Based upon the Fluctuation-Dissipation Theorem of Nyquist-Callen-Welton and the spectrum balancing relationship of Kubo-Martin-Schwinger for operator-valued stochastic noise processes, a two-parameter Quantum Noise-Energy Spectral Density (QN-ESD) is constructed and Fourier-Laplace transformed to obtain its associated Quantum Noise-Autocorrelation Function (QN-ACF). By means of a Taylor series expansion, the quantum thermal noise correlations are linked to Number Theory’s Reimann-Zeta function. From this perspective, a panoply of spectral components and their QN correlations are characterized for three disjoint regions located across the Electromagnetic Spectrum (EMS). The second-moment characterizations and frequency boundaries for each region are specified as a function of two frequency dependent design parameters, viz., a normalized temperature dependent frequency and a quantum receiver cutoff frequency. These boundaries define where the quadrature noise correlation projections of the QN-ESD become noticeably asymmetric-in-frequency. For the operator-valued white quantum noise region, various communication system performance metrics are characterized and their limits are presented and compared via interconnecting parameter relations to their classical communications (CC) counterparts. These metrics include Shannon’s non-additive, but regularized, zero-error capacity in bits/photon-pair, bits/photon, bits/channel-use and bits/sec, the spectral efficiency in bits/sec/Hz and the error-correcting code rate. For long coded messages, Shannon’s capacity quadrature approaches $2 \pi \sqrt {5} / \ln (2) \approx 20$ bits/photon-pair or (10 bits/photon) while for single-photon per channel use transmission it approaches 3 bits/photon-pair. Results are graphically illustrated for the quantum noise spectral density, quantum noise correlations, Shannon’s $\widehat {I}$ - $\widetilde {Q}$ capacities and spectral efficiency.

中文翻译：

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通过噪声量子信道传输经典信息——一种频谱方法

基于 Nyquist-Callen-Welton 的涨落-耗散定理和 Kubo-Martin-Schwinger 对算子值随机噪声过程的频谱平衡关系，构造了一个双参数量子噪声-能量谱密度（QN-ESD）并傅里叶-拉普拉斯变换以获得其关联的量子噪声自相关函数 (QN-ACF)。通过泰勒级数展开，量子热噪声相关性与数论的 Reimann-Zeta 函数相关联。从这个角度来看，对位于电磁频谱 (EMS) 上的三个不相交区域的频谱分量及其 QN 相关性进行了表征。每个区域的二阶矩特征和频率边界被指定为两个与频率相关的设计参数的函数，即，归一化温度相关频率和量子接收器截止频率。这些边界定义了 QN-ESD 的正交噪声相关投影变得明显频率不对称的地方。对于算子价值的白量子噪声区域，表征了各种通信系统性能指标，并通过与经典通信 (CC) 对应物的互连参数关系来呈现和比较它们的限制。这些指标包括以比特/光子对、比特/光子、比特/信道使用和比特/秒为单位的香农的非可加性、但正则化的零错误容量、以比特/秒/赫兹为单位的频谱效率和纠错能力码率。对于长编码消息，香农的容量正交接近 $2 \pi \sqrt {5} / \ln (2) \approx 20 $ bits/photon-pair or (10 bits/photon) 而对于单光子每通道使用传输它接近 3 bits/photon-一对。结果以图形方式说明了量子噪声频谱密度、量子噪声相关性、香农的 $\widehat {I}$ - $\widetilde {Q}$ 容量和频谱效率。