Isoholonomic inequalities and speed limits for cyclic quantum systems

TL;DR

This paper introduces a novel quantum speed limit based on isoholonomic inequalities, providing nontrivial bounds for cyclic quantum evolutions by connecting geometric phase holonomy with system dynamics.

Contribution

It extends isoholonomic inequalities to cyclic quantum systems using a gauge-theoretic framework, establishing a new speed limit applicable to closed trajectories in state space.

Findings

Derived a new quantum speed limit for cyclic evolutions.

Linked holonomy with the temporal behavior of quantum systems.

Extended geometric phase concepts to mixed states with isospectral and isodegenerate conditions.

Abstract

Quantum speed limits set fundamental lower bounds on the time required for a quantum system to evolve between states. Traditional bounds, such as those by Mandelstam-Tamm and Margolus-Levitin, rely on state distinguishability and become trivial for cyclic evolutions where the initial and final states coincide. In this work, we explore an alternative approach based on isoholonomic inequalities, which bound the length of closed trajectories in the state space in terms of their holonomy. Building on a gauge-theoretic framework for mixed-state geometric phases, we extend the concept of isoholonomic inequalities to closed curves of isospectral and isodegenerate density operators. This allows us to derive a new quantum speed limit that remains nontrivial for cyclic evolutions. Our results reveal deep connections between the temporal behavior of cyclic quantum systems and holonomy.