Preparation Of High Dimensional Entangled States Based On Quantum Walk

A quantum walk is the generalization of the classical random walk in the quantum world.Compared with classical random walk,quantum walk has the characteristics of quadratic speedup,so the algorithm based on quantum walk is more efficient.Different from the classical walk,quantum entanglement is another important feature of the quantum walk in addition to the quadratic speedup.The quantum walk with two entangled particles can distinguish all non-isomorphic graphs via different position distributions,whereas that without entanglement can only discriminate part of them.The study of entanglement measurement and diffusion in quantum walks has attracted extensive attention.This thesis focuses on the preparation and measurement of entangled states in the quantum walk.The main work is as follows:(1)The inhomogeneous ordered quantum walk is realized,which achieves maximal entanglement while maintaining quadratic speedup.An inhomogeneous ordered quantum walk model is introduced,in which the coin operation remains unchanged with time evolution,but changes according to position in a certain way.The entanglement degree of coin position is quantified by von Neumann entropy,and the functions of entanglement entropy with the initial state of quantum walk,position dependent coin operation and evolution steps are obtained.The theoretical results show that the inhomogeneous quantum walk can obtain the maximal entanglement while maintaining the quadratic speedup.The inhomogeneous quantum walk is realized in the optical system,the two coin states are represented by the horizontal and vertical polarization states of light,and the discrete positions are represented by the different times at which photons arrive at the detector.The maximal coin position entanglement and quadratic speedup are observed through time multiplexing measurement.The experimental results show that the preparation of the maximal entangled state is realized in the inhomogeneous ordered quantum walk,and the quadratic speedup of wave function diffusion is demonstrated.The experimental setup can be extended to higher dimensions,providing a more achievable method for the preparation of multi-particle high-dimensional entangled states.(2)The properties of spatial distribution and entanglement in two-dimensional quantum walks are further studied.The effects of discrete coin operation and joint coin operation,and the initial states of coins are separable and entangled respectively on the spatial entanglement properties in two-dimensional quantum walk are discussed.The spatial entanglement in two-dimensional quantum walk is quantified by negativity,and the variation function of negativity with the initial state of quantum walk,coin operation and evolution steps are obtained.Through the numerical results,it is found that the spatial entanglement in the two-dimensional quantum walk is related to the number of evolution steps in the quantum walk,the initial state of the coin,and the operation of the coin.The measurement method of spatial entanglement in two-dimensional quantum walk proposed in this paper provides a method for the effective measurement of high-dimensional entanglement,which is suitable for optical and other physical systems.