13 research outputs found
Geometry of entanglement witnesses parameterized by SO(3) group
We characterize a set of positive maps in matrix algebra of 4x4 complex
matrices. Equivalently, we provide a subset of entanglement witnesses
parameterized by the rotation group SO(3). Interestingly, these maps/witnesses
define two intersecting convex cones in the 3-dimensional parameter space. The
existence of two cones is related to the topological structure of the
underlying orthogonal group. We perform detailed analysis of the corresponding
geometric structure.Comment: 10 page
On classical and quantum liftings
We analyze the procedure of lifting in classical stochastic and quantum
systems. It enables one to `lift' a state of a system into a state of
`system+reservoir'. This procedure is important both in quantum information
theory and the theory of open systems. We illustrate the general theory of
liftings by a particular class related to so called circulant states.Comment: 25 page
On multipartite invariant states I. Unitary symmetry
We propose a natural generalization of bipartite Werner and isotropic states
to multipartite systems consisting of an arbitrary even number of d-dimensional
subsystems (qudits). These generalized states are invariant under the action of
local unitary operations. We study basic properties of multipartite invariant
states: separability criteria and multi-PPT conditions.Comment: 9 pages; slight correction
Spectral conditions for positive maps
We provide a partial classification of positive linear maps in matrix algebras which is based on a family of spectral conditions. This construction generalizes 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
Positive maps, positive polynomials and entanglement witnesses
We link the study of positive quantum maps, block positive operators, and
entanglement witnesses with problems related to multivariate polynomials. For
instance, we show how indecomposable block positive operators relate to
biquadratic forms that are not sums of squares. Although the general problem of
describing the set of positive maps remains open, in some particular cases we
solve the corresponding polynomial inequalities and obtain explicit conditions
for positivity.Comment: 17 pages, 1 figur
A class of positive atomic maps
We construct a new class of positive indecomposable maps in the algebra of `d x d' complex matrices. These maps are characterized by the `weakest' positivity property and for this reason they are called atomic. This class provides a new reach family of atomic entanglement witnesses which define important tool for investigating quantum entanglement. It turns out that they are able to detect states with the `weakest' quantum entanglement
From Markovian semigroup to non-Markovian quantum evolution
We provided a class of legitimate memory kernels leading to completely positive trace-preserving dynamical maps. Our construction is based on a simple normalization procedure. Interestingly, when applied to a classical system it gives rise to semi-Markov evolution. Therefore, it may be considered as a quantum version of semi-Markov dynamics which is much more general than Markovian dynamics