Role Of Partial Transpose And Generalized Choi Maps In Quantum Dynamical Semigroups Involving Separable And Entangled States

Abstract

Abstract. Power symmetric matrices defined 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.

Entanglement properties of positive operators with ranges in completely entangled subspaces

“…Any state ρ with support in a QCES with specified product vectors, say P has the form ξ p ξ |ξ⟩⟨ξ| with |ξ⟩ varying over P, all p ξ ≥ 0 with p ξ = 1. One can see such instances, for example, see [24,Remark 2.9]. Members of P may or may not be mutually orthogonal.…”

“…In fact, given an arbitrary state in an arbitrary bipartite quantum system the problem of checking whether the state is separable or entangled is in general a NP-hard problem [12,13,11]. Multiple different methods using various mathematical techniques have been discussed in the paper [24]. One such condition used for checking entanglement is called range criteria, which is as follows.…”