An estimation theoretic approach to quantum realizability problems

Abstract

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. In Chapter 3, we review the 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. In particular, we discover that most properties of quantum states are related to the gradient of this flow, also known as the moment map. Afterwards, Chapter 4 demonstrates how to estimate the value of the moment map of a quantum state by performing a covariant quantum measurement on a large number of identical copies of the quantum state. These measurement schemes for estimating the moment map of a quantum state arise naturally from the decomposition of a large tensor-power representation into its irreducible sub-representations. Then, in Chapter 5, we prove an exact correspondence 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. In particular, by composing these estimation schemes, we derive necessary and sufficient conditions for the existence of a quantum state jointly realizing any finite collection of moment maps. Finally, in Chapter 6 we apply these techniques to the quantum marginals problem which aims to characterize precisely the relationships between the marginal density operators describing the various subsystems of a composite quantum system. We make progress toward an analytic solution to the quantum marginals problem by deriving a complete hierarchy of necessary inequality constraints.