Operational criterion and constructive checks for the separability of low-rank density matrices

Horodecki, Pawel, Lewenstein, Maciej, Vidal, Guifre and Cirac, Ignacio (2000) Operational criterion and constructive checks for the separability of low-rank density matrices. Physical Review a, 62 3: . doi:10.1103/PhysRevA.62.032310

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Author Horodecki, Pawel
Lewenstein, Maciej
Vidal, Guifre
Cirac, Ignacio
Title Operational criterion and constructive checks for the separability of low-rank density matrices
Journal name Physical Review a   Check publisher's open access policy
ISSN 1050-2947
1094-1622
Publication date 2000-09-01
Sub-type Article (original research)
DOI 10.1103/PhysRevA.62.032310
Open Access Status File (Publisher version)
Volume 62
Issue 3
Total pages 10
Place of publication College Park, MD, United States
Publisher American Physical Society
Language eng
Formatted abstract
We consider low-rank density operators ϱ supported on a M×N Hilbert space for arbitrary M and N (M<~N), and with a positive partial transpose (PPT) ϱTA>~0. For rank r(ϱ)<~N we prove that having a PPT is necessary and sufficient for ϱ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank-3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of ϱ and ϱTA satisfies r(ϱ)+r(ϱTA)<~2MN−M−N+2. This separability condition has the form of a constructive check, thus also providing a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.
Keyword Bound Entanglement
Mixed States
Quantum Cryptography
Noisy Channels
Distillation
Q-Index Code C1
Q-Index Status Provisional Code
Institutional Status Non-UQ

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Mathematics and Physics
 
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