Let $p_n$ denote the largest possible cp-rank of an $n\times n$ completely positive matrix. This matrix parameter has its significance both in theory and applications, as it sheds light on the geometry and structure of the solution set of hard optimization problems in their completely positive formulation. Known bounds for $p_n$ are $s_n=\binom{n+1}2-4$, the current best upper bound, and the Drew-Johnson-Loewy (DJL) lower bound $d_n=\big\lfloor\frac{n^2}4\big\rfloor$. The famous DJL conjecture (1994) states that $p_n=d_n$. Here we show $\corr{p_n=\frac {n^2}2 +{\mathcal O}\big(n^{3/2}\big) = 2d_n+{\mathcal O}\big(n^{3/2}\big)}$, and construct counterexamples to the DJL conjecture for all $n\ge {12}$

## Citation

Preprint NI14048-POP, Isaac Newton Institute for Mathematical Sciences, University of Cambridge UK, submitted