Efficient adaptive control strategy for multi-parameter quantum metrology in two-dimensional systems

Abstract

Quantum metrology leverages quantum resources such as entanglement and squeezing to enhance parameter estimation precision beyond classical limits. While optimal quantum control strategies can assist to reach or even surpass the Heisenberg limit, their practical implementation often requires the knowledge of the parameters to be estimated, necessitating adaptive control methods with feedback. Such adaptive control methods have been considered in single-parameter quantum metrology, but not much in multi-parameter quantum metrology so far. In this work, we bridge this gap by proposing an efficient adaptive control strategy for multi-parameter quantum metrology in two-dimensional systems. By eliminating the trade-offs among optimal measurements, initial states, and control Hamiltonians through a system extension scheme, we derive an explicit relation between the estimator variance and evolution time. Through a reparameterization technique, the optimization of evolution times in adaptive iterations are obtained, and a recursive relation is established to characterize the precision improvement across the iterations. The proposed strategy achieves the optimal performance up to an overall factor of constant order with only a few iterations and demonstrates strong robustness against deviations in the errors of control parameters at individual iterations. Further analysis shows the effectiveness of this strategy for Hamiltonians with arbitrary parameter dependence. This work provides a practical approach for multi-parameter quantum metrology with adaptive Hamiltonian control in realistic scenarios.

Figures (5)

• Figure 1: System extension scheme. An ancilla with the same dimension as the probe is introduced, with the unitary evolution $U_{\boldsymbol{\alpha}}$ acts only on the probe. The initial state can be any quantum state of the joint system, and measurements are performed on the joint system.
• Figure 2: Relation between estimation variance and evolution time. The estimation variance of $\widehat{\alpha}_{1}$ exhibits two characteristic time scalings. Fig. (a) shows the time scaling of variance for $\xi_{1}\neq0$, depicted by the orange curve. For $t\ll1/\left|\delta E\right|$, the variance decays quadratically with time. As $t$ increases, the variance oscillates with time and diverges at integer multiples of $2\pi/\left|\delta E\right|$, with the asymptotes plotted by the green dashed lines. The lower envelope of the variance, depicted by the green solid line, decays and rapidly converges to $\xi_{1}/n\xi_{3}$. Fig. (b) illustrates the time scaling of estimation variance for $\xi_{1}=0$, where the variance decays quadratically with time and reaches the Heisenberg scaling.
• Figure 3: Relations between different physical quantities. Arrows schematically denote the relations between different physical quantities occurred in the proposed optimal evolution time scheme, pointing from one quantity to the derived quantity. The upper section, linked by dashed arrows, depicts the no-deviation cases with the errors of all control parameters averaged. The lower section, linked by solid arrows, depicts the practical cases where the errors of the control parameters have random deviation from their average values, manifested by the deviation factor $D_{k}$ for the $k$-th iteration.
• Figure 4: Robustness of a single iteration against deviation in the errors of control parameters. Fig. (a) illustrates the effect of the deviation factor on the estimation precision. The estimation precision decreases as $\dev$ increases. When $\dev$ is sufficiently low, the real estimation precision approaches the optimal precision with precise control parameters, so the ratio $R_{k}$ between the real estimation precision to the estimation precision with average errors in the control parameters drops below $1$, as shown by the left panel in this figure. In the region indicated by the red dashed line in this figure, the estimation precision is almost insensitive to the random deviation of the errors of control parameters. Fig. (b) shows the probability density function of $\dev$, which suggests $\dev$ lies in an interval where the estimation precision is close to that with average errors in the control parameters with a high probability, indicating strong robustness of the optimal evolution time scheme against the deviation in the errors of control parameters.
• Figure 5: Robustness of the optimal evolution time scheme. These figures illustrate the effect of the deviation in $\deltaesk[k]$ on the estimation precision for a total of two (orange curve), three (blue curve), and four (green curve) iterations. The orange, blue, and green dashed lines plot the probabilities that the precision with the deviation in $\deltaesk[k]$ surpasses that without deviation for different iteration numbers. They all exceed $50\%$ and increase with the number of iterations, implying that the real estimation precision actually benefits from the deviation in $\deltaesk[k]$ and becomes better than the expected estimation precision.