{
  "id": "2104.14058",
  "version": "v1",
  "published": "2021-04-29T00:46:58.000Z",
  "updated": "2021-04-29T00:46:58.000Z",
  "title": "On a class of $k$-entanglement witnesses",
  "authors": [
    "Marcin Marciniak",
    "Tomasz Młynik",
    "Hiroyuki Osaka"
  ],
  "comment": "7 pages",
  "categories": [
    "quant-ph"
  ],
  "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 $\\mathbb{C}^3\\otimes\\mathbb{C}^4$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-04-29T00:46:58.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "entanglement witnesses",
      "ppt state",
      "natural question arises",
      "positive maps",
      "positivity properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}