Let $\cal{E}$ be a finitely generated projective bimodule over a (noncommutative) algebra $A$. Moreover, let us assume that $\cal{E}$ is of finite rank $n$. The tensor product $ \cal{E} \otimes_A \cal{E} $ is again finitely generated, projective, and of finite rank. My question is when does it hold that $$ \text{ rank}({\cal E} \otimes_A {\cal E}) \leq n^2 ~ ? $$ To be more specific, I am looking for a set of sufficient (and hopefully natural) conditions for this to hold.

I will volunteer the condition of *locality* as a sufficient condition. Since in this case we have a well defined notion of minimal number of generators, which coincides with the rank, and so, the tensor product should (?) have a set of generators of size less than or equal to that of the square of the original generating set. I may be totally wrong here though!

[Edit] With regard to the definition of the rank of a projective module, I am using the one found here. The algebras I am interested in are sub-algebras of (noncommutative) Hopf algebras over the complex numbers, and so, the homomorphism $\phi$ can just be taken to be the counit $\epsilon$ of the Hopf algebra.