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In extension to this question What are known examples of positive but not completely positive maps? I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. (I don't know a single one.) By M.D. Choi's theorem the size of the matrices involved must grow with $k$, but how fast?

In extension to this question What are known examples of positive but not completely positive maps? I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. (I don't know a single one.) By M.D. Choi's theorem the size of the matrices involved must grow with $k$, but how fast?

In extension to this question What are known examples of positive but not completely positive maps? I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. (I don't know a single one.) By Choi's theorem, https://en.wikipedia.org/wiki/Choi's_theorem_on_completely_positive_maps the size of the matrices involved must grow with $k$, but how fast?

In extension to this question What are known examples of positive but not completely positive maps? I'd like to know, for $k>1$, examples of $k$-positive linear maps of a matrix algebra into itself that are not $k+1$-positive. (I don't know a single one.) By M.D. Choi's theorem the size of the matrices involved must grow with $k$, but how fast?