Role of Partial Transpose and Generalized Choi maps in Quantum Dynamical Semigroups involving Separable and Entangled States

TL;DR

This paper explores the use of partial transpose and generalized Choi maps to analyze quantum dynamical semigroups, providing new criteria for separability and entanglement levels in quantum states.

Contribution

It introduces a Trichotomy Theorem based on generalized Choi maps to distinguish different levels of entanglement breaking in quantum dynamical semigroups.

Findings

Established a new criterion for PPT states using power symmetric matrices.

Proved a Trichotomy Theorem classifying entanglement levels.

Provided examples illustrating the application of generalized Choi maps.

Abstract

Power symmetric matrices defned and studied by R. Sinkhorn (1981) and their generalization by R.B. Bapat, S.K. Jain and K. Manjunatha Prasad (1999) have been utilized to give positive block matrices with trace one possessing positive partial transpose, the so-called PPT states. Another method to construct such PPT states is given, it uses the form of a matrix unitarily equivalent to its transpose obtained by S.R. Garcia and J.E. Tener (2012). Evolvement or suppression of separability or entanglement of various levels for a quantum dynamical semigroup of completely positive maps has been studied using Choi-Jamiolkowsky matrix of such maps and the famous Horodecki's criteria (1996). A Trichotomy Theorem has been proved, and examples have been given that depend mainly on generalized Choi maps and clearly distinguish the levels of entanglement breaking.