2 Decomposable maps and their relation to PPT states

We present two different descriptions of positive partially transposed (PPT) states. One is based on the theory of positive maps while the second description provides a characterization of PPT states in terms of Hilbert space vectors. Our note is based on our previous results presented in [22], [18], and [20]. 1 Definitions and notations In Quantum Computing a characterization of states with positive partial transposition is important problem (see [13]). Recently, some partial results in this direction were obtained (see [6], and [7]). The aim of this note, based on our previous results (see [22], [18], and [20]), is to present two different complete characterization of PPT states. For the sake of convenience, we provide all necessary preliminaries and set up the notation. Let B(H) be the set of all linear bounded operators on a Hilbert space H. We denote the set of all positive elements of B(H) by B(H) +. A state on B(H) is a linear functional φ : B(H) −→ C such that φ(A) ≥ 0 for every A ∈ B(H) + and φ(1l) = 1, where 1l is the unit of B(H). The set of all states on B(H) is denoted by S B(H). For any subset T ⊂ S H we define the dual cone by T d = {A ∈ B(H) : φ(A) ≥ 0 for every φ ∈ T }. It is easy to check that the definition of a state implies B(H) + ⊂ T d for every T ⊂ S B(H). We say that the family T determines the order of B(H) when T d = B(H) +. 1 A linear map Ψ : B(H 1) −→ B(H 2) is called positive if Ψ(B(H 1) +) ⊂ B(H 2) +. For k ∈ IN we consider a map Ψ k : M k ⊗B(H 1) −→ M k ⊗B(H 2) where M k denotes the algebra of k × k-matrices with complex entries and Ψ k = id M k ⊗ Ψ. We say that Ψ is k-positive if the map Ψ k is positive. The map Ψ is said completely positive when Ψ is k-positive for every k ∈ IN. Let us recall that for a finite dimensional Hilbert space L every state φ on B(L) has the form of φ(A) = Tr (̺A), where ̺ is a uniquely determined density matrix, i.e. an element of B(L) + such that …