We show that the stability theorem of the depolarizing channel holds for the
output quantum $p$-R\'enyi entropy for $p \ge 2$ or $p=1$, which is an
extension of the well known case $p=2$. As an application, we present a
protocol in which Bob determines whether Alice prepares a pure quantum state
close to a product state. In the protocol, Alice transmits to Bob multiple
copies of a pure state through a depolarizing channel, and Bob estimates its
output quantum $p$-R\'enyi entropy. By using our stability theorem, we show
that Bob can determine whether her preparation is appropriate.