Dynamical role of quantum signatures in quantum thermometry

Nonequilibrium quantum thermometry is a central topic for both fundamental and technological purposes. It shows advantages against its equilibrium counterpart by reducing the uncertainty associated to the temperature of the bath, and a negligible invasiveness. Here, we analyze how quantum features influence the dynamical speed of single- and two-qubit nonequilibrium thermometers interacting with a bosonic thermal bath. Our investigations exploit Riemannian geometric tools to show that the Riemannian speed of the nonequilibrium thermometer is only indirectly influenced by its quantum features, thus highlighting how the quantum fingerprint seems to be concealed by the classical aspects of thermalization dynamics.

Figure 1

Behavior of the $G_{\lambda \lambda }^{f_M}$ figure of merit as a function of the thermometer-bath time of interaction $\lambda$ for different temperatures. Panels (a)–(d) show the dispersion of pure states for $\alpha =0$ (a), $\alpha =-1/3$ (b), $\alpha =-2/3$ (c), $\alpha =-1$ (d), while panels (e) and (f) show the dispersion of mixed states for $\alpha =-1/2$ (e), and $\alpha =-1$ (f). (a)–(d) Blue curve with medium dashing: ground state; black dotted curve: excited state; green solid curve: maximally coherent state $\left[x\right(0)=1,z(0)=0]$; magenta dashed-dot curve: $x\left(0\right)=1/\sqrt{2},z\left(0\right)=1/\sqrt{2}$; orange curve with large dashing: $x\left(0\right)=1/\sqrt{2},z\left(0\right)=-1/\sqrt{2}$. (e),(f) Blue curve with medium dashing: ground state; black dotted curve: excited state; green solid curve: maximally mixed state $\left[x\right(0)=0,z(0)=0]$; magenta dashed-dot: $x\left(0\right)=1/2,z\left(0\right)=1/2$; orange curve with large dashing: $x\left(0\right)=1/2,z\left(0\right)=-1/2$.

Figure 2

Behavior of the $\sqrt{g_{\lambda \lambda }^{f_M}}$ ($\lambda =0$) quantity for single-qubit states associated to different temperatures [(a) $\alpha =0$, (b) $\alpha =0.3$, (c) $\alpha =1$] vs the purity of the state $r$. Blue curve (with triangles): “classical” states $x\left(0\right)=0$; red solid curve: states with $x\left(0\right)=1$. The points are generated from an even random distribution according to the Haar metric. States are differentiated according to their energies: $Z>0$ (yellow points) and $Z\le 0$ (grey points).

Figure 3

Behavior of $g_{\lambda \lambda }^{f_M}$ for two-qubit states systems with thermalizing maps acting on either a single (a)–(c) or both (d)–(f) subsystems vs the concurrence $C$ of the same states. The panels refer to different thermalization temperatures: in detail, (a),(d) $\alpha =0$, (b),(e) $\alpha =-0.5$, (c),(f) $\alpha =-1$. Grey points represent generic bipartite states, red points represent $X$ states. All points have been generated from an even random distribution according to the Haar metric.

Figure 4

Behavior of $\mathrm{\Delta }G$ for two-qubit systems with thermalizing maps acting on both subsystems vs the concurrence $C$ of the same states. The panels refer to different thermalization temperatures: (a) $\alpha =0$, (b) $\alpha =-0.5$, (c) $\alpha =-1$. In the latter, our numerical analysis has identified states with $\mathrm{\Delta }G\le 0$. Grey points represent generic bipartite states, red points represent $X$ states. All points have been generated from an even random distribution according to the Haar metric.