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Journal of linear and topological algebra

A class of 2-positive entanglement witness in $\mathcal{B}(\mathcal_3(\mathbb{C}), \mathcal_4(\mathbb{C}))$

C. A. Winda

Published Sep 6, 2025• 0 citations
Publisher

Abstract

We give an affirmative answer to the question posed by Yang et al. on the existence of indecomposable 2-positive linear maps from $\mathcal{M}_3(\mathbb{C})$ to $\mathcal{M}_4(\mathbb{C})$. In this study, we construct an explicit class of linear maps in $\mathcal{B}(\mathcal{M}_3(\mathbb{C}), \mathcal{M}_4(\mathbb{C}))$ that are positive, yet fail to be completely positive or completely copositive. Furthermore, we demonstrate that the associated Choi matrices of these maps serve as entanglement witnesses.

DOI
10.71483/jlta.2025.1217080

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We give an affirmative answer to the question posed by Yang et al. on the existence of indecomposable 2-positive linear maps from M3(C)\mathcal{M}_3(\mathbb{C})M3​(C) to M4(C)\mathcal{M}_4(\mathbb{C})M4​(C). In this study, we construct an explicit class of linear maps in B(M3(C),M4(C))\mathcal{B}(\mathcal{M}_3(\mathbb{C}), \mathcal{M}_4(\mathbb{C}))B(M3​(C),M4​(C)) that are positive, yet fail to be completely positive or completely copositive. Furthermore, we demonstrate that the associated Choi matrices of these maps serve as 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.
Optimality of generalized Choi maps in M3M_3M3​
A family of linear positive maps in the algebra of 3×33 \times 33×3 complex matrices proposed recently in Bera et al. arXiv:2212.03807 is further analyzed. It provides a generalization of a seminal Choi nondecomposable extremal map in M3M_3M3​. We investigate when generalized Choi maps are optimal, i.e. cannot be represented as a sum of positive and completely positive maps. This property is weaker than extremality, however, it turns out that it plays a key role in detecting quantum entanglement.
We examine various notions related to the optimality for entanglement witnesses arising from Choi-type positive linear maps. We found examples of optimal entanglement witnesses which are nondecomposable but are not ``nondecomposable optimal entanglement witnesses'' in the sense of Lewenstein, Kraus, Cirac, and Horodecki [Phys. Rev. A 62, 052310 (2000)]. We suggest using the terms PPTES witness and optimal PPTES witness in place of ``nondecomposable entanglement witness'' and ``nondecomposable optimal entanglement witnesses'' in order to avoid possible confusion. Here, PPTES refers entangled states with positive partial transposes. We also found examples of nonextremal optimal entanglement witnesses which are indecomposable.