Studies On Theories And Optical Experiments Of Quantum Tomography Technics

Quantum information including quantum computation is considered as one of the most powerful and promising technologies of 21th century.These quantum technics are usually involved with preparation,manipulation and measurement of quantum states.To observe and calibrate the performance of realized technics in laboratory,quantum state tomography and quantum process tomography,as the basic tools in this field,are developed.Quantum state tomography is the procedure of estimating the density ma-trix of unknown system via some quantum measurements and data processing.Quantum process tomography is the procedure of inserting different quantum states into some un-known process,such as quantum channel or quantum gate,and to interact with it,then estimating the output quantum states and finally inferring the process based on input-output relations.The two of them have a lot in common,and can be categorized as quan-tum tomography technics.Quantum tomography consists of two processes:quantum measurement and reconstructing algorithm,from which different tomography protocols have been developed with their own advantages and drawbacks.Meanwhile,two as-pects are mainly paid attention to in quantum tomography:the accuracy and complexity of estimation.Highly accurate quantum tomography is necessary to quantum compu-tation and other quantum technics,and the scarcity of quantum resource forces us to discover more efficient ways to decrease estimation error.As the Hilbert dimension of quantum systems increases exponentially with the number of qubits,both measurement process and reconstructing algorithm are becoming extremely complex.Above ques-tions are the concerns of this thesis,and some following works have been done for these concerns:1.Since the traditional maximum likelihood estimation in quantum state tomogra-phy tends to assign zero-eigenvalue density matrix,we reviewed the tricky solu-tion of multiplying the likelihood with some hedged function.Also we found in numerical simulations that its performance can be further improved by modifying the value of its parameter.Finally,we implemented this method in an two-qubit optical system to prove its feasibility.2.Accuracy is the eternal topic in metrology field.To optimize the accuracy of quantum state tomography,Sugiyama et al proposed an adaptive protocol based on average-variance-optimality,which has an analytic solution for one-qubit case and dramatically reduces the computational complexity compared to other adap-tive protocols.We implemented this adaptive protocol in our experiment and realized real time feedback and update of measurement bases.High accuracy has been achieved for one-qubit quantum state tomography,which approximates the theoretical limit.3.Quantum technics will keep involving to bigger quantum systems with more qubits,while the exponential growth of the Hilbert space becomes an inevitable problem of quantum state tomography.Recently proposed compressed sensing method sheds light on this problem.This method can reduce the number of mea-surement base needed from O(d2)to O(rdlog2d)for quantum states with low rank r in d.dimensional space,which largely simplifies the process of experi-ments and reconstructing algorithm.However,we found the restricted isotropic property requirement of measurement bases cannot be assured in practice.Thus we proposed to assist compressed sensing with adaptive strategy,which achieves higher estimation accuracy.4.Quantum process tomography has higher complexity than quantum state tomog-raphy,whose process matrix has dA-d2 free parameters.Fortunately,for quan-tum gate widely used in quantum computation,its specialities can be utilized to simplify the estimation task.We proposed a fast quantum gate estimation pro-tocol based on pure states tomography,whose computational complexity is only of order O(d3).Compared with traditional maximum likelihood estimation,our protocol is much faster without evidently sacrificing its accuracy.