Realizing quantum speed limit in open system with a -symmetric trapped-ion qubit

Quantum speed limit (QSL), the lower bound of the time for transferring an initial state to a target one, is of fundamental interest in quantum information processing. Despite that the speed limit of a unitary evolution could be well analyzed by either the Mandelstam–Tamm or the Margolus–Levitin bound, there are still many unknowns for the QSL in open systems. A particularly exciting result is about that the evolution time can be made arbitrarily small without violating the time-energy uncertainty principle, whenever the dynamics is governed by a parity-time ( $\mathcal{PT}$ ) symmetric Hamiltonian. Here we study the QSLs with both $\mathcal{PT}$ and anti- $\mathcal{PT}$ Hamiltonians, and pose the QSL as a brachistochrone problem on a non-Hermitian Bloch sphere. We then use dissipative trapped-ion qubits to construct the Hamiltonians, where the state evolutions reach the QSL governed by a generalized Margolus-Levitin bound of the non-Hermitian system. We find that the evolution time monotonously decreases with the increase of the dissipation strength and exhibits chiral dependence on the Bloch sphere. These results enable a well-controlled knob for speeding up the state manipulation in open quantum systems, which could be used for quantum control and simulation with non-unitary dynamics.