Władysław A. Majewski

On a characterization of positive maps

Drawing on results of Choi, Stormer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras B(H) and B(K), where H and K are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on B(H ⊗ K). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone B(H) + ⊗ B(K) + is given.

On nonlinear Koopman's construction

Abstract We show that there exists a possibility for a nonlinear variant of Koopmans construction. This gives a natural way for introducing nonlinear quantum maps. Applications to quantum chaos are indicated.

On the structure of positive maps. II. Low dimensional matrix algebras

We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the differences and similarities between unital positive maps on M2(C) and M3(C). Our main tool stems from classical Grothendieck theorem on tensor product of Banach spaces and is an older and more general version of Choi-Jamiolkowski isomorphism between positive maps and block positive Choi matrices. It takes into account the correct topology on the latter set that is induced by the uniform topology on positive maps. In this setting, we show that in M2(C) case a large class of nice positive maps can be generated from the small set of maps represented by self-adjoint unitaries, 2Px with x maximally entangled vector and p⊗1 with p rank 1 projector. We indicate problems with passing this result to M3(C) case. Among similarities, in both M2(C) and M3(C) cases any unital positive map represented by self-adjoint unitary is unitarily equivalent to the transposition map. Consequently, ...

On Positive Maps, Entanglement and Quantization

We outline the scheme for quantization of classical Banach space results associated with some prototypes of dynamical maps and we describe the quantization of correlations. A relation between these two areas is discussed.

Transformations between quantum states

Abstract General properties and characterization of transformations between quantum states (in the conventional framework of quantum mechanics) are given. Some classes of such transformations are described in detail. Examples of transformations between quantum states in statistical mechanics are given.

Implementability of Liouville Evolution, Koopman and Banach-Lamperti Theorems in Classical and Quantum Dynamics

We extend the concept of implementability of semigroups of evolution operators associated with dynamical systems to quantum case. We show that such an extension can be properly formulated in terms of Jordan morphisms and isometries on non-commutative Lp spaces. We focus our attention on a non-commutative analog of the Banach-Lamperti theorem.