  This thesis seeks to develop a general method for solving so-called quantum
realizability problems, which are questions of the following form: under which
conditions does there exist a quantum state exhibiting a given collection of
properties? The approach adopted by this thesis is to utilize mathematical
techniques previously developed for the related problem of property estimation
which is concerned with learning or estimating the properties of an unknown
quantum state. Our primary result is to recognize a correspondence between (i)
property values which are realized by some quantum state, and (ii) property
values which are occasionally produced as estimates of a generic quantum state.
Chapter 3 reviews concepts of stability and norm minimization from geometric
invariant theory and non-commutative optimization theory for the purposes of
characterizing the flow of a quantum state under the action of a reductive
group. Chapter 4 demonstrates how the gradient of this flow, also called the
moment map, can be estimated by performing a covariant quantum measurement on a
large number of identical copies of the quantum state. Chapter 5 outlines the
correspondence between between the realizability of a moment map value on one
hand and the asymptotic likelihood it is produced as an estimate on the other
hand. By appropriately composing these moment map estimation schemes, we derive
necessary and sufficient conditions for the existence of a quantum state
jointly realizing any finite collection of moment maps. Chapter 6 applies these
insights to the quantum marginal problem and is a duplication of
arXiv:2211.00685.
