In extension to this question 
[Positive but not completely positive?](https://mathoverflow.net/questions/86550/positive-but-not-completely-positive)
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](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? 



positive not completely positive maps
The "canonical" example of a map that is $k$-positive but not $(k+1)$-positive is the map defined by
$$
\Phi\_k(X) = k\cdot\mathrm{Tr}(X)I\_n - X.
$$


Above, $n$ denotes the size of $X$ (i.e., $X \in M\_n$) and $I\_n$ is the $n \times n$ identity matrix. This map was introduced in "J. Tomiyama. On the geometry of positive maps in matrix algebras II. Linear Algebra Appl., 69:169–177, 1985." (though it had been considered by Choi earlier in the special case when $k = n-1$).


It's not difficult to prove that this map is $k$-positive just by elementary linear algebra. I won't post details here since it's a bit long and messy, but it's almost exactly the same as the proof that starts at the bottom of page 4 of [these notes](http://www.njohnston.ca/wp-content/uploads/2008/12/PEB_2009.pdf).


To show that this map is *not* $(k+1)$-positive (when $k < n$), simply let $\mathbf{v} = \sum\_{i=1}^{k+1} \mathbf{e}\_i \otimes \mathbf{e}\_i \in \mathbb{C}^{k+1} \otimes \mathbb{C}^n$ (here $\{\mathbf{e}\_i\}$ is the standard basis of $\mathbb{C}^{k+1}$ on the first subsystem, and it's just embedded into $\mathbb{C}^n$ in the natural way on the second subsystem). Then compute that
$$
(id\_{k+1} \otimes \Phi\_k)(\mathbf{v}\mathbf{v}^\*) = k(I\_{k+1} \otimes I\_n) - \mathbf{v}\mathbf{v}^\*,
$$
which has $-1$ as an eigenvalue and is thus not positive.




---


A couple of side notes:


1. This map shows that Choi's theorem on complete positivity is optimal in some sense: if $k \geq n$ then $k$-positivity implies complete positivity, but if $k < n$ then $k$-positivity and $(k+1)$-positivity are indeed different sets.
2. In the $k = 1$ case this map comes up frequently in quantum information theory under the name of the "reduction map".


