  The development of large-scale platforms for quantum information requires new
methods for verification and validation of quantum behavior. Quantum tomography
(QT) is the standard tool for diagnosing quantum states, process, and readout
devices by providing complete information. However, QT is limited since it is
expensive to not only implement experimentally, but also requires heavy
classical post-processing of data. In this dissertation, we introduce new
methods for QT that are more efficient to implement and robust to noise and
errors, thereby making QT a more practical tool for current quantum information
experiments.
  The crucial detail that makes these new, efficient, and robust methods
possible is prior information about the quantum system. This prior information
is prompted by the goals of most experiments in quantum information, which
require pure states, unitary processes, and rank-1 POVM operators. Therefore,
most experiments are designed to operate near this ideal regime. We show that
when this is the case, QT can be accomplished with significantly fewer
resources, and produce a robust estimate in the presence of noise and errors.
Moreover, the estimate is also robust if the state is not exactly pure, the
process is not exactly unitary, or the POVM is not exactly rank-1. Such
compelling methods are only made possible by the positivity constraint on
quantum states, processes, and POVMs. This requirement is an inherent feature
of quantum mechanics, but has powerful consequences to QT.
  Since QT is necessarily an experimental tool, we discuss a test of these new
methods in an experimental setting. The physical system is an ensemble of
laser-cooled cesium atoms in the laboratory of Prof. Poul Jessen. Experiments
were conducted by Hector Sosa-Martinez et al. to demonstrate different QT
protocols. We compare the results, and conclude that the new methods are
effective.
