A Geometric Approach To The Speed Limits Of Quantum System Evolution

As a fundamental problem of quantum mechanics,the definition of Quantum Speed Limit(QSL)sets the upper bound of the evolution speed or the lower bound of the evolution time of a quantum system.The study of QSL can not only con-tribute to our better understanding of such basic problems in quantum mechanics,but also has important scientific significance in many research fields of quantum physics.For instance,based on the rapid inversion of quantum states,the quantum communication,quantum computation and quantum simulator can achieve super-fast computing speed.Improving the evolution rate of quantum state can achieve the optimal control of the state and dynamical evolution process of quantum sys-tem.Moreover,it also has important significance in quantum thermodynamics,quantum entropy reduction,entropy production and etc.For the case of the unitary evolution,Mandelstam and Tamm presented the Mandelstam-Tamm(MT)bound in 1945,which related the minimum time a quan-tum system needs to evolve between two orthogonal states with the energy uncer-tainty of the system.And,Margolus and Levitin presented the Margolus-Levitin(ML)bound in 1998,which established the relationship between the minimum evo-lution time and the average energy of the system.The unified quantum speed limit bound for closed system is defined by combining the MT and the ML bounds,and the maximum is considered as the tightest bound.In the open quantum system,due to the interaction between the system and the environment,the quantum system can no longer evolve to the orthogonal state.The MT type and ML type QSL bounds have been established.Many authors have attempted to develop the generation of the QSL to open quantum system,and investigate the relations between QSL and the physical natures of the quantum system,including the role of entanglement in QSL for open dynamics and many body system,the relativistic system,non-equilibrium dynamics and non-Hermitian system,the relation between the maximum interaction speed and QSL in quantum spin systems,the non-Markovianity effect of the environment on accelerating the speed of evolution.QSL has also been studied in the quantum representation of Wigner function.Also,in quantum thermodynamics the maximal rate of entropy production has been derived from the QSL.It has shown that the speed limit could exist in classical systems.Due to the system energy dissipation and decoherence caused by the interaction between the open system and the environment,it is necessary to consider the metric of distance between two distinguishable quantum states to study the quantum speed limit of the open system.The unified quantum speed limit bound of open systems is established by combining the various quantum speed limit bounds under the same metric.However,it is difficult to compare the quantum speed limit bounds under different metrics.The establishment of QSL bound basing on Cauchy-Schwarz inequality or Von-Neumann inequality leaves us a lot of questions worth thinking about,such as:What is the physical property of QSL?Whether the QSL bounds have geometric properties since the quantum mechanics is of geometric properties.Besides the ex-isting MT and ML types of QSL bounds,are there any other undiscovered bounds?Under the Schrodinger equation,what is the relationship between different QSL bounds and the evolution modes of the system?Riemannian manifold is a differential manifold.The Riemannian metric based on the differential geometry has the gauge invariant property.In Riemannian man-ifold,geometric phases can be obtained through the geometric transformation of quantum state vectors.Geometric phase has been paid much attention and ap-plied by the mathematical physicists with its rich topological properties and close relationship with quantum field gauge theory.For example,the quantum gate oper-ation can be realized by using Berry phase and its non-abelian extension.In recent years,the geometric properties of the quantum system has reawakened interest.It is of great significance to study the physical nature of the acceleration of quantum systems by using geometric method to derive a distinct bound and to es-tablish a unified QSL bound under a unified framework.This topic helps us to make better comprehension of the geometric nature of acceleration of quantum system evolution and the phase.It provides potential solutions and approaches to control the geometric evolution of the quantum system,accelerate the geometric quantum computation and so on.Moreover,it has scientific significance in understanding the problem such as quantum entropy reduction,and puts forward more insights into the maximum evolution rate of quantum gate operation,quantum metrology,quan-tum control,adiabatic quantum computing and quantum phase change indicator and etc.So,in this thesis,we investigate and discuss the relevant questions mentioned above.The thesis is divided into six chapters and organized as followsIn the second chapter,we first introduce the gauge transformation theory of quantum states in Riemannian manifold and the geometric structure of manifold space,including Hilbert quantum state space,projection Hilbert space,structure group,projective map and the rays,etc.In addition,we introduce the Fisher-Rao metric,Bures metric and Fubini-study metric,which are three typical Riemannian metrics.In the following,we briefly introduce the Stiefel manifold.Also,the Non-Hermitian quantum system is introduced,the geometric structure of the topological vector space and the rays in the topological vector space are discussed.Using the Fubini-study metric,the geodesic equation is defined.In the third chapter,based on the Fubini-study metric,we establishes the relationship between the shortest evolution time of quantum system and the gen-eralized Pancharatnam.Considering the parallel evolution mode of quantum state vector,we establish a distinct bound besides the MT type and ML type bounds.This distinct bound establishes the relationship between the quantum speed limit time and the geometric phase of the quantum system,and presents the geometric and gauge invariant properties of it.It reflects the influence of evolution mode on evolution curve of quantum state vector in manifold space.The distinct bound can be applied not only to Hermitian quantum systems,but also to non-Hermitian quantum systems.Moreover,the distinct bound can be tighter than the existing MT type and ML type bounds under some cases.In the fourth chapter,we generalize the ML QSL bound.This result shows that the ML QSL bound cannot be established on considering the evolution without the geometric phase accumulation in the manifold space.A type method is to make a geodesic evolution of the quantum state vectors.In the fifth chapter,using the changing rate of phase in the quantum system to characterize the maximum evolution speed of quantum system,we establish a unified bound of QSL.The unified bound puts the existing QSL bounds into a sim-ple unified framework.The unified quantum speed limit bound can be affected by the different evolution modes of quantum state vectors in topological vector space.When using the geodesic joining their projection of a horizontal affine geodesic in the base manifold,the unified bound returns to the MT type bound.Considering the geodesic evolution,we can obtain the generalized ML type QSL bound.Con-sidering the parallel evolution of the quantum state vector,we can get the distinct quantum speed limit bound in the previous chapter.Since the phase is experi-mentally measurable,our unified bound can also be experimentally measurable.In addition,compared with the existing results,our unified bound can provide a tighter upper bound on the evolution time of quantum systems.In the last chapter,we give the brief conclusions drawn from the present studies and take the outlook on the future investigation.