Positive Maps In Matrix Algebras Research Articles

On Kadison-Schwarz Approximation to Positive Maps

We analyze Kadison-Schwarz approximation to positive maps in matrix algebras. This is an analogue of the well known structural physical approximation to positive maps used in entanglement theory. We study several known maps both decomposable (like transposition) and non-decomposable (like Choi map and its generalizations).

On pseudo-stochastic matrices and pseudo-positive maps

Stochastic matrices and positive maps in matrix algebras have proved to be very important tools for analysing classical and quantum systems. In particular they represent a natural set of transformations for classical and quantum states, respectively. Here we introduce the notion of pseudo-stochastic matrices and consider their semigroup property. Unlike stochastic matrices, pseudo-stochastic matrices are permitted to have matrix elements which are negative while respecting the requirement that the sum of the elements of each column is one. They also allow for convex combinations, and carry a Lie group structure which permits the introduction of Lie algebra generators. The quantum analog of a pseudo-stochastic matrix exists and is called a pseudo-positive map. They have the property of transforming a subset of quantum states (characterized by maximal purity or minimal von Neumann entropy requirements) into quantum states. Examples of qubit dynamics connected with ‘diamond’ sets of stochastic matrices and pseudo-positive maps are dealt with.

Spectral properties of circulant positive maps: classical versus quantum**Dedicated to Margarita and Volodia Man’ko for their 150th birthday.

A class of circulant stochastic matrices and positive maps in matrix algebras displaying circulant structure is analyzed. It is shown how the spectral properties of matrices and linear maps are related to positivity conditions. An interesting interplay between the classical and quantum case is revealed.

New Tools for Investigating Positive Maps in Matrix Algebras

We provide a novel tool which may be used to construct new examples of positive maps in matrix algebras (or, equivalently, entanglement witnesses). It turns out that this can be used to prove positivity of several well known maps (such as reduction map, generalized reduction, Robertson map, and many others). Furthermore, we use it to construct a new family of linear maps and prove that they are positive, indecomposable and (nd)optimal.

A class of exposed indecomposable positive maps

Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a class of indecomposable positive maps in the algebra of 2n × 2n complex matrices with n ⩾ 2. It is shown that these maps are exposed and hence define the strongest tool in entanglement theory to discriminate between separable and entangled states.

Exposed positive maps: a sufficient condition

Exposed positive maps in matrix algebras define a dense subset of extremal maps. We provide a sufficient condition for a positive map to be exposed. This is an analogue of a spanning property which guarantees that a positive map is optimal. We analyze a class of decomposable maps for which this condition is also necessary.

Optimal entanglement witnesses from generalized reduction and Robertson maps

We provide a generalization of the reduction and Robertson positive maps in matrix algebras. They give rise to a new class of optimal entanglement witnesses. Their structural physical approximation is analyzed. As a byproduct we provide new examples of PPT (positive partial transpose) entangled states.

Positive maps, doubly stochastic matrices and new family of spectral conditions

We provide partial classification of positive maps in matrix algebras which is based on a family of spectral conditions. This construction recovers many well known examples of positive maps. We displayed interesting connection between positive maps and doubly stochastic matrices.

Geometry of quantum states: New construction of positive maps

We provide a new class of positive maps in matrix algebras. The construction is based on the family of balls living in the space of density matrices of n-level quantum system. This class generalizes the celebrated Choi map and provide a wide family of entanglement witnesses which define a basic tool for analyzing quantum entanglement.

Absolute continuity for operator valued completely positive maps on C∗-algebras

Motivated by applicability to quantum operations, quantum information, and quantum probability, we investigate the notion of absolute continuity for operator valued completely positive maps on C∗-algebras, previously introduced by Parthasarathy [in Athens Conference on Applied Probability and Time Series Analysis I (Springer-Verlag, Berlin, 1996), pp. 34–54]. We obtain an intrinsic definition of absolute continuity, we show that the Lebesgue decomposition defined by Parthasarathy is the maximal one among all other Lebesgue-type decompositions and that this maximal Lebesgue decomposition does not depend on the jointly dominating completely positive map, we obtain more flexible formulas for calculating the maximal Lebesgue decomposition, and we point out the nonuniqueness of the Lebesgue decomposition as well as a sufficient condition for uniqueness. In addition, we consider Radon–Nikodym derivatives for absolutely continuous completely positive maps that, in general, are unbounded positive self-adjoint operators affiliated to a certain von Neumann algebra, and we obtain a spectral approximation by bounded Radon–Nikodym derivatives. An application to the existence of the infimum of two completely positive maps is indicated, and formulas in terms of Choi’s matrices for the Lebesgue decomposition of completely positive maps in matrix algebras are obtained.

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.

A Class of Linear Positive Maps in Matrix Algebras

A class of linear positive, trace preserving maps in Mn is given in terms of affine maps in ℝn2−1 which map the closed unit ball into itself.

A series of absolutely indecomposable positive maps in matrix algebras

We provide examples of positive maps in M n( C) (n ⩾ 4) which cannot be decomposed into a sum of a 2-positive map and a 2-copositive map.

On the geometry of positive maps in matrix algebras

On the geometry of positive maps in matrix algebras