Quantum Fidelity on Krein and S-spaces

The notion of Fidelity for quantum states is a measure of how much two states overlap. In the matrix formalism of quantum mechanics, states are represented by density operators i.e. positive semi-definite matrices with trace equal to 1 in a complex Euclidean space $M_n(\mathbb{C})$. The notion of quantum states in this setting has already started to be considered. We will define an analogous notion of measurement for so-called $J$-states and use it to show that a notion of fidelity holds in the Krein setting. We will also show that there exists an analogous result to the Fuchs-Caves measurement holds in the Krein setting. We will then will extend this definition of fidelity to so-called $U$-quantum states on $S$-spaces. We will demonstrate that the analogous geometric motivation holds in the Krein and $S$-space setting, as holds for quantum fidelity and geometric means of operators.