Quantifying entanglement in ℂ𝑁 ⊗ ℂ𝑁 by analyzing separability in ℂ2 ⊗ ℂ𝑁

Abstract

A fundamental challenge in quantum entanglement is determining whether a given bipartite quantum state is separable or entangled, a problem known to be computationally intractable in general. This thesis focuses on bipartite systems of the form C2 ⊗ CN, consisting of a qubit and a qudit, which offer a rich yet tractable setting for studying entanglement. The central objective of this thesis is to investigate the maximal Schmidt number that an entangled quantum state can attain in CN ⊗CN systems. The Schmidt number is a bona fide measure of entanglement in bipartite systems. Here, we investigate how this measure of entanglement correlates with the structure of separable states in C2 ⊗ CN. To achieve this, the work combines analytical and numerical tools. Here, we examine structured quantum states, constructing families of states with computable or bounded Schmidt number, and apply criteria to assess their entanglement. In particular, we focus on the study of those entangled states that are positive under partial transposition (PPT), also denoted as bound entangled states. By integrating the above approaches, we provide a better characterization of entanglement in low-dimensional bipartite systems and we offer novel insights on how to classify quantum states according to their Schmidt number. Overall, this study advances the characterization of quantum correlations in CN ⊗ CN systems for a particular family of states, but offers a foundation for future investigations into entanglement quantification and separability criteria in generic states.