Some applications of Choi polynomials of linear maps

TL;DR

This paper explores the properties of Choi polynomials in positive linear maps, linking them to entanglement detection and classification of PPT states in quantum information theory.

Contribution

It introduces new methods for constructing indecomposable positive maps and applying them as entanglement witnesses, advancing the understanding of quantum entanglement detection.

Findings

Established a connection between Hermitian biquadratic forms and positive maps.

Constructed indecomposable positive maps for entanglement detection.

Extended analysis to classify edge PPT states in matrix algebras.

Abstract

This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the positivity of these forms and the structure of positive maps. We specifically explore the construction of indecomposable positive maps in matrix algebras, and their application as entanglement witnesses. Our analysis extends to the detection of Positive Partial Transpose (PPT) entangled states and the classification of edge PPT states in $M_m(\mathbb{C}) \otimes M_n(\mathbb{C})$. Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.