Separability of Quantum States via Algebraic Geometry

Abstract

In this thesis, we study the quantum separability problem by taking advantage of various methods in algebraic geometry.

In order to explore the separability of quantum states, we begin with the range criterion for separability. It leads us to examine the condition that $

\psi_1 \rangle \otimes

\psi_2 \rangle \in D$ and $

\overline{\psi_1} \rangle \otimes

\psi_2 \rangle \in E$ for subspaces $D$ and $E$ of a finite-dimensional composite quantum system $\mathcal{H}_A \otimes \mathcal{H}_B$. More explicitly, the following two questions naturally arise : (1) For which conditions there is a nonzero product vector $

\psi_2 \rangle$ in $\mathcal{H}_A \otimes \mathcal{H}_B$ such that $

\psi_2 \rangle \in E$? (2) if it exists, how many such nonzero product vectors in $\mathcal{H}_A \otimes \mathcal{H}_B$ exist up to constant?

We investigate the question (1) and generalize it for the multipartite cases. Moreover, we answer the question (2) so that the upper bound for the number of vectors $

\psi_2 \rangle \in \mathcal{H}_A \otimes \mathcal{H}_B$ satisfying the condition that $

\psi_2 \rangle \in E$ is expected to be sharp for the qubit-qunit case.