Quantum speed limits based on Jensen-Shannon and Jeffreys divergences for general physical processes

TL;DR

This work derives quantum speed limits (QSLs) for finite-dimensional quantum systems undergoing general physical processes by exploiting the square roots of the Jensen-Shannon divergence (QJSD) and the quantum Jeffreys divergence (QJPD). The authors obtain time-integrated bounds on these entropic distinguishability measures, introducing cost functions that depend on extreme eigenvalues of the initial and instantaneous states, and define QSL times that generalize Mandelstam–Tamm-type limits. They specialize to unitary dynamics and to three common open-system channels (depolarizing, phase damping, generalized amplitude damping), providing closed-form expressions and numerical analyses that reveal how the bounds depend on state purity, coherence, and channel parameters. The results offer a scalable, spectrally parsimious framework for QSLs with potential applications in quantum thermodynamics, entropic uncertainty relations, and the study of complexity in quantum many-body dynamics, and suggest routes for experimental validation.

Abstract

We discuss quantum speed limits (QSLs) for finite-dimensional quantum systems undergoing general physical processes. These QSLs were obtained using two families of entropic measures, namely the square root of the Jensen-Shannon divergence, which in turn defines a faithful distance of quantum states, and the square root of the quantum Jeffreys divergence. The results apply to both closed and open quantum systems, and are evaluated in terms of the Schatten speed of the evolved state, as well as cost functions that depend on the smallest and largest eigenvalues of both initial and instantaneous states of the quantum system. To illustrate our findings, we focus on the unitary and nonunitary dynamics of mixed single-qubit states. In the first case, we obtain speed limits $\textit{à la}$ Mandelstam-Tamm that are inversely proportional to the variance of the Hamiltonian driving the evolution. In the second case, we set the nonunitary dynamics to be described by the noisy operations: depolarizing channel, phase damping channel, and generalized amplitude damping channel. We provide analytical results for the two entropic measures, present numerical simulations to support our results on the speed limits, comment on the tightness of the bounds, and provide a comparison with previous QSLs. Our results may find applications in the study of quantum thermodynamics, entropic uncertainty relations, and also complexity of many-body systems.

Figures (7)

• Figure 1: (Color online) Overview of the role played by quantum Jeffreys pseudo-distance (QJPD), ${D_J}({\rho},{\varrho})$ [see Eq. \ref{['eq:0000003']}], and quantum Jensen-Shannon distance (QJSD), ${D_{JS}}({\rho},{\varrho})$ [see Eq. \ref{['eq:0000006']}]. QJPD (blue solid lines) and QJSD (solid red lines) map density matrices of the convex manifold of quantum states $\mathcal{S}$ (gray smooth surface) to nonnegative numbers on the real line $\mathbb{R}$ (black solid line), i.e., ${D_{J,JS}}: ({\rho},{\varrho})\in\mathcal{S} \mapsto \mathbb{R}$. Noteworty, QJSD takes into account the "average state" $\varpi$. QJSD is a faithful metric of quantum states, thus linking the distinguishability between $\rho$ and $\varrho$ to the length of a given path that connects these states (black dash-dotted line). QJPD, however, is not a true distance, but serves as a useful entropic measure.
• Figure 2: (Color online) Density plot of the entropic QSLs ${\tau}_{J,JS}^{\text{QSL}}$ [see Eqs. \ref{['eq:0000035']} and \ref{['eq:0000037']}, respectively], and the normalized relative errors $\widetilde{\delta}_{J,JS}({\tau})$ [see Eq. \ref{['eq:0000033']}], as a function of the dimensionless parameters $\Gamma\tau$ and $r \in [0,1)$, for initial single-qubit states evolving under the depo-la-ri-zing channel [see Sec. \ref{['sec:00000000005B1']}]. Here, we consider $\theta \in [0,\pi]$ and $\phi \in [0,2\pi)$.
• Figure 3: (Color online) Density plot of the entropic QSLs ${\tau}_{J,JS}^{\text{QSL}}$ [see Eqs. \ref{['eq:0000026']} and \ref{['eq:0000027']}, respectively], and the normalized relative errors $\widetilde{\delta}_{J,JS}({\tau})$ [see Eq. \ref{['eq:0000033']}], as a function of the dimensionless parameters $\Gamma\tau$ and $r \in [0,1)$, for initial single-qubit states subject to the phase damping quantum channel [see Sec. \ref{['sec:00000000005B2']}]. Here, we set $\theta = \pi/2$, for all $\phi \in [0,2\pi)$.
• Figure 4: (Color online) Density plot of the entropic QSL ${\tau}_J^{\text{QSL}}$ [see Eq. \ref{['eq:0000026']}], as a function of the dimensionless parameters $\Gamma\tau$ and $r \in [0,1)$, for initial single-qubit states subject to the generalized amplitude damping channel [see Sec. \ref{['sec:00000000005B3']}]. Here, we set $\alpha = \{0,0.1,1\}$ and $\theta = \{0,\pi/4,\pi/2$}, for all $\phi \in [0,2\pi)$.
• Figure 5: (Color online) Density plot of the normalized relative error ${\widetilde{\delta}_J}({\tau})$ [see Eq. \ref{['eq:0000033']}], as a function of the dimensionless parameters $\Gamma\tau$ and $r \in [0,1)$, for initial single-qubit states subject to the generalized amplitude damping channel [see Sec. \ref{['sec:00000000005B3']}]. Here, we set $\alpha = \{0,0.1,1\}$ and $\theta = \{0,\pi/4,\pi/2$}, for all $\phi \in [0,2\pi)$.