  Let $\Omega$ be a $C^2$-smooth bounded pseudoconvex domain in $\mathbb{C}^n$
for $n\geq 2$ and let $\varphi$ be a holomorphic function on $\Omega$ that is
$C^2$-smooth on the closure of $\Omega$. We prove that if
$H_{\overline{\varphi}}$ is in Schatten $p$-class for $p\leq 2n$ then $\varphi$
is a constant function. As a corollary, we show that the
$\overline{\partial}$-Neumann operator on $\Omega$ is not Hilbert-Schmidt.
