On states of quantum theory

In this talk, the generalized quantum states, i.e. positive and normalized linear functionals on $C^{*}-$algebras, are discussed.
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 $C^{*}-$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 talk.