Thermodynamics and the quantum speed limit in the non-Markovian regime

The quantum speed limit for open quantum systems in the non-Markovian regime is analyzed. We provide a lower bound for the time required to transform an initial state to a final state in terms of thermodynamic quantities such as the energy fluctuation, entropy production rate, and dynamical activity. Such bound was already analyzed for Markovian evolution satisfying detailed balance condition. Here we generalize this approach to deal with arbitrary evolution governed by a time-local generator. Our analysis is illustrated by three paradigmatic examples of qubit evolution: amplitude damping, pure dephasing, and the eternally non-Markovian evolution.

Figure 1

Plot of the ratios ${\tau }_{q1}/\tau$ and ${\tau }_{q2}/\tau$ with ${\gamma }_0$ for the amplitude-damping model. Unit of ${\gamma }_0$ is in $1/\mathrm{s}$. The initial states with (a) different and (b) same diagonal entries are taken from Table 1. The dashed and solid lines denote the ratios ${\tau }_{q1}/\tau$ and ${\tau }_{q2}/\tau$, respectively. Here $\lambda =1$. Calculations have been performed with logarithm base 2.

Figure 2

Plot of the ratio ${\tau }_{\mathrm{q}2}/{\tau }_{\mathrm{q}1}$ with coupling strength ${\gamma }_0$ for amplitude-damping model. Unit of ${\gamma }_0$ is in $1/\mathrm{s}$. Here $\lambda =1$.

Figure 3

Plot of the ratios ${\tau }_{q1}/\tau$ and ${\tau }_{q2}/\tau$ with parameter $k$ for pure dephasing model. $k$ is dimensionless. The initial states with (a) different and (b) same diagonal entries are taken from Table 1. Here ${\omega }_0=0.5$. Dashed and solid lines denote the ratios ${\tau }_{q1}/\tau$ and ${\tau }_{q2}/\tau$, respectively. Calculations have been performed with logarithm base 2.

Figure 4

Plot of ${\tau }_{\mathrm{q}2}$ and ${\tau }_{\mathrm{q}1}$ with $\tau$ for eternally non-Markovian evolution. Units of ${\tau }_{\mathrm{q}2}, {\tau }_{\mathrm{q}1}$, and $\tau$ are in seconds.