Modified quantum-speed-limit bounds for open quantum dynamics in quantum channels

The minimal evolution time between two distinguishable states is of fundamental interest in quantum physics. Very recently Mirkin et al. [Phys. Rev. A 94, 052125 (2016)] argued that some of the most common quantum-speed-limit (QSL) bounds which depend on the actual evolution time do not cleave to the essence of the QSL theory as they grow indefinitely but the final state is reached at a finite time in a damped Jaynes-Cummings model. In this paper, we thoroughly study this puzzling phenomenon. We find the inconsistent estimates will happen if and only if the limit of resolution of a calculation program is achieved, through which we propose that the nature of the inconsistency is not a violation of the essence of the QSL theory but an illusion caused by the finite precision in numerical simulations. We also present a generic method to overcome the inconsistent estimates and confirm its effectiveness in both amplitude-damping and phase-damping channels. Additionally, we show special cases which may restrict the QSL bound defined by “quantumness”.

Figure 1

(a) The decoherence function and (b) the trace distance between the evolved state and its final stationary state in an amplitude-damping channel. ${\tau }_{\mathrm{qsl}}^{\mathrm{av}}$ [the blue (middle) line], ${\tau }_{\mathrm{qsl}}^{\mathrm{op}}$ [the red (lowest) line], and ${\tau }_{\mathrm{qsl}}^{\mathrm{quant}}$ [the green (highest) line] as a function of the scaled time $\lambda t$ for (c) a Markovian environment with ${\gamma }_0/\lambda =0.4$ and (d) a non-Markovian environment with ${\gamma }_0/\lambda =20$. The blue (upper) curve in (a) [(b)] means the same Markovian environment and the red (lower) one means the same non-Markovian environment.

Figure 2

(a) The decoherence function and (b) the trace distance between the evolved state and its final stationary state in a phase-damping channel ($s=1$). The evolution of (c) ${\tau }_{\mathrm{qsl}}^{\mathrm{av}}$ and (d) ${\tau }_{\mathrm{qsl}}^{\mathrm{op}}$ in the same channel.

Figure 3

The intrinsic reason which causes the inconsistent estimates.

Figure 4

The three modified QSL times as a function of the scaled time $\lambda t$ for (a) a Markovian environment with ${\gamma }_0/\lambda =0.4$ and (b) a non-Markovian environment with ${\gamma }_0/\lambda =20$ in an amplitude-damping channel. The tightness of the three bounds for (c) a Markovian environment and (d) a non-Markovian environment with the same parameters. The blue (middle) line represents ${\tau }_{\mathrm{qsl}}^{\mathrm{av}}$; the red (lowest) line, ${\tau }_{\mathrm{qsl}}^{\mathrm{op}}$; the green (highest) line, ${\tau }_{\mathrm{qsl}}^{\mathrm{quant}}$.

Figure 5

(a) ${\tau }_{\mathrm{qsl}}^{\mathrm{av}}$, (b) ${\tau }_{\mathrm{qsl}}^{\mathrm{op}}$, and (c,d) their tightness as a function of the scaled time ${\omega }_{c}t$ in a phase-damping channel with $s=1$.