Systems that saturate the Margolus-Levitin quantum speed limit

TL;DR

This work provides a complete purification-based characterization of all finite-dimensional quantum states that saturate the Margolus--Levitin speed limit. It shows that saturation requires the state to live in the direct sum of exactly two energy eigenspaces, with each nonzero eigenvector having a fixed two-level superposition between the ground and a single excited level, and with all such eigenvectors evolving in mutually orthogonal two-dimensional subspaces; this imposes a strict rank bound that excludes faithful states. The authors derive a purity-resolved, tight MT bound for qubits and extend the dual MT bound to mixed states via a time-reversal argument. The results illuminate the geometric and spectral structure underlying quantum speed limits and provide precise criteria for when the bound is tight in both pure and mixed-state dynamics.

Abstract

We provide a complete characterization of all finite-dimensional quantum states that saturate the Margolus-Levitin quantum speed limit at arbitrary Uhlmann-Jozsa fidelity. Employing a purification-based approach, we prove that mixed-state saturation occurs precisely when three structural criteria are fulfilled: the state's support is confined to the sum of two energy eigenspaces (the ground level and a single excited level); each eigenvector of the state with nonzero weight is a fixed superposition of one ground- and one excited-state energy eigenvector (determined by the minimizer of the objective function identified by Giovannetti et al.) and all such eigenvectors evolve in mutually orthogonal two-dimensional subspaces. These requirements impose a strict rank bound, ruling out saturation by any faithful state. For qubit systems, we derive a purity-resolved and tight Margolus-Levitin bound that reduces to the pure-state result in the limit of unit purity. Through a time-reversal argument, we further extend the dual Margolus-Levitin quantum speed limit to mixed states and establish the corresponding saturation conditions.

Figures (3)

• Figure 1: Graph of the function $\alpha$, which appears in the numerator of the Margolus--Levitin bound, shown as a function of the fidelity $\delta$ between the initial and final states. As evident from the plot, the function is strictly decreasing. A proof of this property is provided in Appendix \ref{['app: A']}.
• Figure 2: Plots of the objective functions corresponding to the fidelities $\delta = 0.9, 0.7, 0.5, 0.3, 0.1$. The $z$-coordinates of their minima are indicated along the horizontal axis. A proof that the objective function for a given fidelity possesses a unique minimum is provided in Appendix \ref{['app: B']}.
• Figure 3: During the evolution, the Bloch vector precesses about the $z$-axis (vertical dashed line) with fixed inclination and angular speed $\omega$. The projections of the initial Bloch vector $\boldsymbol{r}$ and final Bloch vector $\boldsymbol{r}_\tau$ onto the $xy$-plane subtend an angle $\theta$, yielding an evolution time $\tau=\theta/\omega$.

Theorems & Definitions (2)

• Remark 1
• Remark 2