Indecomposable Exposed Positive Bi-linear Maps Between Two by Two Matrices
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Indecomposable Exposed Positive Bi-linear Maps Between Two by Two Matrices Seung-Hyeok Kye1
Received: 19 September 2017 / Revised: 28 December 2017 / Accepted: 2 January 2018 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018
Abstract Positive bi-linear maps between matrix algebras play important roles to detect tri-partite entanglement by the duality between bi-linear maps and tri-tensor products. We exhibit indecomposable positive bi-linear maps between 2 × 2 matrices which generate extreme rays in the cone of all positive bi-linear maps. In fact, they are exposed, and so detect entanglement of positive partial transpose (PPT) whose volume is nonzero. Keywords Positive multi-linear maps · Exposed · Indecomposable · Separable · Partial transpose Mathematics Subject Classification (2010) 46L07 · 15A30 · 81P15
1 Introduction Positive linear maps in matrix algebras are indispensable to distinguish entanglement from separability, by the duality [8, 16] between bi-partite separable states and positive linear maps. The convex cone consisting of positive linear maps is very complicated even in the cases when the domain and range are low-dimensional matrices, and its whole structure is far from being understood. Typical examples of positive maps from m × m matrices into n × n matrices are of the forms X → V ∗ XV ,
X → W ∗ X t W,
(1)
of X and m × n matrices V and W . A sum of the first (respectively with the transpose second) type is called completely positive (respectively completely copositive), and a sum Xt
Seung-Hyeok Kye
[email protected] 1
Department of Mathematics and Institute of Mathematics, Seoul National University, Seoul 151-742, Korea
S.-H. Kye
of them is said to be decomposable. It is known that every positive map from Mm into Mn is decomposable if and only if (m, n) is (2, 2), (2, 3) or (3, 2) [4, 25, 26], and many examples of indecomposable positive linear maps are known. See the survey article [18]. Recall that a positive map is said to be exposed (respectively extremal) if it generates an exposed (respectively extreme) ray in the convex cone of all positive linear maps. Among various examples of indecomposable positive linear maps, exposed maps attract special interests in quantum information theory. First of all, indecomposable exposed maps detect PPT entanglement of nonzero volume. Furthermore, exposed maps are enough to determine separability, by the classical Straszewicz’s theorem: exposed points form a dense subset in the set of all extreme points. See [23, Theorem 18.6]. In the case of decomposable maps, the maps in (1) are known to be exposed [20, 28], and there are no other exposed decomposable maps. Nevertheless, there are very few known examples of exposed indecomposable positive linear maps in the literature. Some variants [2] of the Choi map [5] on M3 had been proved to be exposed in [9]. Note that the Choi map itself is not exposed, even though it is extremal. On the other hand, Robertson’s map [21, 22] on M4 wa
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