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Quantum fidelities are cornerstone metrics in quantum sciences, widely used to quantify the similarity of quantum states. Despite its prominence and age, several of its properties remain unexplored. This thesis advances our understanding of fidelities and its generalizations by addressing open problems and introducing novel frameworks rooted in quantum information theory and Riemannian geometry.
This thesis focuses mainly on three of problems. In the first part, we resolve the problem of maximizing average fidelity over finite ensembles of quantum states. By constructing a semidefinite program to compute the maximum average fidelity and deriving scalable fixed-point algorithms, we demonstrate significant improvements in computational runtime. We also derive novel bounds and expressions for near-optimal states, which are exact in special cases, such as when the ensemble consists of commuting states. These results provide new tools for applications, including Bayesian quantum tomography, where they address outstanding challenges.
In the second part, we extend the concept of fidelity by introducing a family of generalized fidelities based on the Riemannian geometry of the Bures–Wasserstein manifold. This framework unifies and generalizes several existing quantum fidelities, including Uhlmann-, Holevo-, and Matsumoto-fidelity, and preserves their celebrated properties. Through a rigorous mathematical treatment, we establish invariance and covariance properties, derive an Uhlmann-like theorem, and discuss some possible generalizations of quantum divergences.
In the third part, we study the problem of projecting positive matrices to certain convex and compact sets with respect to Bures distance (or equivalently Uhlmann fidelity). These convex and compact sets are defined by quantum channels, and for certain channels, most importantly partial trace, we derive a closed-form for the projection.
Using the closed form for partial-trace projection we demonstrate various applications for our results including quantum process tomography and random state generation. Moreover, these results also endow the pretty good measurement and Petz recovery map with novel geometric and operational interpretations.
The existence of closed-form for specific channels is related to the saturation of the data processing inequality (DPI) for fidelity. Thus our results also provide explicit examples for the saturation of DPI for fidelity.
Together, these contributions provide a comprehensive study of quantum fidelities bridging foundational theory and practical applications.