In this article, we are proposing a closed-form solution for the capacity of the single quantum channel. The Gaussian distributed input has been considered for the analytical calculation of the capacity. In our previous couple of papers, we invoked models for joint quantum noise and the corresponding received signals; in this current research, we proved that these models are Gaussian mixtures distributions. In this paper, we showed how to deal with both of cases, namely (I)the Gaussian mixtures distribution for scalar variables and (II) the Gaussian mixtures distribution for random vectors. Our target is to calculate the entropy of the joint noise and the entropy of the received signal in order to calculate the capacity expression of the quantum channel. The main challenge is to work with the function type of the Gaussian mixture distribution. The entropy of the Gaussian mixture distributions cannot be calculated in the closed-form solution due to the logarithm of a sum of exponential functions. As a solution, we proposed a lower bound and a upper bound for each of the entropies of joint noise and the received signal, and finally upper inequality and lower inequality lead to the upper bound for the mutual information and hence the maximum achievable data rate as the capacity. In this paper reader will able to visualize an closed-form capacity experssion which make this paper distinct from our previous works. These capacity experssion and coresses ponding bounds are calculated for both the cases: the Gaussian mixtures distribution for scalar variables and the Gaussian mixtures distribution for random vectors as well.

翻译：本文提出了单量子信道容量的闭式解。在容量的解析计算中，我们考虑了高斯分布输入。在前几篇论文中，我们引入了联合量子噪声及其对应接收信号的模型；在本研究中，我们证明了这些模型属于高斯混合分布。本文展示了如何处理两种情况：即（I）标量变量的高斯混合分布，以及（II）随机向量的高斯混合分布。我们的目标是通过计算联合噪声的熵和接收信号的熵，进而推导量子信道的容量表达式。主要挑战在于处理高斯混合分布的函数形式。由于指数函数之和的对数特性，高斯混合分布的熵无法通过闭式解直接计算。为此，我们分别提出了联合噪声熵和接收信号熵的下界与上界，最终通过上界不等式和下界不等式得到互信息的上界，从而获得最大可达数据率即信道容量。本文读者将能够观察到闭式容量表达式，这使本文区别于我们之前的工作。这些容量表达式及其对应的边界均针对两种情形进行了计算：标量变量的高斯混合分布与随机向量的高斯混合分布。