Algebraic Geometry tools for the study of entanglement: an application to spin squeezed states

TL;DR

This paper reviews algebraic geometry methods for tensor and polynomial decomposition, applying them to quantum physics, specifically analyzing spin squeezed states and their entanglement properties using Sylvester's algorithm.

Contribution

It introduces algebraic geometry tools to analyze quantum entanglement in spin states, including calculating ranks and decompositions for complex quantum states.

Findings

Calculated symmetric rank and border rank of spin squeezed states.

Demonstrated the application of Sylvester's algorithm to quantum state decomposition.

Highlighted differences in rank properties for Fock states.

Abstract

A short review of Algebraic Geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics. Examples of application to assemblies of indistinguishable two-level bosonic atoms are discussed using modern formulations of the classical Sylvester's algorithm for the decomposition of homogeneous polynomials in two variables. In particular, the symmetric rank and symmetric border rank of spin squeezed states is calculated as well as their Schr\"odinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different.