Criteria of positivity for linear maps constructed from permutation pairs

Abstract

In this paper, we show that a $D$-type map ${\Phi }_D:M_n\to M_n$ with $D=(n-2)I_n+P_{{\pi }_1}+P_{{\pi }_2}$ induced by a pair $\{{\pi }_1,{\pi }_2\}$ of permutations of $(1,2,...,n)$ is positive if $\{{\pi }_1,{\pi }_2\}$ has property (C). The property (C) is characterized for $\{{\pi }_1,{\pi }_2\}$, and an easy criterion is given for the case that ${\pi }_1={\pi }^p$ and ${\pi }_2={\pi }^q$, where $\pi$ is the permutation defined by $\pi (i)=i+1$ mod $n$ and $1\le p<q\le n$.

References in corpus (2)

• Spectral conditions for positive maps
• Positive finite rank elementary operators and characterizing entanglement of states