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arXiv:1201.5995 · Journal of Physics A Mathematical and Theoretical

A class of exposed indecomposable positive maps

Gniewomir Sarbicki, Dariusz Chruściński

Published Dec 5, 2012• 19 citations
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Abstract

Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a class of indecomposable positive maps in the algebra of 2 n × 2 n complex matrices with n ⩾ 2. It is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states.

DOI
10.1088/1751-8113/46/1/015306

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Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a class of indecomposable positive maps in the algebra of 2 n × 2 n complex matrices with n ⩾ 2. It is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states.
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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 Mm(C)⊗Mn(C)M_m(\mathbb{C}) \otimes M_n(\mathbb{C})Mm​(C)⊗Mn​(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.