  We investigate the optimal rate of convergence in the multidimensional normal
approximation of vector-valued Wiener-Ito integrals of which components all
belong to the same fixed Wiener chaos. Combining Malliavin calculus, Stein's
method for normal approximation and method of cumulants, we obtain the optimal
rate of convergence with respect to a suitable smooth distance. As
applications, we derive the optimal rates of convergences for complex
Wiener-Ito integrals, vector-valued Wiener-Ito integrals with kernels of step
functions and vector-valued Toeplitz quadratic functionals.
