On the notion of positive rank The positive rank of a square matrix is defined in Theorem $3$ of "Expressing Combinatorial Optimization Problems by Linear Programs" by Mihalis Yannakakis as follows: given a $n\times n$ matrix $A$, the positive rank $rank_{\Bbb R}^+(A)$ is the smallest $m$ such that $A=LR$ for a non-negative $n\times m$ matrix $L$, and non-negative $m\times n$ matrix $R$.

This concept is valuable in communication complexity, since it was shown that if $rank_{\Bbb R}^+(A)$ and $rank_{\Bbb R}(A)$ could be quasi-polynomially related for a $0/1$ matrix $A$, then the log-rank conjecture holds.

Is there an example of a $0/1$ matrix $A$ whose positive rank is strictly smaller than the dimension $m$ of any positive decomposition of $A$ into $0/1$ matrices $L,R$? 

I think Theorem 1.1 in http://cjtcs.cs.uchicago.edu/articles/2016/2/cj16-02.pdf answers this.


  Do we know that if non-negative rank and real rank are not quasipolynomially related then the log-rank conjecture fails?


The converse is direct.
This is not an answer, but another version, NOT equivalent to the original of this excellent, but perhaps hard to grasp question.
Define the bipartite clique partition number of a graph as the least number of its complete bipartite subgraphs that cover each edge exactly once.
Similarly define the support of the fractional bipartite clique partition number of a graph as the least number of its complete bipartite subgraphs that have a positive weighing such that for each edge the sum of the weights of the subgraphs containing it is exactly one.
The question is whether these two parameters are always equal or not.

I find it quite surprising that while fractional version are studied a lot, I've never heard of studying the size of the support of the fractional version.
I also wonder whether this latter parameter has any applications.
Also, the question could be asked about several other parameters.