dorsal/arxiv
View SchemaThe Jamio{\l}kowski isomorphism and a conceptionally simple proof for the correspondence between vectors having Schmidt number $k$ and $k$-positive maps
| Authors | Kedar S. Ranade, Mazhar Ali |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0702255 |
| URL | https://arxiv.org/abs/quant-ph/0702255 |
| DOI | 10.1007/s11080-007-9062-2 |
| Journal | Open Sys. Inf. Dyn. 14 (No. 4, Dec. 2007), 371 - 378 |
Abstract
Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and more generally as witnesses for states having Schmidt number not greater than k. It is known that such witnesses are related to k-positive maps. In this article we propose a new proof for the correspondence between vectors having Schmidt number k and k-positive maps using Jamiolkowski's criterion for positivity of linear maps; to this aim, we also investigate the precise notion of the term "Jamiolkowski isomorphism". As consequences of our proof we get the Jamiolkowski criterion for complete positivity, and we find a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act.
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"abstract": "Positive maps which are not completely positive are used in quantum\ninformation theory as witnesses for convex sets of states, in particular as\nentanglement witnesses and more generally as witnesses for states having\nSchmidt number not greater than k. It is known that such witnesses are related\nto k-positive maps. In this article we propose a new proof for the\ncorrespondence between vectors having Schmidt number k and k-positive maps\nusing Jamiolkowski\u0027s criterion for positivity of linear maps; to this aim, we\nalso investigate the precise notion of the term \"Jamiolkowski isomorphism\". As\nconsequences of our proof we get the Jamiolkowski criterion for complete\npositivity, and we find a special case of a result by Choi, namely that\nk-positivity implies complete positivity, if k is the dimension of the smaller\none of the Hilbert spaces on which the operators act.",
"arxiv_id": "quant-ph/0702255",
"authors": [
"Kedar S. Ranade",
"Mazhar Ali"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s11080-007-9062-2",
"journal_ref": "Open Sys. Inf. Dyn. 14 (No. 4, Dec. 2007), 371 - 378",
"title": "The Jamio{\\l}kowski isomorphism and a conceptionally simple proof for the correspondence between vectors having Schmidt number $k$ and $k$-positive maps",
"url": "https://arxiv.org/abs/quant-ph/0702255"
},
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