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 $M_n(\mathbb{C})$. Felipe-Sosa and Felipe (2022) introduced the notion of quantum states on certain Krein spaces with indefinite metric induced by a fundamental symmetry $J$, calling these $J$-states. We define an analogous notion of measurement for $J$-states to the regular quantum theory and use it to show that a notion of fidelity holds in the Krein setting. We also show that an analogous result to the Fuchs-Caves measurement holds in this setting. Following the developments of Bag, Rohilla, and Trivedi (2024), we then extend this definition of fidelity to $U$-quantum states on $S$-spaces. We demonstrate that the analogous geometric motivation holds in the Krein and $S$-space setting, as holds for quantum fidelity and geometric means of operators.