A toric varieties approach to geometrical structure of multipartite states

TL;DR

This paper explores the geometric structure of multipartite quantum states using toric varieties, providing a systematic framework that generalizes existing models like the Segre variety.

Contribution

It introduces a novel approach to represent multipartite quantum states via toric varieties, linking algebraic geometry with quantum information theory.

Findings

Multipartite states correspond to toric varieties constructed from polytopes.

Projective toric varieties represent spaces of separable states.

The approach generalizes the complex multi-projective Segre variety.

Abstract

We investigate the geometrical structures of multipartite states based on construction of toric varieties. In particular, we describe pure quantum systems in terms of affine toric varieties and projective embedding of these varieties in complex projective spaces. We show that a quantum system can be corresponds to a toric variety of a fan which is constructed by gluing together affine toric varieties of polytopes. Moreover, we show that the projective toric varieties are the spaces of separable multipartite quantum states. The construction is a generalization of the complex multi-projective Segre variety. Our construction suggests a systematic way of looking at the structures of multipartite quantum systems.