Geometric derivation of the quantum speed limit

The Mandelstam-Tamm and Margolus-Levitin inequalities play an important role in the study of quantum-mechanical processes in nature since they provide general limits on the speed of dynamical evolution. However, to date there has been only one derivation of the Margolus-Levitin inequality. In this paper, alternative geometric derivations for both inequalities are obtained from the statistical distance between quantum states. The inequalities are shown to hold for unitary evolution of pure and mixed states, and a counterexample to the inequalities is given for evolution described by completely positive trace-preserving maps. The counterexample shows that there is no quantum speed limit for nonunitary evolution.

Figure 1

Nonunitary evolution of the central qubit that violates both the Mandelstam-Tamm and Margolus-Levitin bounds: (a) All qubits are prepared in the state $|+\rangle$; (b) each qubit is entangled simultaneously with the central qubit using a controlled-$Z$ (cz) gate; (c) Hadamard gates are applied to the satellite qubits, which creates a GHZ state of size $N+1$; (d) phase shifts $\varphi$ are applied to the satellite qubits and (e) a second set of Hadamard gates are applied to the satellite qubits; (f) $N$ simultaneous cz gates disentangle the satellite qubits from the central qubit, leaving it with an accumulated phase shift of $N\varphi$.