Global Sensing and Its Impact for Quantum Many-Body Probes with Criticality

Quantum sensing is one of the key areas that exemplify the superiority of quantum technologies. Nonetheless, most quantum sensing protocols operate efficiently only when the unknown parameters vary within a very narrow region, i.e., local sensing. Here, we provide a systematic formulation for quantifying the precision of a probe for multiparameter global sensing when there is no prior information about the parameters. In many-body probes, in which extra tunable parameters exist, our protocol can tune the performance for harnessing the quantum criticality over arbitrarily large sensing intervals. For the single-parameter sensing, our protocol optimizes a control field such that an Ising probe is tuned to always operate around its criticality. This significantly enhances the performance of the probe even when the interval of interest is so large that the precision is bounded by the standard limit. For the multiparameter case, our protocol optimizes the control fields such that the probe operates at the most efficient point along its critical line. Finally, it is shown that even a simple magnetization measurement significantly benefits from our global sensing protocol.

Figure 1

(a) The phase diagram of an Ising model in the presence of a skew magnetic field (see Ref. [92] for details). (b) By optimizing the control field, the probe is shifted in its phase diagram to operate around the criticality, where the quantum Fisher information is maximum. (c) For multiparameter sensing, the optimization of the probe also displaces the sensing area along the critical line.

Figure 2

(a) Average uncertainty $g(B_z)$ as function of $B_z/J$ for different values of interval widths $\mathrm{\Delta }{h}_z$ and centers $h_z^{\mathrm{cen}}$. The optimal probe can always be found by tuning the control field such that $g(B_z^{*})={\min }_{B_z}[g(B_z)]$. (b) $g(B_z^{*})$ (shown by markers) and its corresponding fitting function $a{L}^{-b}+c$ (solid lines) as a function of $L$ for various choices of $\mathrm{\Delta }{h}_z$. (c) Fitting coefficients $a$ and $b$ versus the controlled field $B_z/J$. (d) $g(B_z)$ is plotted as a function of $\mathrm{\Delta }{h}_z/J$ for various choices of $B_z$.

Figure 3

(a) $g(B_x,B_z)$ as a function of control fields $(B_x,B_z)$ for the case of an almost local sensing $\mathrm{\Delta }{h}_x=\mathrm{\Delta }{h}_z=0.02\mathrm{J}$. (b) By increasing the widths to $\mathrm{\Delta }{h}_x=\mathrm{\Delta }{h}_z=0.3\mathrm{J}$, one finds a nontrivial optimal probe along its critical line. The total system is $L=16$, and centers are chosen to be $h_x^{\mathrm{cen}}=h_z^{\mathrm{cen}}=0.02\mathrm{J}$. The dashed red lines represent the critical line, and the numbers show the value of $g(B_x,B_z)$ at that point in the phase diagram. The global minimum is depicted in orange.

Figure 4

(a) $\mathrm{Tr}[{\mathcal{F}}_C(\mathit{h}|{\mathit{B}}^{*}{)}^{-1}]$ as a function of $(h_x+B_x^{*})/J$ and $(h_z+B_z^{*})/J$. (b) $\mathrm{Tr}[{\mathcal{F}}_C(\mathit{h}|{\mathit{B}}^{*c}{)}^{-1}]$ as a function of $(h_x+B_x^{*c})/J$ and $(h_z+B_z^{*c})/J$. The probe optimized with control field ${\mathit{B}}^{*c}$ provides smaller uncertainty. Other values are $L=10$, $h_x^{\mathrm{cen}}=0.5\mathrm{J}$, $h_z^{\mathrm{cen}}=0.7\mathrm{J}$, $\mathrm{\Delta }{h}_x=\mathrm{\Delta }{h}_z=0.2\mathrm{J}$, $B_x^{*}=1.39\mathrm{J}$, $B_z^{*}=-0.39\mathrm{J}$, $B_x^{*c}=-0.39\mathrm{J}$, and $B_z^{*c}=0.44\mathrm{J}$.