Characterization of Ppt States and Measures of Entanglement

A detailed characterization of PPT states, both in the Heisenberg and in the Schrödinger picture, is given. Measures of entanglement are defined and discussed in details. Illustrative examples are provided. 1. Preliminaries In this section we compile some basic facts on the theory of positive maps on C∗-algebras. To begin with, let A and B be C∗-algebras (with unit), Ah = {a ∈ A; a = a∗} the set of all selfadjoint elements in A, A+ = {a ∈ Ah; a ≥ 0} the set of all positive elements in A, and S(A) the set of all states on A, i.e. the set of all linear functionals φ on A such that φ(1) = 1 and φ(a) ≥ 0 for any a ∈ A+. In particular (Ah,A) is an ordered Banach space. We say that a linear map α : A → B is positive if α(A+) ⊂ B+. The theory of positive maps on non-commutative algebras can be viewed as a jig-saw-puzzle with pieces whose exact form is not well known. On the other hand, as we address this paper to a readership interested in quantum mechanics and quantum information theory, in this section, we will focus our attention on some carefully selected basic concepts and fundamental results in order to facilitate access to main problems of that theory. Furthermore, the relations between the theory of positive maps and the entanglement problem will be indicated. We begin with a very strong notion of positivity: the so called complete positivity (CP). Namely, a linear map τ : A → B is CP iff (1) τn : Mn(A) → Mn(B); [aij ] 7→ [τ(aij)] is positive for all n. Here, Mn(A) stands for n× n matrices with entries in A. To explain the basic motivation for that concept we need the following notion: an operator state of C∗-algebra A on a Hilbert space K, is a CP map τ : A → B(K), where B(K) stands for the set of all bounded linear operators on K. Having this concept we can recall the Stinespring result, [37], which is a generalization of GNS construction and which was the starting point for a general interest in the concept of complete positivity. Theorem 1. ([37]) For an operator state τ there is a Hilbert space H, a ∗representation (∗-morphism) π : A → B(H) and a partial isometry V : K → H for which (2) τ(a) = V ∗π(a)V. A nice and frequently used criterion for CP can be extracted from Takesaki book [42]: 1 2 W LADYS LAW A. MAJEWSKI, TAKASHI MATSUOKA, AND MASANORI OHYA Criterion 2. Let A and B be C∗-algebras. A linear map φ : A → B is CP if and only if (3) n ∑ i,j=1 y∗ i φ(x ∗ i xj)yj ≥ 0 for every x1, ..., xn ∈ A, y1, ..., yn ∈ B, and every n ∈ N. Up to now we considered linear positive maps on an algebra without entering into the (possible) complexity of the underlying algebra. The situation changes when one is dealing with composed systems (for example in the framework of open system theory). Namely, there is a need to use the tensor product structure. At this point, it is worth citing Takesaki’s remark [42]:“...Unlike the finite dimensional case, the tensor product of infinite dimensional Banach spaces behaves mysteriously.” He had in mind “topological properties of Banach spaces” , i.e.: “ cross norms in the tensor product are highly non-unique.” But from the point of view of composed systems the situation is, even, more mysterious as finite dimensional cases are also obscure. To explain this point, let us consider positive maps defined on the tensor product of two C∗-algebras, τ : A⊗B → A⊗B. But now the question of order is much more complicated. Namely, there are various cones determining the order structure in the tensor product of algebras (cf. [48]) (4) Cinj ≡ (A⊗ B) ⊇, ...,⊇ Cβ ⊇, ...,⊇ Cpro ≡ conv(A ⊗ B) and correspondingly in terms of states (cf [32]) (5) S(A ⊗ B) ⊇, ...,⊇ Sβ ⊇, ...,⊇ conv(S(A) ⊗ S(B)). Here, Cinj stands for the injective cone, Cβ for a tensor cone, while Cpro for the projective cone. The tensor cone Cβ is defined by the property: the canonical bilinear mappings ω : Ah ×Bh → (Ah ⊗Bh, Cβ) and ω∗ : Ah ×B∗ h → (Ah ⊗B∗ h, C∗ β) are positive. The cones Cinj , Cβ , Cpro are different unless either A, or B, or both A and B are abelian (so a finite dimension does not help very much!). This feature is the origin of various positivity concepts for non-commutative composed systems and it was Stinespring who used the partial transposition (transposition tensored with identity map) for showing the difference between Cβ and Cinj and Cpro (see [48], also [21]). Clearly, in dual terms, the mentioned property corresponds to the fact that the set of separable states conv(S(A) ⊗ S(B)) is different from the set of all states and that there are various special subsets of states if both subsystems are truly quantum. In his pioneering work on Banach spaces, Grothendieck [19] observed the links between tensor products and mapping spaces. A nice example of such links was provided by Størmer [38]. To present this result we need a little preparation. Let A denote a norm closed self-adjoint subspace of bounded operators on a Hilbert space K containing the identity operator on K. T will denote the set of trace class operators on B(H). x → x denotes the transpose map of B(H) with respect to some orthonormal basis. The set of all linear bounded (positive) maps φ : A → B(H) will be denoted by B(A,B(H)) (B(A,B(H))+ respectively). Finally, we denote by A ⊙ T the algebraic tensor product of A and T (algebraic tensor product of two vector spaces is defined as its ∗-algebraic structure when the factor CHARACTERIZATION OF PPT STATES AND MEASURES OF ENTANGLEMENT 3 spaces are ∗-algebras; so the topological questions are not considered) and denote by A⊗̂T its Banach space closure under the projective norm defined by (6) ||x|| = inf{ n ∑ i=1 ||ai||||bi||1 : x = n ∑ i=1 ai ⊗ bi, ai ∈ A, bi ∈ T}, where || · ||1 stands for the trace norm. Now, we are in a position to give (see [38]) Lemma 3. There is an isometric isomorphism φ → φ̃ between B(A,B(H)) and (A⊗̂T)∗ given by (7) (φ̃)( n ∑