Unified speed limits in classical and quantum dynamics via temporal Fisher information

Abstract

The importance of Fisher information is increasing in nonequilibrium thermodynamics, as it has played a fundamental role in trade-off relations such as thermodynamic uncertainty relations and speed limits. In this work, we investigate temporal Fisher information, which measures the temporal information content encoded in probability distributions, for both classical and quantum systems. We establish that temporal Fisher information is bounded from above by physical costs, such as entropy production in classical Langevin and Markov processes and the variance of interaction Hamiltonians in open quantum systems. Conversely, temporal Fisher information is bounded from below by statistical distances (e.g., the Bhattacharyya arccos distance), leading to classical and quantum speed limits that constrain the minimal time required for state transformations. We perform numerical simulations on two quantum dot models to validate the obtained bounds. Our work provides a unified perspective on speed limits from the point of view of temporal Fisher information in both classical and quantum dynamics.

Figures (4)

• Figure 1: Illustration of the relationship between the geodesic and the dynamics. The point $P(0)$ represents the initial position, while $P(\tau)$ represents the position after time evolution. The blue curve depicts the geodesic, the shortest path connecting $P(0)$ and $P(\tau)$. The purple dashed curve represents the trajectory of the time evolution of the system dynamics.
• Figure 2: Illustration of quantum dot models. (a) A two-level system with energy gap $\epsilon$ is coupled to an electrode with inverse temperature $\beta$, allowing electrons to be exchanged between them. (b) A double quantum dot model, where the left and right quantum dots are coupled. The left and right systems are regarded as the system and environment, respectively.
• Figure 3: Numerical simulation of a quantum dot coupled to the electrode. The dynamics is given by a two-state Markov chain with transition rates $W_{10}=W_{01}=1$. Two initial conditions are considered: (a) $P(0)=[1,0]$ and (b) $P(0)=[0.6,0.4]$. The red solid line shows the Bhattacharyya arccos distance $\mathcal{L}_P(P(0),P(\tau))$ between the initial and final states, the blue dotted line shows the upper bound derived from entropy production (cf. Eq. \ref{['eq:speedlimit_Markov']}), and the orange dotted line shows the upper bound derived from dynamical activity (cf. Eq. \ref{['eq:CSL_dynamical_activity']}).
• Figure 4: Numerical simulation of a double quantum dot model for two Coulomb repulsion cases: (a) the non-interacting case, $V_L=V_R=0$, and (b) the interacting case, $V_L=V_R=5$. The red solid line shows the distance $\widetilde{\mathcal{L}}_{D}([\rho_{S}(0)],[\rho_{S}(\tau)])$ between the initial and final states, and the blue dotted line shows the upper bound (cf. Eq. \ref{['eq:SL_general_open_interaction']}). The other parameters are $\epsilon_L = \epsilon_R = 1$ and $g=0.5$.