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AIP conference proceedings

Indecomposable Positive Maps and a New Family of Inseparable PPT States

Sudhavathani Simon, S. P. Rajagopalan, R. Simon

Published Jan 1, 2006• 2 citations
Publisher

Abstract

We present a new class of entangled bipartite states. These remain positive under partial transposition, and hence live outside the jurisdiction of the Peres‐Horodecki separability criterion. Their inseparability is demonstrated using Choi‐type indecomposable positive maps. It is shown that these states are not isolated, but occupy a finite volume in the bipartite state space. It is further shown that in the space of positive maps there exists a neighborhood around the Choi map with the property that every map in this neighborhood is indecomposable.

DOI
10.1063/1.2400879

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We present a new class of entangled bipartite states. These remain positive under partial transposition, and hence live outside the jurisdiction of the Peres‐Horodecki separability criterion. Their inseparability is demonstrated using Choi‐type indecomposable positive maps. It is shown that these states are not isolated, but occupy a finite volume in the bipartite state space. It is further shown that in the space of positive maps there exists a neighborhood around the Choi map with the property that every map in this neighborhood is indecomposable.
We construct a class of quantum bipartite $d\ensuremath{\bigotimes}d$ states which are positive under partial transposition (PPT states). This class is invariant under the maximal commutative subgroup of U(d)\mathrm{U}(d)U(d) and contains as special cases many well-known examples of PPT states. States from our class provide criteria for testing the indecomposability of positive maps. Such maps are crucial for constructing entanglement witnesses.
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.
We construct a family of positive but not completely positive linear maps acting on M4(C)M_4(\mathcal{C})M4​(C), obtained as a natural extension of Kye’s indecomposable maps defined for M3(C)M_3(\mathcal{C})M3​(C). We rigorously prove positivity of these maps on 'X' states and employ them to identify a one-parameter family of bipartite quantum states living in the vicinity of maximally mixed state to be entangled even though states are positive under partial transposition (PPT). This provides an explicit example of a family of quantum states with both bound entangled states and free entangled states in 4⊗44 \otimes 44⊗4 systems, a regime that remains less characterized compared to lower-dimensional cases. The proposed maps detect entanglement and reveal new structural features of the PPT entangled region in higher dimensions. Our results extend the applicability of positive-map–based entanglement detection and contribute to the systematic understanding of bound entanglement beyond the 3⊗33\otimes 33⊗3 and 2⊗42 \otimes 42⊗4 systems. In addition, we show that generalized Choi maps simply can not detect well known PPT entangled states for 2⊗42 \otimes 42⊗4 systems.
We study robustness of bipartite entangled states that are positive under partial transposition (PPT). It is shown that almost all PPT entangled states are unconditionally robust, in the sense, both inseparability and positivity are preserved under sufficiently small perturbations in its immediate neighborhood. Such unconditionally robust PPT entangled states lie inside an open PPT entangled ball. We construct examples of such balls whose radii are shown to be finite and can be explicitly calculated. This provides a lower bound on the volume of all PPT entangled states. Multipartite generalization of our constructions is also outlined.