $B_+^{(ii)}$ are obtained from our approach by setting $n=2$ and $m=1500$, respectively. The parameters are chosen as $\theta _1=\theta _2=1/2$ and $\theta _3=\sqrt{2}/2$ with $t=0.1$ and $N=2$.