Quantum Speed Limit for Physical Processes

The evaluation of the minimal evolution time between two distinguishable states of a system is important for assessing the maximal speed of quantum computers and communication channels. Lower bounds for this minimal time have been proposed for unitary dynamics. Here we show that it is possible to extend this concept to nonunitary processes, using an attainable lower bound that is connected to the quantum Fisher information for time estimation. This result is used to delimit the minimal evolution time for typical noisy channels.

Figure 1

Relative difference between bound on $\mathcal{D}$ for single-qubit dephasing (14) and the exact result, as a function of the dimensionless time $\gamma \tau$, for different values of $⟨\Delta {\widehat{Z}}^2⟩$ (different initial states), with $r=8$.

Figure 2

Lower bound (solid curves) on time for separable, symmetric state with $⟨\Delta {\widehat{\mathcal{Z}}}^2⟩=1$ to reach $\mathcal{D}\simeq 94\%$ of the maximal distance ($F_B=1\%$), measured in dimensionless units $\gamma \tau$ as a function of number of qubits $N$, calculated numerically from (18). Results from exact calculations [46] are plotted for comparison (dashed curves). For the black (blue) curves, $r=8$, for the light gray (green) curves, $r=400$; the asymptotes are proportional to $1/N$ and $1/\sqrt{N}$.