Quantum Speed Limit Quantified by the Changing Rate of Phase

The quantum speed limit is important in determining the minimum evolution time of a quantum system, and thus is essential for quantum community. In this Letter, we derive a novel unified quantum speed limit bound for Hermitian and non-Hermitian quantum systems. The bound is quantified by the changing rate of phase of the quantum system, which represents the transmission mode of the quantum states over their evolution. The bound leads to further insights beyond the previous bounds on concrete evolution modes of the quantum system, such as horizontal or parallel transition or horizontal joining of the two quantum states in Hilbert space. The bound is linked to the feasibility of the evolutions of the state vectors, and provides a tighter upper bound. In addition, the generalized Margolus-Levitin bound is discussed.

Figure 1

Sketch of quantum system evolution in the Hilbert space $\mathscr{H}$ with the projection of its evolution paths in base manifold $ℬ$.

Figure 2

$ϵ$ as a function $p_0$ [panel (a)] and $\beta /k$ [panel (b)] in the initial state. The solid red lines denote our results from Eq. (15) [panel (a)] and Eq. (18) [panel (b)]. The dashed blue lines denote the results from Eq. (25) and Eq. (31) of Ref. [47]. The other parameters for panel (a) are $E_1=18$, $E_2=40$ and those for panel (b) are $k=40$, $E_1=18$, $\beta =(\alpha -1)/4$.