Applications of differential geometry and representation theory to description of quantum correlations

TL;DR

This thesis explores how differential geometry and representation theory can unify and characterize various quantum correlations, extending the separability problem to broader types of quantum states with symmetry considerations.

Contribution

It introduces a unified framework for analyzing different quantum correlations using symmetry groups and polynomial criteria, generalizing the separability problem.

Findings

Explicit polynomial characterizations for pure states.

Complete analytical descriptions for certain mixed states.

Polynomial criteria for detecting correlations.

Abstract

One of the most important questions in quantum information theory is the so-called separability problem. It involves characterizing the set of separable (or, equivalently entangled) states among mixed states of a multipartite quantum system. In this thesis we study the generalization of this problem to types of quantum correlations that are defined in a manner analogous to entanglement. We start with the subset of set of pure states of a given quantum system and call states belonging to the convex hull of this subset "non-correlated" states. Consequently, the states laying outside the convex hull are referred to as "correlated". In this work we focus on cases when there exist a symmetry group that preserves the class of "non-correlated" pure states. The presence of symmetries allows to obtain a unified treatment of many types of seemingly unrelated types of correlations. We apply our general results to particular types of correlations: (i) entanglement of distinguishable particles, (ii) particle entanglement of bosons, (iii) "entanglement" of fermions, (iv) non-convex-Gaussian correlations in fermionic systems, (v) genuine multiparty entanglement, and finally (vi) refined notions of bipartite entanglement based on the concept of the Schmidt number. We investigate the natural problems and questions concerning the correlations defined above: (I) We provide explicit polynomial characterization of various types of correlations for pure states. (II) We examine cases in which it is possible to give a complete analytical characterization of correlated mixed states. (III) We derive a variety of polynomial criteria for detection of correlations in mixed states.(IV) We use the above criteria and the technique of measure concentration to study typical properties of correlations on sets of isospectral density matrices.