Quantum Incapacity beyond No-Cloning and PPT Mechanisms

Abstract

We show an explicit qutrit channel whose private and quantum capacities both vanish, although it is neither antidegradable nor positive under partial transposition (PPT). This resolves two longstanding open problems in quantum information theory: whether zero quantum capacity can occur outside the PPT and antidegradable classes, and whether antidegradability is the only nontrivial mechanism that forces the private capacity to vanish. For qutrit systems $A$ and $B$, the channel is \[ Λ_{A\to B}(X)= \frac{1}{2}X+\frac{1}{4}\left(\operatorname{Tr}(X)\mathbb{1}_{B}-X^{\mathsf T} \right). \] We use the established Bogoliubov--Kubo--Mori/relative-entropy comparison results to show that its optimized coherent and private information vanish at every blocklength, and hence $P(\hat{I}›)=Q(\hat{I}›)=0$. Nevertheless, the Choi state of $\hat{I}›$ is not PPT, and $\hat{I}›$ is not antidegradable. The underlying mechanism is an observable-level relaxation of antidegradability. Specifically, for any input state and receiver observable, the complementary output can provide an unbiased expectation value of the receiver output with no larger variance. Our results show that this mechanism is strictly weaker than antidegradability and implies the complete less-noisy order of the complement. Therefore, the constructed channel establishes a new class of zero-capacity channels beyond the conventional PPT and no-cloning mechanisms.

23 pages. v2 adds results on observable-level relaxation of antidegradability