The Trajectory Ensemble Theory Of Quantum Speed Limits

Quantum speed limits(QSL)is the maximum evolution speed of a quantum system.It has potential applications in quantifying and controlling the quantum coherence,protecting the quantum information from decoherence caused by the environmental noise and so on.It also sets the limits for parameter estimation of the quantum measurement.The unified bound of the MT bound and the ML bound for closed systems was firstly obtained.The MT bound is related to the energy uncertainty and the ML bound is related to the energy average.With development of the quantum mechanics,interactions between quantum systems and the environment have drawn wide attention,the concept of quantum speed limits is generalized to the open system,and the generalized unified bound of the MT and the ML bound is obtained as well.Based on rediscovery to the geometry properties of the quantum system,a new QSL bound was derived based on the geometric property.As an equivalent description of the conventional quantum mechanics,QSL in the quantum phase space is thus researched.After establishment of the quantum mechanics,Wigner introduced the Wigner distribution function to modify the quantum effect of the statistical thermodynam-ics and it is the beginning of the theory of the quantum phase space.It is still a challenge to solve problems with strong coupling,high dimensionality and nonlin-earity by using the conventional quantum mechanics.The advantages of the theory of quantum phase space lie in two aspects:one is to transform operator operation into algebraic operation,which avoids the complex calculation.The second is that motion equation of the quantum trajectory can be established to simulate some quantum processes based on theory of the quasi probability distribution function,and thus to describe quantum effect of the quantum system with abundant semi-classical pictures.The theory of quantum phase space has been extensively used in multiple fields,such as nonlinear physics,statistical physics,quantum optics and collision theory,etc.The distribution function of the quantum phase space is the foundation of the quantum phase space theory,it not only displays quantum effect in the phase space,but also provides a bridge from quantum mechanics to classical mechanics.Based on this,quantum speed limit in the quantum phase space is researched in this thesis with the trajectory ensemble,and the contribution of single trajectory to the QSL is discussed,by which a potential method to deal with the quantum speed limit of the high-dimensional complicated quantum system is proposed.The specific contents and conclusions are as follows.Based on the Wasserstein-1-distance,the expression of the quantum speed limit with the single energy surface entangled trajectory ensemble in the phase space is derived.The quantum speed limit is regarded as superposition of the average of the energy flow of the connecting harmonic oscillators over the phase space.In this theoretical framework,two classical models(the time-dependent harmonic oscillator and the undriven harmonic oscillator coupled to a thermal bath)are studied,and the results are in good agreement with the previous results.In addition,we also analyze the influence of different trajectory subsets on QSL.As the trajectory subset is farther away from the center of the phase space,the more consistent the speed direction of members in a subset,the greater the contribution of the trajectory subset to the QSL.Based on the Fubini-study measurement,the expression of the quantum speed limit with the multiple potential energy surfaces hopping trajectory ensemble in quantum phase space is derived.Different from the above trajectories,trajectories in this ensemble can hop between or among the potential energy surfaces.They collectively determine the mechanical quantities of the quantum system.Under diabatic and adiabatic representations,we study Tully's single crossing model and discuss the contribution of the coherent speed and population speed to the QSL.Besides,the influence of single trajectory on the QSL is analyzed.Under the dia-batic representation,the QSL is mainly affected by the population speed.Although there exist negative values in the contribution of single trajectory to the QSL,the total effect is positive.Under the adiabatic representation,the contribution of pop-ulation rate and coherence rate is near.The contribution of each trajectory to the change of the imaginary part of the coherence is pretty uniform.We also study the acceleration potential of a two-level system driven across avoided crossing.These regions are of great significance to quantum information process.We can find that the system has the acceleration potential under Markovian dynamics.In different modes,the acceleration potential shows different behaviors.Under strong driving,the acceleration potentials of the exact evolution and the two approximate evolutions are different.The difference is due to the approximation of Hamiltonian,which leads to the loss of some quantum information.