In this paper, the generalized quantum states, i.e. positive and normalized linear functionals on $C^{\ast }$-algebras, are studied. First, we study normal states, i.e. states which are represented by density operators, and singular states, i.e. states cannot be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the Gelfand–Neumark–Segal (GNS) construction, i.e. the fundamental result of Gelfand, Neumark and Segal on the representation theory of $C^{\ast }$-algebras, and theory of projections. Second, it is given an application in quantum information theory. We study covariant cloners, i.e. quantum channels in the Heisenberg and the Schrödinger pictures which are covariant by shifting, and it is shown that the optimal cloners cannot have a singular component. Finally, we discuss on the representation of pure states in the sense of the Gelfand–Pettis integral. We also give physical interpretations and examples in different sections of this work.

在本文中，广义量子态，即正的和归一化的线性泛函$C^{＊}$-代数，进行了研究。首先，我们研究了正常状态，即由密度算子表示的状态，以及奇异状态，即不能由密度算子表示的状态。它给出了一种通过应用 Gelfand-Neumark-Segal (GNS) 构造将有界线性泛函分解为量子态的方法，即 Gelfand、Neumark 和 Segal 在表示论方面的基本结果$C^{＊}$-代数和投影理论。其次，给出了它在量子信息论中的应用。我们研究了协变克隆子，即海森堡和薛定谔图像中的量子通道，它们通过移位而协变，结果表明最优克隆子不能有奇异分量。最后，我们讨论了 Gelfand-Pettis 积分意义上的纯态表示。我们还在这项工作的不同部分给出了物理解释和示例。