Quantum Incapacity beyond No-Cloning and PPT Mechanisms

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 prove that, for every tensor power and every finite-dimensional quantum reference system, the complementary output dominates the receiver output in relative entropy. Consequently, we show that the optimized private and coherent information vanish at every blocklength, and hence $P(Λ)=Q(Λ)=0$. The proof uses a nonpositive, adjoint-preserving signed lift whose weighted defect admits an exact rank-three completely positive factorization. Applying established Bogoliubov--Kubo--Mori/relative-entropy comparison results and complete less-noisy tensorization results then yields the stated order at every tensor power, leading to zero quantum and private capacities. Meanwhile, we show that the channel is neither PPT nor antidegradable. The channel therefore realizes exact quantum and private incapacity beyond the standard PPT-based bound-entanglement mechanism and the no-cloning mechanism.

Source: arXiv:2607.24693v1 - http://arxiv.org/abs/2607.24693v1 PDF: https://arxiv.org/pdf/2607.24693v1 Original Link: http://arxiv.org/abs/2607.24693v1