Please use this identifier to cite or link to this item:
Title: Properties and Construction of Extreme Bipartite States Having Positive Partial Transpose
Authors: Chen, L. 
Doković, D.Ž.
Issue Date: Oct-2013
Citation: Chen, L., Doković, D.Ž. (2013-10). Properties and Construction of Extreme Bipartite States Having Positive Partial Transpose. Communications in Mathematical Physics 323 (1) : 241-284. ScholarBank@NUS Repository.
Abstract: We consider a bipartite quantum system HA ⊗ HB with M = Dim HA and N = Dim HB. We study the set ε of extreme points of the compact convex set of all states having positive partial transpose (PPT) and its subsets εr = {ρ ∈ ε: rank ρ = 1}. Our main results pertain to the subsets εr M, N of εr consisting of states whose reduced density operators have ranks M and N, respectively. The set ε1 is just the set of pure product states. It is known that εr M, N = ∅ for 1 < r ≤ min(M, N) and for r = MN. We prove that also εMN-1 M, N = ∅. Leinaas, Myrheim and Sollid have conjectured that εM+N-2 M, N ≠ ∅ for all M, N > 2 and that εr M, N = ∅ for 1 ≤ r ≤ M + N - 2. We prove the first part of their conjecture. The second part is known to hold when min(M, N) = 3 and we prove that it holds also when min(M, N) = 4. This is a consequence of our result that εN+1 M, N = ∅ if M, N > 3. We introduce the notion of "good" states, show that all pure states are good and give a simple description of the good separable states. For a good state ρ ∈ εM+N-2 M, N, we prove that the range of ρ contains no product vectors and that the partial transpose of ρ has rank M + N - 2 as well. In the special case M = 3, we construct good 3 × N extreme states of rank N + 1 for all N ≥ 4. © 2013 Springer-Verlag Berlin Heidelberg.
Source Title: Communications in Mathematical Physics
ISSN: 00103616
DOI: 10.1007/s00220-013-1770-6
Appears in Collections:Staff Publications

Show full item record
Files in This Item:
There are no files associated with this item.


checked on Jun 29, 2022


checked on Jun 29, 2022

Page view(s)

checked on Jun 23, 2022

Google ScholarTM



Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.