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A Geometrical Derivation of a Family of Quantum Speed Limit Results

We derive a family of quantum speed limit results in time independent systems with pure states and a finite dimensional state space, by using a geometric method based on right invariant action functionals on SU(N). The method relates speed limits for implementing quantum gates to bounds on orthogonality times. We reproduce the known result of the Margolus-Levitin theorem, and a known generalisation of the Margolis-Levitin theorem, as special cases of our method, which produces a rich family of other similar speed limit formulas corresponding to positive homogeneous functions on su(n). We discuss the general relationship between speed limits for controlling a quantum state and a system's time evolution operator.