Mathematical methods in quantum state tomography

Author(s)

Douglas Soares Gonçalves

• Document type: Doctoral Thesis
• Press: Campinas, SP.
• Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
• Defense date: 2013-02-22

Examining board members

Márcia Aparecida Gomes Ruggiero; Sandra Augusta Santos; Amir Ordacgi Caldeira; Reinaldo Oliveira Vianna; Renato Portugal

• Advisor: Carlile Campos Lavor; Márcia Aparecida Gomes Ruggiero

Abstract

Preparation, manipulation and characterization of quantum states are essential tasks for quantum computation. In Quantum State Tomography, the aim is to find an estimate for the density matrix associated to an ensemble of identically prepared quantum systems, based on the measurement outcomes. This is an important procedure in quantum information and computation applied for instance, to verify the fidelity of a prepared state or in quantum process tomography. In this thesis we study mathematical methods applied to problems that raise from the reconstruction of quantum states. In the Maximum Likelihood Estimation we present two methods to solve the optimization problems of this approach. The first one is based on a reparameterization of the density matrix and, in this case, we prove the equivalence of local solutions of the related unconstrained optimization problem. In the second one, related to multinomial likelihoods, we prove the global convergence of the method under weaker assumptions than those of literature. We also discuss two formulations to the case of quantum state tomography with incomplete measurements: Maximum Entropy and Variational Quantum Tomography. We propose a new formulation for the second one in order to have a similar behavior to the Maximum Entropy approach, keeping the linear semidefinite positive programming structure. Furthermore, in order to solve other optimization problems over the density matrices space besides the Quantum State Tomography, we present a Projected Gradient method which shows a good performance in preliminary numerical tests. We also briefly talk about the implementation of a Bayesian inference scheme, through Monte Carlo Markov chains methods, to the density matrix estimation problem (AU)

• FAPESP's process: 09/08027-0 - Global optimization applied to the tomography of quantum states
• Grantee: Douglas Soares Gonçalves
• Support Opportunities: Scholarships in Brazil - Doctorate