arXiv:1406.4954

Indecomposability of entanglement witnesses constructed from any permutations

Xiaofei Qi, Jinchuan Hou

Published Jun 19, 2014• 0 citations

Abstract

Let $n\geq 2$ and $Φ_{n,t,π}: M_n({\mathbb C}) \rightarrow M_n({\mathbb C})$ be a linear map defined by $Φ_{n,t,π}(A)=(n-t)\sum_{i=1}^nE_{ii}AE_{ii}+t\sum_{i=1}^nE_{i,π(i)}AE_{i,π(i)}^†-A$, where $0\leq t\leq n$, $E_{ij}$s are the matrix units and $π$ is a non-identity permutation of $(1,2,\cdots,n)$. Denote by $\{{ F}_s: s=1,2\ldots, k\}$ the set of all minimal cycles of $π$ and $l(π)=\max\{\# { F}_s: s=1,2,\ldots,k\}$ the length of $π$. It is shown that the Hermitian matrix $W_{n,t,π}$ induced by $Φ_{n,t,π}$ is an indecomposable entanglement witness if and only if $π^2\not={\rm id}$ (the identity permutation) and $0

DOI: 10.48550/arxiv.1406.4954

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arXiv:1406.4954

Indecomposability of entanglement witnesses constructed from any permutations

Xiaofei Qi, Jinchuan Hou

Published Jun 19, 2014• 0 citations

arXiv HTML PDF

Abstract

Let

n\geq 2

and

Φ_{n,t,π}: M_n({\mathbb C}) \rightarrow M_n({\mathbb C})

be a linear map defined by

Φ_{n,t,π}(A)=(n-t)\sum_{i=1}^nE_{ii}AE_{ii}+t\sum_{i=1}^nE_{i,π(i)}AE_{i,π(i)}^†-A

, where

0\leq t\leq n

E_{ij}

s are the matrix units and

π

is a non-identity permutation of

(1,2,\cdots,n)

. Denote by

\{{ F}_s: s=1,2\ldots, k\}

the set of all minimal cycles of

π

and

l(π)=\max\{\# { F}_s: s=1,2,\ldots,k\}

the length of

π

. It is shown that the Hermitian matrix

W_{n,t,π}

induced by

Φ_{n,t,π}

is an indecomposable entanglement witness if and only if

π^2\not={\rm id}

(the identity permutation) and $0

DOI: 10.48550/arxiv.1406.4954

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Katarzyna Siudzińska

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Katarzyna Siudzińska

arXiv:2202.04046Feb 8, 20220 citations

Let

n\geq 2

and

Φ_{n,t,π}: M_n({\mathbb C}) \rightarrow M_n({\mathbb C})

be a linear map defined by

Φ_{n,t,π}(A)=(n-t)\sum_{i=1}^nE_{ii}AE_{ii}+t\sum_{i=1}^nE_{i,π(i)}AE_{i,π(i)}^†-A

, where

0\leq t\leq n

E_{ij}

s are the matrix units and

π

is a non-identity permutation of

(1,2,\cdots,n)

. Denote by

\{{ F}_s: s=1,2\ldots, k\}

the set of all minimal cycles of

π

and

l(π)=\max\{\# { F}_s: s=1,2,\ldots,k\}

the length of

π

. It is shown that the Hermitian matrix

W_{n,t,π}

induced by

Φ_{n,t,π}

is an indecomposable entanglement witness if and only if

π^2\not={\rm id}

(the identity permutation) and $0

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})

\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.