Extractable information capacity in sequential measurements metrology

The conventional formulation of quantum sensing is based on the assumption that the probe is reset to its initial state after each measurement. In a very distinct approach, one can also pursue a sequential measurement scheme in which time-consuming resetting is avoided. In this situation, every measurement outcome effectively comes from a different probe, yet is correlated with other data samples. Finding a proper description for the precision of sequential measurement sensing is very challenging as it requires the analysis of long sequences with exponentially large outcomes. Here, we develop a recursive formula and an efficient Monte Carlo approach to calculate the Fisher information, as a figure of merit for sensing precision, for arbitrary lengths of sequential measurements. Our results show that the value of the Fisher information initially increases nonlinearly with the number of measurements and then asymptotically saturates to a linear scaling. This transition, which fundamentally constrains the extractable information about the parameter of interest, is directly linked to the finite memory of the probe when it undergoes multiple sequential measurements. Based on these findings, we establish a figure of merit to determine the optimal measurement sequence length and exemplify our results in three different physical systems.

Figure 1

We compare final states ${\varphi }^{\left(n_{\mathrm{seq}}\right)}$ and ${\theta }^{\left(n_{\mathrm{seq}}\right)}$ resulting from two distinct random initial states ${\rho }_0={\varphi }_0$ and ${\rho }_0={\theta }_0$ following the same quantum trajectory. (a) Fidelity ${\langle \mathcal{F}\rangle }_{\mathrm{traj}}:={\langle \mathcal{F}({\varphi }^{\left(n_{\mathrm{seq}}\right)},{\theta }^{\left(n_{\mathrm{seq}}\right)})\rangle }_{\mathrm{traj}}$ as a function of $n_{\mathrm{seq}}$ for several unitary operators. (b) Ratio between the largest and second-largest singular values $s_1$ and $s_2$ as a function of $n_{\mathrm{seq}}$ for several unitary operators. We consider a system size of $N=6$.

Figure 2

(a) FI increment $\mathrm{\Delta }{\mathcal{F}}_B^{\left(n_{\mathrm{seq}}\right)}$ as a function of $n_{\mathrm{seq}}$ for various $B$. (b) Averaged fidelity ${\langle \mathcal{F}\rangle }_{\mathrm{traj}}$ as a function of $n_{\mathrm{seq}}$ for various $B$. Dashed lines divide the curves into a nontrivial (memory) and constant (memoryless) dependence on $n_{\mathrm{seq}}$. (c) ${\mathcal{F}}_B^{\left(n_{\mathrm{seq}}\right)}$ as a function of $n_{\mathrm{seq}}$ for several values of $B$.

Figure 3

JC model case: (a) $\mathrm{\Delta }{\mathcal{F}}_{\mathrm{\Omega }}^{\left(n_{\mathrm{seq}}\right)}$ as a function of $n_{\mathrm{seq}}$ for various $\mathrm{\Omega }$; (b) ${\mathcal{F}}_{\mathrm{\Omega }}^{\left(n_{\mathrm{seq}}\right)}$ as a function of $n_{\mathrm{seq}}$ for several $\mathrm{\Omega }$. Nonunitary case: (c) $\mathrm{\Delta }{\mathcal{F}}_{\kappa }^{\left(n_{\mathrm{seq}}\right)}$ as a function of $n_{\mathrm{seq}}$ for different values of $\kappa$; (d) ${\mathcal{F}}_{\kappa }^{\left(n_{\mathrm{seq}}\right)}$ as a function of $n_{\mathrm{seq}}$ for various $\kappa$.

Figure 4

Relative error between the Monte Carlo FI ${\mathcal{F}}^{\text{MC}}$ and the exact FI ${\mathcal{F}}^{\text{exact}}$ as a function of the number of Monte Carlo trajectories for the (a) spin chain, (b) light-matter interaction, and (c) nonunitary dynamics examples. The relative error decreases to around 1% as the number of trajectories increases for all three cases. Monte Carlo FI ${\mathcal{F}}^{\text{MC}}$ as a function of the number of Monte Carlo trajectories for a fixed sequential measurement number $n_{\mathrm{seq}}=100$, for the (d) spin chain, (e) light-matter interaction, and (f) nonunitary dynamics cases. A clear convergence of ${\mathcal{F}}^{\text{MC}}$ towards a stationary value emerges as the number of Monte Carlo trajectories increases. Exact FI and Monte Carlo FI as a function of the number of sequential measurements $n_{\mathrm{seq}}$ for various parameter values, for the (g) spin chain, (h) light-matter interaction, and (i) nonunitary dynamics scenarios. The curves closely overlap, demonstrating the accuracy of our procedure.

Figure 5

(a) Gain ${\mathcal{F}}_B^{\left(n_{\mathrm{seq}}\right)}/n_{\mathrm{seq}}$ as a function of $n_{\mathrm{seq}}$ for different values of $B$. (b) $n_{\mathrm{seq}}^{*}$ as a function of $B$.

Figure 6

Inverse of the Monte Carlo FI ${\mathcal{F}}^{\text{MC}}$ as a function of $n_{\mathrm{seq}}$ for several resetting times $t_{\mathrm{reset}}$: (a) spin chain case, (b) Jaynes-Cummings case, and (c) nonunitary dynamics case. We consider the measurement time as $t_{\mathrm{meas}}=10\tau$, where $\tau$ is the free-evolution time between measurements. In particular, we chose $J\tau =4$ for the spin chain case, $\omega \tau =2\pi$ for the Jaynes-Cummings case, and $J\tau =1$ for the nonunitary case, respectively. Other parameters are shown in the figure.

Figure 7

(a)–(d) The occupation probability of the field as a function of its number state for different $n_{\mathrm{seq}}$. (e)–(h) The Wigner function for the field state for different $n_{\mathrm{seq}}$ instances. This figure demonstrates that the field, under Jaynes-Cummings interaction, is likely to collapse into a single number state after the qubit is measured repeatedly.