On equality of complete positivity and complete copositivity of positive map

Subject Areas : Operator theory

C. A. Winda ¹ , N. B. Okelo ² , O. Ongati ³

1 - Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Kenya
2 - Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Kenya
3 - Department of Pure and Applied Mathematics, Jaramogi Oginga Odinga University of Science and Technology, Kenya

Received: 2022-01-08 Accepted : 2022-05-30 Published : 2022-06-01

Keywords: 2-positivity, Choi matrix, completely positivity,

Abstract :

‎In this paper we construct a $2$-positive map from $\ma_4(\Complex)$ to $\ma_5(\Complex)$ and state the conditions under which the map is positive and completely positive (copositivity of positive)‎. ‎The construction allows us to create a decomposable map, where the Choi matrix of complete positivity is equal to the Choi matrix of complete copositivity‎.

References:

[1] M. D. Choi, Completely positive maps on complex matries, Linear Algebra and its Applications. 10 (1975), 285-290.

[2] W. Hall, Constructions of indecomposable positive maps based on a new criterion for indecomposability, arXiv:quant-ph/0607035v1, (2006), 1-11.

[3] W. Hall, A new criterion for indecomposability of positive maps a new criterion for indecomposability of positive maps a new criterion for indecomposability of positive maps, J. Phys. A. 39 (2006), 14119-14131.

[4] F. Hiai, D. Petz, Introduction to Matrix Analysis and Applications, Springer, New Delhi, 2014.

[5] H. J. Kim, S. H. Kye, Indecomposable extreme positive linear maps in matrix algebras, London Math. Soc. 26 (1994), 575-581.

[6] A. Kossakowski, A class of linear positive maps in matrix algebras, Open Sys. Inform. Dyn. 10 (2003), 213-220.

[7] H. Osaka, Indecomposable positive maps in low dimensional matrix algebras, Linear Algebra and its Applications. 153 (1991), 73-83.

[8] A. G. Robertson, Positive prjection on C∗−algebra and extremal positive maps, J. London Math. Soc. 32 (2) (1985), 133-140.

[9] A. G. Robertson, Automorphisms of spin factors and the decomposition of positive maps, Quart. J. Math. Oxford. 34 (1983), 87-96.

[10] E. Stomer, Decomposable positive maps on C∗-algebras, Amer. Math. Soc. 86 (1982), 402-404.

[11] W. Tang, On positive linear maps between matrix algebras, Linear Algebra and its Applicatlons. 79 (1986), 33-44.

[12] C. A. Winda, N. B. Okelo, O. Ongati, Choi matrices of 2-positive maps on positive semidefinite matrices, Asian Research Journal of Mathematics. 16 (4) (2020), 60-71.