Entanglement and the lower bounds on the speed of quantum evolution

The concept of quantum speed limit-time (QSL) was initially introduced as a lower bound to the time interval that a given initial state ${\psi }_I$ may need so as to evolve into a state orthogonal to itself. Recently [V. Giovanetti, S. Lloyd, and L. Maccone, Phys. Rev. A 67, 052109 (2003)] this bound has been generalized to the case where ${\psi }_I$ does not necessarily evolve into an orthogonal state, but into any other ${\psi }_F$. It was pointed out that, for certain classes of states, quantum entanglement enhances the evolution “speed” of composite quantum systems. In this work we provide an exhaustive and systematic QSL study for pure and mixed states belonging to the whole 15-dimensional space of two qubits, with ${\psi }_F$ a not necessarily orthogonal state to ${\psi }_I$. We display convincing evidence for a clear correlation between concurrence, on the one hand, and the speed of quantum evolution determined by the action of a rather general local Hamiltonian, on the other one.

Figure 1

$\tau ∕T_{L.\mathit{\text{Bound}}}$ for pure states. a) Pure states that evolve to an orthogonal one and b) pure states for which $F_{\mathit{min}}∊[0.35,0.4]$. The points $P_1$ and $P_2$ represent the fastest states corresponding to each of the families determined by Eq. (17). For the range of values of $F_{\mathit{min}}$ considered here, these fast states correspond to $F_{\mathit{min}}=0.35$.

Figure 2

Concurrence for the fastest pure states compatible with a given value of $F_{\mathit{min}}$ as given by Eq. (22). The horizontal and vertical lines cross at the point corresponding to the fastest states (points $P_1$ and $P_2$) of Fig. 1b.

Figure 3

$\tau ∕T_{L.P_i}$ for the fastest pure states compatible with a given value of $F_{\mathit{min}}$ as given by Eq. (24) for $\tau ∕T_{L.P_1}$ and by Eq. (25) for $\tau ∕T_{L.P_2}$. The horizontal and vertical lines cross at two different points corresponding to the fastest states (points $P_1$ and $P_2$) of Fig. 1b. The upper crossing corresponds to $P_2$ and the lower one to $P_1$.

Figure 4

$\tau ∕T_{L.\mathit{\text{Bound}}}$ for mixed states that evolve to several fixed ($F=0.95$ and $F=0.8$) and minimum ($F_{\mathit{min}}=0.925$ and $F_{\mathit{min}}=0.825$) values of the fidelity. See text for details.

Figure 5

$\tau ∕T_{L.\mathit{\text{Bound}}}$ for MEMS with Hamiltonian $H_{\mathit{II}}$ and the corresponding average for the IH case. See text for details.