  In the present paper, we consider the Cauchy problem of the system of
quadratic derivative nonlinear Schr\"odinger equations for the spatial
dimension $d=2$ and $3$. This system was introduced by M. Colin and T. Colin
(2004). The first author obtained some well-posedness results in the Sobolev
space $H^{s}$. But under some condition for the coefficient of Laplacian, this
result is not optimal. We improve the bilinear estimate by using the nonlinear
version of the classical Loomis-Whitney inequality, and prove the
well-posedness in $H^s$ for $s\ge 1/2$ if $d=2$, and $s>1/2$ if $d=3$.
