Isoholonomic inequalities and speed limits for cyclic quantum systems

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 state space trajectories 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 new quantum speed limits that remain nontrivial for cyclic evolutions. Our results reveal deep connections between the temporal behavior of cyclic quantum systems and holonomy.