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 describe the quantization of correlations as well. A relation between these two areas is discussed

On Kolgomorov-Sinai entropy and its quantization

In the paper we present the new approach to Kolmogorov-Sinai entropy and its quantization. Our presentation stems from an application of the Choquet theory to the theory of decompositions of states and therefore, it resembles our rigorous description of entanglement of formationComment: 10 page

Does quantum chaos exist? (A quantum Lyapunov exponents approach.)

We shortly review the progress in the domain of deterministic chaos for quantum dynamical systems. With the appropriately extended definition of quantum Lyapunov exponent we analyze various quantum dynamical maps. It is argued that, within Quantum Mechanics, irregular evolution for properly chosen observables can coexist with regular and predictable evolution of states.Comment: Latex, 28 page

A note on Stormer condition for decomposability of positive maps

We present a partial characterization of matrices in M_n(\cA)^+ satisfying the St{\o}rmer condition.Comment: 5 page

Some remarks on separability of states

Several problems concerning separable states are clarified on the basis of Choi's scheme and old Kadison and Tomiyama results. Moreover, we generalize Terhal's construction of positive maps.Comment: 12 page

Quantum correlations; quantum probability approach

This survey gives a comprehensive account of quantum correlations understood as a phenomenon stemming from the rules of quantization. Centered on quantum probability it describes the physical concepts related to correlations (both classical and quantum), mathematical structures, and their consequences. These include the canonical form of classical correlation functionals, general definitions of separable (entangled) states, definition and analysis of quantumness of correlations, description of entanglement of formation, and PPT states. This work is intended both for physicists interested not only in collection of results but also in the mathematical methods justifying them, and mathematicians looking for an application of quantum probability to concrete new problems of quantum theory.Comment: Revised version, Minor improvements. Typos fixe

On the origin of non-decomposable maps

The Radon-Nikodym formalism is used to study the structure of the set of positive maps from $\mathcal{B}(\mathcal{H})$ into itself, where $\mathcal{H}$ is a finite dimensional Hilbert space. In particular, this formalism was employed to formulate simple criteria which ensure that certain maps are non decomposable. In that way, a recipe for construction of non decomposable maps was obtained.Comment: Incorrect example given in subsection 5.5 was remove

On positive maps in quantum information

Using Grothendieck approach to the tensor product of locally convex spaces we review a characterization of positive maps as well as Belavkin-Ohya characterization of PPT states. Moreover, within this scheme, \textit{ a generalization of the idea of Choi matrices for genuine quantum systems will be presented}.Comment: Paper dedicated to the memory of Viacheslav "Slava" Belavkin. arXiv admin note: text overlap with arXiv:1005.394

On a characterization of 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.Comment: a report based on previous result

On decomposability of positive maps between M2M_2 and MnM_n

A map \phi:M_m(\bC)\to M_n(\bC) is decomposable if it is of the form $\phi=\phi_1+\phi_2$ where $\phi_1$ is a CP map while $\phi_2$ is a co-CP map. A partial characterization of decomposability for maps \phi: M_2(\bC) \to M_3(\bC) is given.Comment: 7 page