Stability theorem of depolarizing channels for the minimal output quantum Rényi entropies

The stability theorem of the depolarizing channel states that if a state is close to achieving the minimal/maximal output value of a certain quantity through the channel, then it must be close to an input state giving the minimal/maximal value. We show that the stability theorem of the depolarizing channel holds for the output quantum p-Rényi entropy for p ≥ 2 or p = 1, which is an extension of the 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-Rényi entropy. By using our stability theorem, we show that Bob can determine whether her preparation is appropriate.