semidefinite representation
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Author(s):  
Hamza Fawzi

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.


2015 ◽  
Vol 465 ◽  
pp. 347-362 ◽  
Author(s):  
J. William Helton ◽  
Jiawang Nie ◽  
Jeremy S. Semko

2013 ◽  
Vol 7 (6) ◽  
pp. 1387-1397 ◽  
Author(s):  
Peter J. C. Dickinson ◽  
Gabriele Eichfelder ◽  
Janez Povh

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