New examples of extremal positive linear maps

Abstract

New families of nonnegative biquadratic forms that have 8, 9 or 10 real zeros in P 2 × P 2 are constructed. These are the first examples with 8, 9 or 10 real zeros. It is known that nonnegative biquadratic forms with finitely many real zeros can have at most 10 zeros; our examples show that the upper bound is obtained. Such biquadratic forms define positive linear maps that are not completely positive. Our constructions are explicit, and moreover we are able to determine which of the examples are extremal.

New examples of entangled states on $\mathbb{C}^3 \otimes \mathbb{C}^3$

“…In what follows we describe our construction of families of entanglement witnesses, which we call "the method of prescribing the zero set", first discovered in [4]. We construct positive maps by extending Choi's approach, described in (2.3), from the real symmetric matrices to Hermitian matrices.…”

“…This paper aims to to shed some light on the geometry of the SEP cone by constructing new entanglement witnesses for states on C 3 ⊗ C 3 . The main contribution is in using the families of positive maps on M sa 3 from [4] to construct explicit examples of new entanglement witnesses and consequently obtain new entangled states. By Choi's isomorphism, this is equivalent to the construction of witnesses of non-entanglement breaking maps.…”

“…In mathemathics, positive maps are key notions in C * algebras (see [21]) and in connection with cone optimization, where positive maps act between real symmetric matrices (following the ideas of Lasserre and Parrilo, polynomial optimization problems can be relaxed and transferred into semidefinite optimization problems [15,3]). The relevance of positive maps in QIT was observed by the Horodecki group [12] and since then there are many examples of positive maps that are not completely positive in the literature (for an overview we refer the interested reader to the references in [4]). It is straightforward to extend examples from M sym 3 (R) to M sa 3 (C) however, the complexifications are in general not extremal.…”

“…This paper deals with the smallest dimension where the partial transposition does not detect all entangled states. Section 2 describes our method of constructing new entanglement witnesses -it builds on the families of positive maps on M sa 3 from [4]. We demonstrate the strength of our method by constructing a 5-parameter family of positive maps that amalgamates all the generalizations of Choi's map in the literature.…”

“…In [4] we constructed new families of nonnegative forms p Ψ on P 2 (C) × P 2 (C) which have 8, 9 or 10 real zeros. This ensures that the associated positive maps Ψ : M sa 3 → M sa 3 are significantly different from the Choi map (2.2) and its generalizations considered in Example 2.2, whose corresponding forms have 7 real zeros.…”