Kind of entanglement that speeds up quantum evolution

The “speed” of unitary quantum evolution was recently shown to be connected to entanglement in multipartite quantum systems. Here, we discuss a tighter version of the Mandelstam-Tamm uncertainty relation that depends on the Fisher information. The passage time is estimated by a lower bound that depends inversely proportional to the square root of the Fisher information. This leads to a better understanding of the origin of a fast quantum time-evolution of entangled states.

Figure 1

The two-qubit example that is discussed in Sec. 3b is plotted for $x=\sqrt{2}-1$. The solid curves from top to bottom are $D_{\rho }\left(t\right)$, $E_{\rho }\left(t\right)$, and $T_{\rho }\left(t\right)$. The black, dotted curve shows ${\cos }^2{\left(\Delta H\right)}_{\rho }t/\hslash$ from the Mandelstam-Tam inequality (1). The orange, dashed curve above the shading represents the improved version ${\cos }^2\sqrt{\mathcal{F}(\rho ,H)}t/2$ from Eqs. (9), (10), and (14), which are clearly tighter than Eq. (1).

Figure 2

Illustration of the example in Sec. IIIC. (a) The expectation value ${\langle \Pi \rangle }_{\rho \left(t\right)}$ with $\Pi =\left|0\rangle \langle 0\right|$ is plotted for $x=3/4$. The dashed, orange curves are the generalized lower and upper bounds from Eqs. (19) and (21), respectively. These bounds can be compared to similar bounds (black, dotted curve) on the basis of Eq. (13). (b) The same example for the survival probabilities $T_{\rho }\left(t\right)$ and $D_{\rho }\left(t\right)$, which coincide in this example. Again, we compare them to ${\cos }^2{\left(\Delta H\right)}_{\rho }t/\hslash$ from Eq. (1) (black, dotted curve) and ${\cos }^2\sqrt{\mathcal{F}(\rho ,H)}t/2$ from Eq. (14) (orange, dashed curve). In contrast to (a), the latter bound is tight for a small time interval.