scholarly journals Approximating the set of separable states using the positive partial transpose test

2010 ◽  
Vol 51 (4) ◽  
pp. 042202 ◽  
Author(s):  
Salman Beigi ◽  
Peter W. Shor
Keyword(s):  
Positive Partial Transpose ◽  
Separable States ◽  
Partial Transpose

scholarly journals The Set of Separable States has no Finite Semidefinite Representation Except in Dimension $$3\times 2$$

2021 ◽  
Author(s):  
Hamza Fawzi
Keyword(s):  
Hilbert Space ◽  
Semidefinite Programming ◽  
Finite Size ◽  
Semialgebraic Sets ◽  
Simple Description ◽  
Positive Partial Transpose ◽  
Separable States ◽  
Semialgebraic Set ◽  
Partial Transpose ◽  
Semidefinite Representation

AbstractGiven integers $$n \ge m$$ n ≥ m , let $$\text {Sep}(n,m)$$ Sep ( n , m ) be the set of separable states on the Hilbert space $$\mathbb {C}^n \otimes \mathbb {C}^m$$ C n ⊗ C m . It is well-known that for $$(n,m)=(3,2)$$ ( n , m ) = ( 3 , 2 ) the set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values of n and m the set $$\text {Sep}(n,m)$$ Sep ( n , m ) has no semidefinite programming description of finite size. As $$\text {Sep}(n,m)$$ Sep ( n , m ) is a semialgebraic set this provides a new counterexample to the Helton–Nie conjecture, which was recently disproved by Scheiderer in a breakthrough result. Compared to Scheiderer’s approach, our proof is elementary and relies only on basic results about semialgebraic sets and functions.

scholarly journals Global Geometric Difference Between Separable and Positive Partial Transpose States

2014 ◽  
Vol 21 (04) ◽  
pp. 1450009
Author(s):  
Kil-Chan Ha ◽  
Seung-Hyeok Kye
Keyword(s):  
Interior Point ◽  
Convex Set ◽  
Convex Combination ◽  
Extreme Points ◽  
Positive Partial Transpose ◽  
Separable States ◽  
Positive Maps ◽  
Geometric Difference ◽  
Partial Transpose ◽  
Ppt States

In the convex set of all 3 ⊗ 3 states with positive partial transposes, we show that one can take two extreme points whose convex combinations belong to the interior of the convex set. Their convex combinations may be even in the interior of the convex set of all separable states. In general, we need at least mn extreme points to get an interior point by their convex combination, for the case of the convex set of all m ⊗ n separable states. This shows a sharp distinction between PPT states and separable states. We also consider the same questions for positive maps and decomposable maps.

scholarly journals Postponing Distillability Sudden Death in a Correlated Dephasing Channel

Entropy ◽  
2020 ◽  
Vol 22 (8) ◽  
pp. 827
Author(s):  
Guanghao Xue ◽  
Liang Qiu
Keyword(s):  
Sudden Death ◽  
Quantum Channel ◽  
Positive Partial Transpose ◽  
Partial Correlations ◽  
Partial Transpose ◽  
Correlated Channel

We investigated the dynamics of a two-qutrit system in a correlated quantum channel. The partial correlations between consecutive actions of the channel can effectively postpone the phenomenon of distillability sudden death (DSD) and broaden the range of the time cutoff that indicates entanglement of the positive partial transpose states. Particularly, the negativity of the system will revive and DSD will disappear in the fully correlated channel.

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