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Suppose I have a matrix $A \in \mathcal{M}_{n,n}$ and for fun let's normalize is so that $tr(A) = 1$. Then with the identity matrix $I \in \mathcal{M}_{k,k}$, I'm interested in what happens when we look at

$tr((A \otimes I) (I \otimes A))$

for different $k \leq n$, where we can just think of the tensor as the normal Kronecker product.

For example, if $k = n$, we'd get $tr(A \otimes A) = tr(A)^2 = 1$, and if $k = 1$ we'd just get $tr(A^2)$. I'd like to understand how varying $k$ might move between $[tr(A^2), tr(A)^2]$. Is there some known inequality that addresses this?

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