Operational Definition Of Quantum Speed Limit

Coherence and entanglement are important resources in quantum information and quantum technology which are widely interested.However,in open system,the existence of decoherence limits the lifetime of these quantum resources.Currently,reduce the operation time is a common method in general for this problem with the current experimental technique.The shortest time for performing a quantum operation or the minimum time for the system complete the fixed evolution is referred to as the quantum speed limit.Quantum speed limit is not only a subject in quantum information theory,but also a fundamental concept in quantum mechanics,which aims to study the maximum speed or shortest evolution time for a fixed target in any system.The traditional methods in the studies of quantum speed limit are generally based on the construction of mathematical inequalities on the evolution time,however,these inequalities cannot tell if a chosen initial state can fulfill the target or not,and the attainability is another challenge for it.Here we provide an operational approach for the definition of quantum speed limit,which utilizes the set of reachable states to define the quantum speed limit.This definition can clarify if an initial state can fulfill the target and its attainability is guaranteed by definition.The main research of this paper are as follows: For the time-independent Hamiltonians,quantum speed limit is inverse-proportional to the difference between the highest and lowest energies,and the attainability doesn't require a zero state energy,which indicates that it can be used as an indicator for quantum phase transitions.For the time-dependent Hamiltonians,the study on two-level system Landau-Zener model confirms the fact that quantum speed limit should not be a function of time.In open system,the study on spontaneous emission shows that a lousy purity can speed up the realization of target.