Algebraic Geometry tools for the study of entanglement: an application to spin squeezed 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 Schrödinger-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.

8 pages, 1 figure

References in corpus (4)

• Nonlinear atom interferometer surpasses classical precision limit
• Squeezing and entanglement in a Bose-Einstein condensate
• General criterion for the entanglement of two indistinguishable particles
• On tensors of border rank $l$ in $\C^{m\times n\times l}$

Cited by in corpus (14)

• The Hitchhiker guide to: Secant Varieties and Tensor Decomposition
• Geometry of spin coherent states
• On the partially symmetric rank of tensor products of W-states and other symmetric tensors
• Border rank is not multiplicative under the tensor product
• Waring, tangential and cactus decompositions
• A link between Quantum Entanglement, Secant varieties and Sphericity
• Identifiability and singular locus of secant varieties to Grassmannians
• Skew-Symmetric Tensor Decomposition
• Real and complex rank for real symmetric tensors with low ranks
• Semialgebraic decomposition of real binary forms of a given degree's space
• Entanglement in the symmetric subspace: mapping multipartite to bipartite states
• Waring decompositions of the product of two quadrics: the small rank cases
• On real Waring decompositions of real binary forms
• Collineation varieties of tensors