Explicit Pfaffian Formula for Amplitudes of Fermionic Gaussian Pure States in Arbitrary Pauli Bases

The explicit computation of amplitudes for fermionic Gaussian pure states in arbitrary Pauli bases is a long-standing challenge in quantum many-body physics, with significant implications for quantum tomography, experimental studies, and quantum dynamics. These calculations are essential for analyzing complex properties beyond traditional measures, such as formation probabilities, global entanglement, and entropy in non-standard bases, where exact and computationally efficient methods remain underdeveloped. In addition to these physical applications, having explicit formulas is crucial for optimizing negative log-likelihood functions in quantum tomography, a key task in the NISQ era. In this work, we present an explicit Pfaffian formula (Theorem 1) for determining these amplitudes in arbitrary Pauli bases, utilizing a matrix whose structure reflects the qubit parity. Additionally, we introduce a recursive relation (Theorem 2) that connects amplitudes for systems with varying qubit numbers, enabling scalable computations for large systems. Together, these results provide a versatile framework for studying global entanglement, Shannon-R\'enyi entropies, formation probabilities, and performing efficient quantum tomography, thereby significantly expanding the computational toolkit for analyzing complex quantum systems. Finally, we utilize our formalism to determine the post-measurement entanglement entropy, reflecting how local measurements alter entanglement, and compare the outcomes with conformal field theory predictions.