Related papers: Spectral conditions for positive maps
Physical transformations are described by linear maps that are completely positive and trace preserving (CPTP). However, maps that are positive (P) but not completely positive (CP) are instrumental to derive separability/entanglement…
A class of unital qubit maps displaying diagonal unitary and orthogonal symmetries is analyzed. Such maps already found a lot applications in quantum information theory. We provide a complete characterization of this class of maps showing…
We study learning of indexed families from positive data where a learner can freely choose a hypothesis space (with uniformly decidable membership) comprising at least the languages to be learned. This abstracts a very universal learning…
If a tuple of matrices has a common invariant subspace, its projective joint spectrum has an algebraic component. In general, the converse is not true, and there might be algebraic components in the projective joint spectrum without…
The Choi representation of completely positive (CP) maps, i.e. quantum channels is often used in the context of quantum information and computation as it is easy to work with. It is a correspondence between CP maps and quantum states also…
Quantum states that remain separable (i.e., not entangled) under any global unitary transformation are known as absolutely separable and form a convex set. Despite extensive efforts, the complete characterization of this set remains largely…
We present a general scheme that allows for construction of scalar separability criteria from positive but not completely positive maps. The concept is based on a decomposition of every positive map $\Lambda$ into a difference of two…
Algebras generated by strictly positive matrices are described up to similarity, including the commutative, simple, and semisimple cases. We provide sufficient conditions for some block diagonal matrix algebras to be generated by a set of…
We use open quantum system techniques to construct one-parameter semigroups of positive maps and apply them to study the entanglement properties of a class of 16-dimensional density matrices, representing states of a 4x4 bipartite system.
Linear maps preserving pure states of a quantum system of any dimension are characterized. This is then used to establish a structure theorem for linear maps that preserve separable pure states in multipartite systems. As an application, a…
Maps that are not completely positive (CP) are often useful to describe the dynamics of open systems. An apparent violation of complete positivity can occur because there are prior correlations of the principal system with the environment,…
There are two ways to turn a categorical model for pure quantum theory into one for mixed quantum theory, both resulting in a category of completely positive maps. One has quantum systems as objects, whereas the other also allows classical…
In this paper, we consider all possible variants of Choi matrices of linear maps, and show that they are determined by non-degenerate bilinear forms on the domain space. We will do this in the setting of finite dimensional vector spaces. In…
Two kinds of maps that describe evolution of states of a subsystem coming from dynamics described by a unitary operator for a larger system, maps defined for fixed mean values and maps defined for fixed correlations, are found to be quite…
We use a new idea that emerged in the examination of exposed positive maps between matrix algebras to investigate in more detail the difference between positive maps on $M_2(C)$ and $M_3(C)$. Our main tool stems from classical Grothendieck…
A partial description of the structure of positive unital maps $\phi: M_2(\bC) \to M_{n+1}(\bC)$ ($n\geq 2$) is given.
A spectrahedron is the positivity region of a linear matrix pencil and thus the feasible set of a semidefinite program. We propose and study a hierarchy of sufficient semidefinite conditions to certify the containment of a spectrahedron in…
We present a spectral analysis for matrix scaling and operator scaling. We prove that if the input matrix or operator has a spectral gap, then a natural gradient flow has linear convergence. This implies that a simple gradient descent…
In this thesis work, we have studied the role of positive and completely positive maps in detecting entanglement.
Entanglement of states and Complete Positivity of maps are concepts that have achieved physical importance with the recent growth of quantum information science. They are however mathematically relevant whenever tensor products of complex…