Semigroup of positive maps for qudit states and entanglement in tomographic probability representation

TL;DR

This paper explores the structure of positive maps for qudit states using stochastic matrices, their relation to Lie groups, and their application in describing entanglement and Bell inequalities within a tomographic probability framework.

Contribution

It introduces a semigroup structure for positive maps on qudits, connects them to Lie groups, and applies this to analyze entanglement and Bell inequalities in quantum tomography.

Findings

Positive maps form semigroups with dense intersections with Lie groups.

Qudit states are described by spin tomograms on a simplex.

Entangled states and Bell inequalities are analyzed using stochastic semigroup properties.

Abstract

Stochastic and bistochastic matrices providing positive maps for spin states (for qudits) are shown to form semigroups with dense intersection with the Lie groups $IGL(n, \mathbb{R})$ and $GL(n, \mathbb{R})$ respectively. The density matrix of a qudit state is shown to be described by a spin tomogram determined by an orbit of the bistochastic semigroup acting on a simplex. A class of positive maps acting transitively on quantum states is introduced by relating stochastic and quantum stochastic maps in the tomographic setting. Finally, the entangled states of two qubits and Bell inequalities are given in the framework of the tomographic probability representation using the stochastic semigroup properties.