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Journal of Mathematical Analysis and Modeling

Completely positive map from M4(C) to M5(C) on positive semidefinite Matrices

Winda C. Akatch, Benard Okelo, Omolo Ongati

Published Apr 1, 2022• 0 citations
Publisher

Abstract

Positive maps are essential in the description of quantum systems. However, characterization of the structure of the set of all positive maps is a challenge in mathematics and mathematical physics. We construct a linear positive map from M4 to M5 and state the conditions under which they are positive and completely positive (copositivity of positive).

DOI
10.48185/jmam.v3i1.441

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Positive maps are essential in the description of quantum systems. However, characterization of the structure of the set of all positive maps is a challenge in mathematics and mathematical physics. We construct a linear positive map from M4 to M5 and state the conditions under which they are positive and completely positive (copositivity of positive).
Several investigations have been done on positive maps on their algebraic structures with more emphasis on completely positive maps. In this study, we have described the structure of the Choi matrices for 2-positive maps on positive semidefinite matrices and the conditions for complete positivity of positive linear maps from n to n+1. The motivation behind these objectives is work done by Majewski and Marciniak on the structure of positive maps ф from Mn to Mn + 1(2 ≥ 2) between matrix algebras.
Abstract A ∗\ast∗-linear map Φ\PhiΦ between matrix spaces is positive if it maps positive semidefinite matrices to positive semidefinite ones, and is called completely positive if all its ampliations In⊗ΦI_n\otimes \PhiIn​⊗Φ are positive. In this article, quantitative bounds on the fraction of positive maps that are completely positive are proved. A main tool is the real algebraic geometry techniques developed by Blekherman to study the gap between positive polynomials and sums of squares. Finally, an algorithm to produce positive maps that are not completely positive is given.
An alternative, geometrical proof of a known theorem concerning the decomposition of positive maps of the matrix algebra has been presented. The premise of the proof is the identification of positive maps with operators preserving the Lorentz cone in four dimensions, and it allows to decompose the positive maps with respect to those preserving the boundary of the cone. In addition, useful conditions implying complete positivity of a map of have been given, together with a sufficient condition for complete positivity of an extremal Schwarz map of . Lastly, following the same geometrical approach, a description in topological terms of maps that are simultaneously completely positive and completely copositive has been presented.
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A Class of Extremal Positive Maps in 3×33 \times 33×3 Matrix Algebras