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

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.

References in corpus (3)

• Alternative commutation relations, star products and tomography
• Qubit-portraits of qudit states and quantum correlations
• Bell's inequalities in the tomographic representation

Cited by in corpus (7)

• An introduction to the tomographic picture of quantum mechanics
• On pseudo-stochastic matrices and pseudo-positive maps
• Minimal sets of dequantizers and quantizers for finite-dimensional quantum systems
• Stochastic evolution of finite level systems: classical vs. quantum
• Geometrical Structures for Classical and Quantum Probability Spaces
• Entropic and information inequality for nonlinearly transformed two-qubit X-states
• Discretization of the density matrix as a nonlinear positive map and entanglement