  We investigate the relationship between quadratic gravity and a restricted
Weyl symmetry where a gauge parameter $\Omega(x)$ of Weyl transformation
satisfies a constraint $\Box \Omega = 0$ in a curved space-time. First, we
briefly review a model with a restricted gauge symmetry on the basis of QED
where a $U(1)$ gauge parameter $\theta(x)$ obeys a similar constraint $\Box
\theta = 0$ in a flat Minkowski space-time, and explain that the restricted
gauge symmetry removes one on-shell mode of gauge field, which together with
the Feynman gauge leaves only two transverse polarizations as physical states.
Next, it is shown that the restricted Weyl symmetry also eliminates one
component of a dipole field in quadratic gravity around a flat Minkowski
background, leaving only a single scalar state. Finally, we show that the
restricted Weyl symmetry cannot remove any dynamical degrees of freedom in
static background metrics by using the zero-energy theorem of quadratic
gravity. This fact also holds for the Euclidean background metrics without
imposing the static condition.
