Characterization of qubit chains by Feynman probes

We address the characterization of qubit chains and assess the performances of local measurements compared to those provided by Feynman probes, i.e., nonlocal measurements realized by coupling a single-qubit register to the chain. We show that local measurements are suitable to estimate small values of the coupling and that a Bayesian strategy may be successfully exploited to achieve optimal precision. For larger values of the coupling Bayesian local strategies do not lead to a consistent estimate. In this regime, Feynman probes may be exploited to build a consistent Bayesian estimator that saturates the Cramér-Rao bound, thus providing an effective characterization of the chain. Finally, we show that ultimate bounds to precision, i.e., saturation of the quantum Cramér-Rao bound, may be achieved by a two-step scheme employing Feynman probes followed by local measurements.

Figure 1

Fisher information for the local measurement $F_m^L(\lambda ;x_0)$ (solid lines) and for the Feynman probe $F_m^P(\lambda ;x_0)$ (dashed lines) as a function of $\lambda =\nu t$. The data refer to a chain of $s=10$ spins for different values of the measured or plugging site: $m=1$ (green lines), $m=2$ (orange lines), and $m=3$ (blue lines). (a) Initial condition set to $x_0=1$; (b) $x_0=2$. In both frames the dot-dashed black line represents the quantum Fisher information.

Figure 2

A schematic representation of the Feynman quantum computer.

Figure 3

$P_{\lambda }^L\left(m\right|x_0)$ (top plot) and $P_{\lambda }(\uparrow |m,x_0)$ (bottom plot) as defined in Eqs. (25) and (31), respectively, as a function of the parameter $\lambda$. The excitation is initially localized in the first site $x_0=1$ of a chain of length $s=10$. Three different values of $m$ are considered: $m=1$ (solid black line), $m=2$ (dashed red line), and $m=3$ (blue dotted line).

Figure 4

(top) Success probability in a single trial for LM and FP as defined in Eqs. (25) and (31), respectively; (bottom) Bayesian probability $P_B\left(\lambda \right|\mathrm{\Omega })$ defined in Eq. (33). The plots refer to simulations of (left column) LM on site $m=1$, (middle column) LM on site $m=2$, and (right column) FP coupled to site $m=2$. In the bottom plots, we use the true values ${\lambda }_T=1$ (solid black line), 2 (dashed red line), and 3 (dotted green line). The vertical lines are a guide for the eye marking the values of ${\lambda }_T$.

Figure 5

Variance for the LM (blue squares), FP (red circles), and combined FP+LM (black triangles) estimators as a function of the number $M$ of repeated simulated measurements for ${\lambda }_T=3$. Each plot refers to a different value of the measured (or plugging) site $m$. The blue dashed line and the red dot-dashed one highlight the limit imposed by the CR bound for LM and FP measurements, respectively. The green shaded area delimits the region forbidden by the quantum CR bound.