Failure of the Asymptotic Equipartition Property for Quantum Channels

RESUMO

We establish two results on the asymptotic equipartition property (AEP) for quantum channels. First, the AEP formulated using smoothing over completely positive trace-preserving maps fails in general. Second, the AEP with smoothing over subchannels (completely positive trace-nonincreasing maps) is equivalent to the strong converse for parallel quantum channel Stein's lemma, to a sharp threshold for the channel hockey-stick divergence, and to an AEP for the channel Lorenz divergence. The counterexample reveals a surprising separation between channel and subchannel smoothing. Although states and classical channels admit a sharp, dimension-independent completion bound, which we establish, the corresponding single-shot gap for quantum channels is unbounded already for qubits. This separation persists asymptotically: we construct qutrit channel pairs with finite max-relative entropy whose CPTP-smoothed rate strictly exceeds the regularized channel relative entropy. The asymptotic gap between channel and subchannel smoothing is unbounded across the family, even when the subchannel approximation error decays exponentially. To establish the equivalences, we develop a uniform filter that converts Lorenz smoothing into a single nearby subchannel. The four conjectured formulations thus share a common asymptotic threshold, whose validity for general channel pairs remains open. Together, these results separate this threshold question from the obstruction imposed by exact trace preservation.