Rényi信道容量在非信号辅助信道模拟下的单调性
Monotonicity of the Rényi channel capacity under non-signaling assisted channel simulation
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中文总结 AI 辅助
本文证明非信号关联无法提高经典信道任意阶Rényi容量，采用勒让德变换变分法，并指出对随机编码、球填充及强逆指数均无影响。
中文摘要 AI 辅助
我们探讨发送方与接收方共享的非信号关联是否能提高经典信道的Rényi容量。非信号箱构成的类别严格大于共享随机性辅助的编码器-解码器对，因此基于独立预处理和后处理的常规论证并不充分；定义此类箱的约束对箱是线性的，而Rényi容量对信道的依赖是非线性的。我们通过基于勒让德变换的变分方法解决了这一不匹配，该方法将容量重新表述为对一族仿射泛函的极值，其中信道以线性方式出现。将此方法与针对香农容量和端点阶数的独立论证相结合，我们证明了任何非信号箱都不能在任何阶数下提高Rényi容量，从零阶到香农容量直至无穷大处的极限阶。因此，非信号模拟不能提高容量以下的随机编码或球填充指数，也不能降低容量以上的强逆指数。
英文摘要
We ask whether non-signaling correlations shared between the sender and receiver can increase the Rényi capacity of a classical channel. Non-signaling boxes form a strictly larger class than that of encoder and decoder pairs assisted by shared randomness, so the usual argument based on independent preprocessing and postprocessing is insufficient, and the constraints defining such boxes are linear in the box, while the Rényi capacity depends on the channel nonlinearly. We resolve this mismatch with a variational approach based on the Legendre transform, which recasts the capacity as an extremum over a family of affine functionals in which the channel appears linearly. Combining this method with separate arguments for the Shannon capacity and endpoint orders, we prove that no non-signaling box can raise the Rényi capacity at any order, from order zero through the Shannon capacity to the limiting order at infinity. Consequently, non-signaling simulation cannot increase the random-coding or sphere-packing exponent below capacity, or decrease the strong-converse exponent above capacity.
发表机构
- Centre for Quantum Technologies, National University of Singapore（新加坡国立大学量子技术中心）
- Department of Electrical and Computer Engineering, National University of Singapore（新加坡国立大学电气与计算机工程系）
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