  Quantum state tomography (QST) aims at estimating a quantum state from
averaged quantum measurements made on copies of the state. Most quantum
algorithms rely on QST at some point and it is a well explored topic in the
literature, mostly for mixed states. In this paper we focus on the QST of a
pure quantum state using parallel unentangled measurements. Pure states are a
small but useful subset of all quantum states, their tomography requires fewer
measurements and is essentially a phase recovery problem. Parallel unentangled
measurements are easy to implement in practice because they allow the user to
measure each qubit individually. We propose two sets of quantum measurements
that one can make on a pure state as well as the algorithms that use the
measurements outcomes in order to identify the state. We also discuss how those
estimates can be fined tuned by finding the state that maximizes the likelihood
of the measurements with different variants of the likelihood. The performances
of the proposed three types of QST methods are validated by means of detailed
numerical tests.
