  Quantum states naturally decay under noise. Many earlier works have
quantified and demonstrated lower bounds on the decay rate, showing exponential
decay in a wide variety of contexts. Here we study the converse question: are
there uniform upper bounds on the ratio of post-noise to initial information
quantities when noise is sufficiently weak?
  In several scenarios, including classical, we find multiplicative converse
bounds. However, this is not always the case. Even for simple noise such as
qubit dephasing or depolarizing, mutual information may fall by an unbounded
factor under arbitrarily weak noise. As an application, we find families of
channels with non-zero private capacity despite arbitrarily high probability of
transmitting an arbitrarily good copy of the input to the environment.
