On a class of $k$-entanglement witnesses
- URL: http://arxiv.org/abs/2104.14058v4
- Date: Wed, 21 Dec 2022 17:07:04 GMT
- Title: On a class of $k$-entanglement witnesses
- Authors: Marcin Marciniak, Tomasz M{\l}ynik, Hiroyuki Osaka
- Abstract summary: Recently, Yang at al. showed that each 2-positive map acting from $mathcalM_3(mathbbC)$ into itself is decomposable.
We construct a positive maps between matrix algebras whose $k$-positivity properties can be easily controlled.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Recently, Yang at al. showed that each 2-positive map acting from
$\mathcal{M}_3(\mathbb{C})$ into itself is decomposable. It is equivalent to
the statement that each PPT state on $\mathbb{C}^3\otimes\mathbb{C}^3$ has
Schmidt number at most 2. It is a generalization of Perez-Horodecki criterion
which states that each PPT state on $\mathbb{C}^2\otimes\mathbb{C}^2$ or
$\mathbb{C}^2\otimes\mathbb{C}^3$ has Schmidt rank 1 i.e. is separable. Natural
question arises whether the result of Yang at al. stays true for PPT states on
$\mathbb{C}^3\otimes\mathbb{C}^4$. This question can be considered also in
higher dimensions. We construct a positive maps which is suspected for being a
counterexample. More generally, we provide a class of positive maps between
matrix algebras whose $k$-positivity properties can be easily controlled.
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