量子信道渐近等分性质的失效
Failure of the Asymptotic Equipartition Property for Quantum Channels
浏览论文内容
中文总结 AI 辅助
本研究证明量子信道渐近等分性质在CPTP平滑下失效，但子信道平滑版本等价于强逆等阈值，并构造反例揭示信道与子信道平滑的渐近无界分离。
中文摘要 AI 辅助
我们建立了关于量子信道渐近等分性质（AEP）的两个结果。首先，使用完全正迹保持映射进行平滑所表述的AEP在一般情况下失效。其次，使用子信道（完全正迹非增映射）进行平滑的AEP等价于并行量子信道Stein引理的强逆、信道冰球棒散度的尖锐阈值以及信道Lorenz散度的AEP。该反例揭示了信道平滑与子信道平滑之间令人惊讶的分离。尽管态和经典信道具有我们建立的尖锐且维度无关的完备界，但量子信道相应的单发间隙在量子比特情形下已无界。这种分离在渐近意义下持续存在：我们构造了具有有限最大相对熵的qutrit信道对，其CPTP平滑速率严格超过正则化信道相对熵。信道平滑与子信道平滑之间的渐近间隙在该族中无界，即使子信道近似误差指数衰减。为建立这些等价性，我们发展了一种均匀滤波器，将Lorenz平滑转换为单个邻近子信道。因此，四个猜想表述共享一个共同的渐近阈值，其对于一般信道对的有效性仍待解决。这些结果共同将该阈值问题与精确迹保持所施加的障碍分离开来。
英文摘要
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.
发表机构
- Faculty of Mathematics, Technion–Israel Institute of Technology（以色列理工学院数学系）
机构由 AI 辅助整理，请以论文原文为准。
