The Set of Separable States has no Finite Semidefinite Representation Except in Dimension 3 × 2
Communications in Mathematical Physics
Springer Berlin Heidelberg
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Fawzi, H. (2021). The Set of Separable States has no Finite Semidefinite Representation Except in Dimension 3 × 2. Communications in Mathematical Physics, 386 (3), 1319-1335. https://doi.org/10.1007/s00220-021-04163-2
Abstract: Given integers n≥m, let Sep(n, m) be the set of separable states on the Hilbert space Cn⊗Cm. It is well-known that for (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 Sep(n, m) has no semidefinite programming description of finite size. As 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.
External DOI: https://doi.org/10.1007/s00220-021-04163-2
This record's URL: https://www.repository.cam.ac.uk/handle/1810/326673