M. A. Seifi Mirjafarlou

arXiv2025

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-Rényi 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.

quant-ph↗

arXiv2024

Generalization of Balian-Brezin decomposition for exponentials with linear fermionic part

Fermionic Gaussian states have garnered considerable attention due to their intriguing properties, most notably Wick's theorem. Expanding upon the work of Balian and Brezin, who generalized properties of fermionic Gaussian operators and states, we further extend their findings to incorporate Gaussian operators with a linear component. Leveraging a technique introduced by Colpa, we streamline the analysis and present a comprehensive extension of the Balian-Brezin decomposition (BBD) to encompass exponentials involving linear terms. Furthermore, we introduce Gaussian states featuring a linear part and derive corresponding overlap formulas. Additionally, we generalize Wick's theorem to encompass scenarios involving linear terms, facilitating the expression of generic expectation values in relation to one and two-point correlation functions. We also provide a brief commentary on the applicability of the BB decomposition in addressing the BCH (Zassenhaus) formulas within the $\mathfrak{so}(N)$ Lie algebra.

math-ph↗