可退化量子信道上私有通信的指数强逆定理
An exponential strong converse for private communication over degradable quantum channels
- Cornell University（康奈尔大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对可退化量子信道，证明私有容量在超相干信息速率下指数强逆，通过隐私测试重叠界与符号平均构造实现，并推广至反退化及擦除信道。
AI中文摘要：
我们证明了每个有限维可退化量子信道的无辅助私有容量的指数强逆定理。我们使用单一误差准则：实际消息-估计-环境状态与理想状态之间的不保真度，理想状态由均匀分布、完全相关且与环境独立的消息组成。在严格高于信道最大相干信息的每个速率下，对该理想状态的优化保真度随信道使用次数指数衰减。当理想环境状态固定为实际环境边缘状态时，同样的界也成立。该证明适用于任意混合态编码器和任意集体解码器。其主要算子成分是隐私测试的重叠界：具有公共子系统维度$d_C$的两种接收器布局的重叠至多为$d_C/M$，其中$M$是密钥大小，与两个屏蔽维度无关。我们将此界与arXiv:2608.01308中提出的符号平均构造以及单滤波器归约为对称态相结合。高斯滤波器将公共接收器子系统替换为受控维度的系统，原始相关性可借助量子边信息从该系统近似恢复。该滤波器保持精确的交换对称性，均匀平滑熵估计限制了任意相关信道输入的输出维度。我们还在相同联合准则下获得了反退化信道在零速率处的指数强逆定理，并因此在整个参数范围内获得了量子擦除信道私有容量的指数强逆定理。
英文摘要：
We prove an exponential strong converse for the unassisted private capacity of every finite-dimensional degradable quantum channel. We use a single error criterion: the infidelity between the actual message-estimate-environment state and an ideal state consisting of uniformly distributed, perfectly correlated messages that are independent of the environment. At every rate strictly above the channel's maximum coherent information, the optimized fidelity to this ideal state decays exponentially in the number of channel uses. The same bound holds when the ideal environment state is fixed to the actual environment marginal. The proof applies to arbitrary mixed-state encoders and arbitrary collective decoders. Its main operator ingredient is an overlap bound for privacy tests: two receiver placements with a common subsystem of dimension $d_C$ have overlap at most $d_C/M$, where $M$ is the key size, independently of both shield dimensions. We combine this bound with the signed averaging construction presented in arXiv:2608.01308 and a single-filter reduction to symmetric states. A Gaussian filter replaces the common receiver subsystem by a system of controlled dimension from which the original correlations can be approximately recovered using quantum side information. The filter preserves exact exchange symmetry, and a uniform smooth-entropy estimate bounds its output dimension for arbitrary correlated channel inputs. We also obtain an exponential strong converse at zero for antidegradable channels under the same joint criterion and, consequently, for the private capacity of the quantum erasure channel throughout its parameter range.
