Andrzej Kossakowski

Completely positive dynamical semigroups of N‐level systems

We establish the general form of the generator of a completely positive dynamical semigroup of an N‐level quantum system, and we apply the result to derive explicit inequalities among the physical parameters characterizing the Markovian evolution of a 2‐level system.

Properties of Quantum Markovian Master Equations

In this paper we give an essentially self-contained account of some general structural properties of the dynamics of quantum open Markovian systems. We review some recent results regarding the problem of the classification of quantum Markovian master equations and the limiting conditions under which the dynamical evolution of a quantum open system obeys an exact semigroup law (weak coupling limit and singular coupling limit). We discuss a general form of quantum detailed balance and its relation to thermal relaxation and to microreversibility.

Quantum detailed balance and KMS condition

A definition of detailed balance for quantum dynamical semigroups is given, and its close connection with the KMS condition is investigated.

N‐level systems in contact with a singular reservoir. II

We study a model of an N‐level atom coupled linearly to an infinite free boson bath whose time correlation functions are Gaussian. We prove that, in the limit when the decay time of the correlations of the bath goes to zero, the reduced dynamics of the atom is given by a completely positive dynamical semigroup. By varying the interaction parameters, any such semigroup can be obtained in the limit. We also discuss the formally analogous situation of an N‐level atom whose Hamiltonian contains an external fluctuating Gaussian stationary contribution.

Measures of non-Markovianity: Divisibility versus backflow of information

We analyze two recently proposed measures of non-Markovianity: one based on the concept of divisibility of the dynamical map and the other one based on distinguishability of quantum states. We provide a toy model to show that these two measures need not agree. In addition, we discuss possible generalizations and intricate relations between these measures.

The Bloch-Vector Space for N-Level Systems: the Spherical-Coordinate Point of View

Bloch-vector spaces for N-level systems are investigated from the spherical-coordinate point of view in order to understand their geometrical aspects. We present a characterization of the space by using the spectra of (orthogonal) generators of SU(N). As an application, we find a dual property of the space which provides an overall picture of the space. We also provide three classes of quantum-state representations based on actual measurements and discuss their state-spaces.

On the Structure of Entanglement Witnesses and New Class of Positive Indecomposable Maps

We construct a new class of positive indecomposable maps in the algebra of d x d complex matrices. Each map is uniquely characterized by a cyclic bistochastic matrix. This class generalizes Choi map for d = 3. It provides a new reach family of indecomposable entanglement witnesses which define important tool for investigating quantum entanglement.

Spectral Conditions for Positive Maps

We provide partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions. This construction generalizes the celebrated Choi example of a map which is positive but not completely positive. It is shown how the spectral conditions enable one to construct linear maps on tensor products of matrix algebras which are positive but only on a convex subset of separable elements. Such maps provide basic tools to study quantum entanglement in multipartite systems.

Circulant states with positive partial transpose

We construct a large class of quantum dxd states which are positive under partial transposition (so called PPT states). The construction is based on certain direct sum decomposition of the total Hilbert space displaying characteristic circular structure - that is why we call them circulant states. It turns out that partial transposition maps any such decomposition into another one and hence both original density matrix and its partially transposed partner share similar cyclic properties. This class contains many well-known examples of PPT states from the literature and gives rise to a huge family of completely new states.

A Class of Linear Positive Maps in Matrix Algebras II

We provide a class of linear trace preserving positive maps on matrix algebras which is a generalization of that in [7]. A systematic construction by means of spectra of generators of SU(n) is discussed.