Connecting measurement invasiveness to optimal metrological scenarios

The connection between the Leggett-Garg inequality and optimal scenarios from the point of view of quantum metrology is investigated for perfect and noisy general dichotomic measurements. In this context, we show that the Fisher information can be expressed in terms of quantum temporal correlations. This connection allows us to associate scenarios with relatively high Fisher information to scenarios in which the Leggett-Garg inequality is violated. We thus demonstrate a qualitative and, to some extent, quantitative link between measurement invasiveness and metrological performance. Finally, we illustrate our results by using a specific model for spin systems.

Figure 1

Plots of the Fisher information $F$ (solid magenta line, scaled to the right vertical axis), the quantum Fisher information ${\mathcal{F}}_Q$ (dashed magenta line, scaled to the right vertical axis), the absolute value of the two-time correlation $C$ (dotted black line, scaled to the left vertical axis) and the absolute value of $K_{\mathrm{LG}}$ (solid black line, scaled to the left vertical axis), as a function of $\theta$, for (a) $b=1$ and (b) $b=0.99$. The LGI violation region (relative to the left vertical axis) is shaded in light gray. All the plotted quantities are dimensionless.

Figure 2

Plots of the Fisher information $F$ (solid magenta line), the quantum Fisher information ${\mathcal{F}}_Q$ (dashed magenta line), and the absolute value of the Leggett-Garg parameter $|K_{\mathrm{LG}}|$ (inset, solid black line), as a function of $b$ for (a) $\theta /\pi =0.95$ and (b) $\theta /\pi =0.34$. In the insets, the LGI violation region is shaded in light gray; note that the LGI is violated in the interval $0.94\le b\le 1$ for case (a), while it is never violated for case (b). All the plotted quantities are dimensionless.

Figure 3

Normalized Fisher information, $F/{\mathcal{F}}_Q$, versus the absolute value of the Leggett-Garg parameter $|K_{\mathrm{LG}}|$. The dashed gray lines are contours at fixed $b$ for all $\theta \in [0,\pi /2]$. Specific contours at $b=0.5$, 0.7, 0.9, 0.99, and 1.0 are highlighted as solid dark gray lines. The solid magenta line connects the points at the optimal $\theta$ maximizing the LGI violation, while the dashed magenta line connects the points at the $\theta$ maximizing instead the Fisher information. The LGI violation region is shaded in light gray. All the plotted quantities are dimensionless.