  In the present paper, we consider the Cauchy problem of the system of
quadratic derivative nonlinear Schr\"odinger equations. This system was
introduced by M. Colin and T. Colin (2004). The first and second authors
obtained some well-posedness results in the Sobolev space
$H^{s}(\mathbb{R}^d)$. We improve these results for conditional radial initial
data by rewriting the system radial form.
