Revivals Of Discrete-time Quantum Walk And Its Entanglement Property With Disorder

Quantum walk,as an extension of classical random walk,is not only a theoretical basis for designing the efficient quantum algorithms,but also a very useful platform for modeling various quantum phenomena.In this thesis,the manipulation of quantum walk,the effects of disorder on the transmission and entanglement are investigated.It is expected that the results are helpful to provide the theoretical foundation to model the dynamics of complex physical systems.The paper firstly presents the formalism theories of discrete time quantum walk and continuous time quantum walk on the basis of classical random walks,gives an outline of the experimental principles in different physical systems to perform quantum walk,and uses the decoherence in coin space of discrete time quantum walk on one-dimensional infinite lattices as an example to illustrate that the quantum coherences are the essential difference between quantum walk and classical random walk.Then the phase modified quantum walk,that is to say the quantum walk obtain a linear position-dependent phase at each enveloping step,is studied.The phase modification makes the quantum walk display the characteristics of quantum resonance and dynamical localization.When the phase is rational,the numerical results and the revival theorem show that the quantum walk periodically returns to the initial position with high probability.We generalize the proving process of revival theorem to include two cases that determinant of coin operator being 1 and-1.When the phase is irrational,it can be made rational approximation by the method of continued fraction expansion,or the quantum walk localize the initial position with certain probabilities.Finally,we consider the discrete-time quantum walk on one-dimensional infinite lattices with open boundary and finite lattices with reflecting boundary,respectively,and introduce static and dynamical disorders by breaking the links randomly to investigate the effects of disorders on the transmission properties and the dynamical entanglement between the coin degree of freedom and position degree of freedom.When the lattice is infinite,it is shown that the probability distribution of static percolation quantum walk can change from a coherent behavior at short time to Anderson localization at longer time,while the dynamical percolation quantum walk change to a classical diffusive behavior.Meanwhile,the static percolation makes the entanglement be less than that of ideal case and fluctuate irregularly around a certain value.The situation is very different for the dynamical percolation that the entanglement increases smoothly with the time steps and can exceed the constant value in ideal case at some time.For quantum walk with reflecting boundary,both static and dynamical disorders can enhance the entanglement for non-local initial state,while only dynamical disorder can do for local initial state.Moreover,the dynamical disorder can produce maximum entanglement at long time limit regardless of the strength of disorder.