Stable Subspaces of Positive Maps of Matrix Algebras

Abstract

We study stable subspaces of positive extremal maps of finite dimensional matrix algebras that preserve trace and matrix identity (so-called bistochastic maps). We have established the existence of the isometric-sweeping decomposition for such maps. As the main result of the paper, we have shown that all extremal bistochastic maps acting on the algebra of matrices of size 3x3 fall into one of the three possible categories, depending on the form of the stable subspace of the isometric-sweeping decomposition. Our example of an extremal atomic positive map seems to be the first one that handles the case of that subspace being non-trivial. Lastly, we compute the entanglement witness associated with the extremal map and specify a large family of entangled states detected by it.

Merging of positive maps: A construction of various classes of positive maps on matrix algebras

“…It was shown by Miller and Olkiewicz in [21] that the map (3.2) is a bistochastic exposed nondecomposable positive map. This is a basic example.…”

“…It was proved in [20] that maps (1.1) are exposed. Further examples of exposed maps are given in [8,11,4,5,32,21]. Geometric approach to exposed maps was presented in [18].…”

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

“…The density matrix has the Schmidt number at least r + 1 if and only if there exists an r-positive linear map Λ : M n → M n such that (I ⊗ Λ)(ρ) ≥ 0. Remarkable progress on the topic has been made in recent years as can be gauged from ( [17], [19], [20], [21], [22], [23], [24], [44], [52], [55], [58], [60], [62], [63], [76], [85]) The name "partially entanglement breaking " has been associated with maps having Schmidt number < n by some authors. We shall not go into details in this paper.…”

“…See ( [12], [13], [14], [15]) for more details. One may observe this tendency in [11], [12], [14], [15], [17], [18], [19], [20], [21], [22], [23], [24], [26], [36], [38], [45], [47], [56], [57], [62], [66], [73], [74], [76], [77], [78], [89], for instance. Such maps have been tested for more properties like separability, entanglement and Schmidt numbers and used to construct Entanglement witnesses, Entanglement breaking or partially entanglement breaking channels by some of them.…”

Generalizing Choi-Like Maps

“…Although there is no systematic prescription of indecomposable maps, there are numerous examples scattered across literature [19,16,17,18,24,7,8,9,23,5,15,14]. An interesting approach to generalization of (1.1) was presented in [1].…”