Study On Low-rank Maximum Likelihood Estimation For Quantum State Tomography

Efficient and reliable characterization of properties of a quantum system is needed for any quantum information processing task,while such are the goals of quantum tomography,broadly classified into state tomograph and process tomograph.So,the quantum state tomograph plays a significant role in quantum information processing.The maximum likelyhood estimator(MLE)for quantum tomograph is a popular estimation strategy.However,the existing MLE models in the literature either neglect the case that the unknown true quantum state is low rank or consider a low-rank MLE model under the assumption that an imperfect measurement result is represented as an additive Gaussian noisy measurement.Among others,this assumption is inapplicable to the setting of high-dimensional experiments with limited data for a quantum system.In this thesis,for the low-rank true quantum state,we propose without such an assumption a rank-constrained MLE model and a rank-regularized MLE model which are closer to the experiment setting of a quantum system.Owing to the high nonlinearity of log-likelihood function and the combinatorial property of rank function,the solution of proposed models is not an easy task.Starting from the equivalent DC constrained formulation of the rank-constraint optimization model and the equivalent MPEC reformulation of the rank-regularized optimization model,we employ the upper Lipschitzian property of the perturbation mapping for the rankconstrained density matrix set to verify that the penalized problems,yielded by the DC constraint and the equilibrium constraint,are exact in a global sense,and then obtain the equivalent DC programs for the two classes nonconvex and nonsmooth problems.By leveraging the structure of the equivalent DC programs,Chapter 3 and 4 propose an inertial MM method for solving them and present the corresponding global convergence analysis.In addition,we also design an inertial MM method for solving the factored formulation of the equivalent DC program for the rank-constrained optimization model.The obtained convergence results improve and extend the result in [38] on the inertial heavy-ball method for the nonconvex nonsmooth composite optimization problems.In addition,we also test the performance of the proposed inertial MM methods with some synthetic data.Numerical results show that the equivalent DC program of the proposed rank-constrained model indeed yield the estimator with a desirable trace distance as well as a lower rank,and moreover,when a tight upper estimation is available for the rank of true density matrix,the factored formulation of the equivalent DC program has a remarkable advantage for the quantum tomograph with a high qubit in computation.