Solidity of three quantum speed limits in few and many-body systems and their crossover

The quantum speed limit is a fundamental concept that governs the evolution of quantum systems, and there is an increasing demand for experimental certifications of these limits. In this study, we present a comprehensive set of experimental schemes encompassing both few-body systems (qubits and qutrits) and many-body systems (one-dimensional chains and two-dimensional square lattices) to examine how unitary time evolution is constrained by the Mandelstam and Tamm (MT), Margolus and Levitin (ML), and dual ML bounds. We rigorously demonstrate that the fidelity, denoted as ||, is strictly limited by a unified bound, which is the maximum among the MT, ML, and dual ML bounds. Moreover, our experimental setup allows us to observe crossovers between these three bounds. We provide a clear explanation for why the fidelity is bounded by the MT limit in short time intervals. For the qubit case, we analytically establish that it consistently satisfies Popoviciu's inequality, which is positioned on the semicircle of the phase diagram. By considering a unique initial pure state—a linear combination of two Fock states with maximal and minimal energy separately—we construct pathways within the phase diagram to illustrate how initially loosely bounded dynamics gradually become sufficiently constrained. Our research offers a comprehensive guide that outlines the expected features in experiments related to quantum speed limits, especially when applied to multi-qubit platforms that involve long-range couplings. This work contributes significantly to our understanding of the boundaries governing quantum evolution and provides valuable insights for experimental certification in quantum systems.