scholarly journals A Decomposition Theorem for Positive Maps, and the Projection Onto a Spin Factor

2016 ◽  
Vol 118 (1) ◽  
pp. 106 ◽  
Author(s):  
Erling Størmer
Keyword(s):  
Decomposition Theorem ◽  
Positive Map ◽  
Matrix Algebras ◽  
Spin Factor ◽  
Positive Maps

It is shown that each positive map between matrix algebras is the sum of a maximal decomposable map and an atomic map which is both optimal and co-optimal. The result is studied in detail for the projection onto a spin factor.

scholarly journals Indecomposable Positive Maps in Matrix Algebras

1988 ◽  
Vol 31 (3) ◽  
pp. 308-317 ◽  
Author(s):  
Kôtarô Tanahashi ◽  
Jun Tomiyama
Keyword(s):  
Positive Map ◽  
Matrix Algebras ◽  
Positive Maps

AbstractWe prove that Choi's map in M3 cannot be written as the sum of a 2-positive map and a 2-copositive map. We also provide other examples of positive maps in Mn which cannot be written as the sum of an n-positive map and a 2-copositive map.

scholarly journals Stable Subspaces of Positive Maps of Matrix Algebras

2015 ◽  
Vol 22 (02) ◽  
pp. 1550011 ◽  
Author(s):  
Marek Miller ◽  
Robert Olkiewicz
Keyword(s):  
Large Family ◽  
Entangled States ◽  
Entanglement Witness ◽  
Positive Map ◽  
Matrix Algebras ◽  
Finite Dimensional ◽  
Positive Maps ◽  
Matrix Identity ◽  
Stable Subspace ◽  
Algebra Of Matrices

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 3×3 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.

scholarly journals New Tools for Investigating Positive Maps in Matrix Algebras

2013 ◽  
Vol 71 (2) ◽  
pp. 163-175 ◽  
Author(s):  
Justyna P. Zwolak ◽  
Dariusz Chruściński
Keyword(s):  
Matrix Algebras ◽  
Positive Maps

scholarly journals ON THE INDEX AND DILATIONS OF COMPLETELY POSITIVE SEMIGROUPS

1999 ◽  
Vol 10 (07) ◽  
pp. 791-823 ◽  
Author(s):  
WILLIAM ARVESON
Keyword(s):  
Bounded Operator ◽  
Positive Semigroup ◽  
Completely Positive Maps ◽  
Completely Positive ◽  
Positive Semigroups ◽  
Matrix Algebras ◽  
Positive Maps ◽  
Completely Positive Semigroups ◽  
Numerical Index ◽  
Completely Positive Semigroup

It is known that every semigroup of normal completely positive maps P = {Pt:t≥ 0} of ℬ(H), satisfying Pt(1) = 1 for every t ≥ 0, has a minimal dilation to an E0 acting on ℬ(K) for some Hilbert space K⊇H. The minimal dilation of P is unique up to conjugacy. In a previous paper a numerical index was introduced for semigroups of completely positive maps and it was shown that the index of P agrees with the index of its minimal dilation to an E0-semigroup. However, no examples were discussed, and no computations were made. In this paper we calculate the index of a unital completely positive semigroup whose generator is a bounded operator [Formula: see text] in terms of natural structures associated with the generator. This includes all unital CP semigroups acting on matrix algebras. We also show that the minimal dilation of the semigroup P={ exp tL:t≥ 0} to an E0-semigroup is is cocycle conjugate to a CAR/CCR flow.

On Kadison-Schwarz Approximation to Positive Maps

2020 ◽  
Vol 27 (03) ◽  
pp. 2050016
Author(s):  
Dariusz Chruściński ◽  
Farrukh Mukhamedov ◽  
Mohamed Ali Hajji
Keyword(s):  
Matrix Algebras ◽  
Positive Maps ◽  
Physical Approximation

We analyze Kadison-Schwarz approximation to positive maps in matrix algebras. This is an analogue of the well known structural physical approximation to positive maps used in entanglement theory. We study several known maps both decomposable (like transposition) and non-decomposable (like Choi map and its generalizations).

On the geometry of positive maps in matrix algebras

1983 ◽  
Vol 184 (1) ◽  
pp. 101-108 ◽  
Author(s):  
Toshiyuki Takasaki ◽  
Jun Tomiyama
Keyword(s):  
Matrix Algebras ◽  
Positive Maps

scholarly journals On the structure of positive maps between matrix algebras

2007 ◽  
Author(s):  
Władysław A. Majewski ◽  
Marcin Marciniak
Keyword(s):  
Matrix Algebras ◽  
Positive Maps
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