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A unified fidelity-based framework for quantum speed limits in open quantum systems

Tristán M. Osán, Yanet Álvarez, Mariela Portesi, Pedro Walter Lamberti

TL;DR

This work addresses how quantum speed limits (QSLs) for open quantum systems can be derived from a unified, fidelity-based geometric framework. By proving that ML- and MT-type QSLs are invariant under any monotone differentiable reparametrization of a chosen fidelity, the authors show that all such QSLs depend only on the selected fidelity measure, not on the specific metric form, thereby unifying fidelity- and metric-based formulations. They systematically derive ML/MT bounds for multiple fidelities—Bures, super-, operator-, and alternative fidelities—and demonstrate how existing QSLs in the literature arise as special cases, including a damped Jaynes–Cummings model as a concrete testbed. Numerical results reveal regime-dependent behavior: in Markovian dynamics a plateau emerges where the QSL matches the evolution time for certain bounds, while non-Markovian backflow can tighten or modify these bounds. Overall, the paper provides a coherent, general foundation for fidelity-based QSLs in open systems and clarifies that improvements must come from new fidelity definitions rather than reparametrizations of existing ones.

Abstract

In this work, we investigate the role of functionals of generalized fidelity measures in deriving quantum speed limits (QSLs) within a geometric approach. We establish a general theoretical framework and show that, once a specific generalized fidelity is selected, the resulting Margolus-Levitin and Mandelstam-Tamm QSLs for both unitary and nonunitary (Lindblad-type) dynamics depend solely on the chosen fidelity measure. We prove that any monotone, differentiable reparametrization of the chosen fidelity yields exactly the same Margolus-Levitin and Mandelstam-Tamm-type QSL, rendering the QSLs invariant under such transformations and thereby unifying fidelity- and metric-induced geometric formulations. This result highlights the limitations of improving QSLs through functional transformations of fidelity and indicates that genuine improvements must arise from alternative fidelity definitions. We further show that several QSLs reported in the literature are encompassed by our framework and discuss possible extensions based on other generalized fidelity measures beyond those explicitly analyzed. In addition, we consider the damped Jaynes-Cummings model as a concrete physical setting to explore the behavior of the QSLs discussed in this work and analyze their regime-dependent features.

A unified fidelity-based framework for quantum speed limits in open quantum systems

TL;DR

This work addresses how quantum speed limits (QSLs) for open quantum systems can be derived from a unified, fidelity-based geometric framework. By proving that ML- and MT-type QSLs are invariant under any monotone differentiable reparametrization of a chosen fidelity, the authors show that all such QSLs depend only on the selected fidelity measure, not on the specific metric form, thereby unifying fidelity- and metric-based formulations. They systematically derive ML/MT bounds for multiple fidelities—Bures, super-, operator-, and alternative fidelities—and demonstrate how existing QSLs in the literature arise as special cases, including a damped Jaynes–Cummings model as a concrete testbed. Numerical results reveal regime-dependent behavior: in Markovian dynamics a plateau emerges where the QSL matches the evolution time for certain bounds, while non-Markovian backflow can tighten or modify these bounds. Overall, the paper provides a coherent, general foundation for fidelity-based QSLs in open systems and clarifies that improvements must come from new fidelity definitions rather than reparametrizations of existing ones.

Abstract

In this work, we investigate the role of functionals of generalized fidelity measures in deriving quantum speed limits (QSLs) within a geometric approach. We establish a general theoretical framework and show that, once a specific generalized fidelity is selected, the resulting Margolus-Levitin and Mandelstam-Tamm QSLs for both unitary and nonunitary (Lindblad-type) dynamics depend solely on the chosen fidelity measure. We prove that any monotone, differentiable reparametrization of the chosen fidelity yields exactly the same Margolus-Levitin and Mandelstam-Tamm-type QSL, rendering the QSLs invariant under such transformations and thereby unifying fidelity- and metric-induced geometric formulations. This result highlights the limitations of improving QSLs through functional transformations of fidelity and indicates that genuine improvements must arise from alternative fidelity definitions. We further show that several QSLs reported in the literature are encompassed by our framework and discuss possible extensions based on other generalized fidelity measures beyond those explicitly analyzed. In addition, we consider the damped Jaynes-Cummings model as a concrete physical setting to explore the behavior of the QSLs discussed in this work and analyze their regime-dependent features.

Paper Structure

This paper contains 18 sections, 7 theorems, 154 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Bures fidelity
  3. Alternative fidelity measures
  4. Analytical results
  5. Quantum speed limits in a physical model
  6. Bures fidelity
  7. Super-fidelity
  8. Operator fidelity
  9. Alternative fidelity
  10. Analysis and discussion
  11. Analytical results
  12. Numerical results
  13. Bures fidelity
  14. Super fidelity
  15. Operator fidelity
  16. ...and 3 more sections

Key Result

Proposition 1

Let $D(x)$ be a bounded function defined on the domain $[0,1]$, assumed to be continuous, monotonically decreasing, and differentiable. Furthermore, suppose that its inverse function $D^{-1}(y)$ exists on the image of $D$ and is also differentiable. Consider the following functional composition: where $\mathcal{F}(\rho_0, \rho_t)\equiv \mathcal{F}(t)$ denotes a suitable measure of fidelity betwee

Figures (8)

  • Figure 1: ML and MT QSLs $\tau_{\text{\tiny QSL}}^B$ given by Eq. \ref{['eq:BuresQSL']}, as a function of $\gamma_0/\omega_0$ for the excited initial state given by $\rho_0 = |1\rangle \langle 1|$. The evolution time was set as $\tau=1$, $\lambda = 20$, and $\omega_0=1$. The horizontal dashed line represents the actual driving time $\tau=1$.
  • Figure 2: ML and MT QSLs $\tau_{\text{\tiny QSL}}^B$ given by Eq. \ref{['eq:BuresQSL']}, as a function of $\gamma_0/\omega_0$ for the excited initial state given by $\rho_0 = |1\rangle \langle 1|$. The evolution time was set as $\tau=1$, $\lambda = 50$, and $\omega_0=1$. The horizontal dashed line represents the actual driving time $\tau=1$.
  • Figure 3: ML-type QSL $\tau_{\text{\tiny QSL}}^s$ given by Eq. \ref{['eq:superfidelMLQSL']}, as a function of $\gamma_0/\omega_0$ for different initial states of the form $\rho_0=(p/2)\mathbb I+(1-p)\,|1\rangle\!\langle 1|$. The evolution time was set as $\tau=1$, $\lambda = 20$, and $\omega_0=1$. The horizontal dashed line represents the actual driving time $\tau=1$.
  • Figure 4: ML-type QSL $\tau_{\text{\tiny QSL}}^s$ given by Eq. \ref{['eq:superfidelMLQSL']}, as a function of $\gamma_0/\omega_0$ for different initial states of the form $\rho_0=(p/2)\mathbb I+(1-p)\,|1\rangle\!\langle 1|$. The evolution time was set as $\tau=1$, $\lambda = 50$, and $\omega_0=1$. The horizontal dashed line represents the actual driving time $\tau=1$.
  • Figure 5: MT-type QSL $\tau_{\text{\tiny QSL}}^o$ given by Eq.\ref{['eq:operfidelMTQSL']}, as a function of $\gamma_0/\omega_0$ for different initial states of the form $\rho_0=(p/2)\mathbb I+(1-p)\,|1\rangle\!\langle 1|$. The evolution time was set as $\tau=1$, $\lambda = 20$, and $\omega_0=1$.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Proposition 1
  • Theorem 1
  • Corollary 1: Immediate
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5