$J$-states and quantum channels between indefinite metric spaces

In the present work, we introduce and study the concepts of state and quantum channel on spaces equipped with an indefinite metric. Exclusively, we will limit our analysis to the matricial framework. As it will be confirmed below, from our research it is noticed that, when passing to the spaces with indefinite metric, the use of the adjoint of a matrix with respect to the indefinite metric is required in the construction of states and quantum channels; which prevents us to consider the space of matrices of certain order $M_{n}(\mathbb{C})$ as a $C^{\ast}$-algebra. In our case, this adjoint is defined through a $J$-metric, where the matrix $J$ is a fundamental symmetry of $M_{n}(\mathbb{C})$. In our paper, for quantum operators, we include the general setting in the which, these operators map $J_{1}$-states into $J_{2}$-states, where $J_{2}\neq \pm J_{1}$ are two arbitrary fundamental symmetries. In the middle of this program, we carry out a study of the completely positive maps between two different positive matrices spaces by considering two different indefinite metrics on $\mathbb{C}^{n}$.