The partition bound subsumes all other bounds except those from information theory. arxiv.org/pdf/0910.4266v2.pdf. The internal information theory subsumes all the bounds except the partition bound arxiv.org/abs/1204.1505. The external information complexity is same as communication complexity up to a log factor. (I forget the reference for this one, but I think you get deduce it using the technique from boazbarak.org/Papers/directsum.pdf)

A more recent reference by Lee and Shraibman, also covers the methods to prove lower bound on quantum communication complexity research.rutgers.edu/~troyjlee/survey_plain.pdf

Thanks. but my question is if there is superexponential gap for $rank(A)$ and $rank^{\alpha}(A)$, where $\alpha>1$ but not $\alpha\neq\infty$.

will the problem be easier when allowing overlap? About the rectangle, yes, the black squares are covered by two rectangles

OK, here "the same" means they are shown to be same, but may use completely different definitions, like $IP=PSPACE$.