Quantum Speed Limits Based on Schatten Norms: Universality and Tightness

Abstract: We present two families of quantum speed limits (QSLs) for finite-dimensional quantum systems undergoing a general physical process. These QSLs were obtained using Schatten $\alpha$-norms, firstly exploiting the geometric features of the space of quantum states, and secondly employing the Holder's inequality for matrix norms. In particular, for the case of single-qubit states, we find that the geometric QSL is independent of the Schatten norm chosen, thus revealing a universality behavior of such quantifiers. Furthermore, we provide a comparison of these quantum speed limits with existing paradigmatic QSLs in literature, thus showing that the latter results represent particular cases of a general class of QSLs related to Schatten $\alpha$-norms. Noteworthy, we address necessary and sufficient conditions for the tightness of the quantum speed limit that mostly depends on the populations and quantum coherences of the evolved single-qubit state, and also present a geometric interpretation for these set of conditions. Finally, we compare the two QSL obtained for the dynamics of single-qubit states, also presenting an inequality between them that has a clear geometrical meaning.