Trace and tensor products

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

tr((A⊗I)(I⊗A))tr((A \otimes I) (I \otimes A))

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

For example, if k=nk = n, we’d get tr(A⊗A)=tr(A)2=1tr(A \otimes A) = tr(A)^2 = 1, and if k=1k = 1 we’d just get tr(A2)tr(A^2). I’d like to understand how varying kk might move between [tr(A2),tr(A)2][tr(A^2), tr(A)^2]. Is there some known inequality that addresses this?

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