Information Fragility or Robustness Under Quantum Channels

Information stored in quantum states naturally decays under noise. Many earlier works have quantified and demonstrated lower bounds on decay rates of quantum relative entropy, 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 when noise is sufficiently weak? In several scenarios, including when a quantum channel induces decay of relative entropy to its own fixed point, we find such multiplicative converse bounds. However, there are other cases in which the ratio can fall by an arbitrary extent. 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.