Trace distance between fermionic Gaussian states from a truncation method

In this paper, we propose a novel truncation method for determining the trace distance between two Gaussian states in fermionic systems. For two fermionic Gaussian states, characterized by their correlation matrices, we consider the von Neumann entropies and dissimilarities between their correlation matrices and truncate the correlation matrices to facilitate trace distance calculations. Our method exhibits notable efficacy in two distinct scenarios. In the first scenario, the states have small von Neumann entropies, indicating finite or logarithmic-law entropy, while their correlation matrices display near-commuting behavior, characterized by a finite or gradual nonlinear increase in the trace norm of the correlation matrix commutator relative to the system size. The second scenario encompasses situations where the two states are nearly orthogonal, with a maximal canonical value difference approaching 2. To evaluate the performance of our method, we apply it to various compelling examples. Notably, we successfully compute the subsystem trace distances between low lying eigenstates of Ising and XX spin chains, even for significantly large subsystem sizes. This is in stark contrast to existing literature, where subsystem trace distances are limited to subsystems of approximately ten sites. With our truncation method, we extend the analysis to subsystems comprising several hundred sites, thus expanding the scope of research in this field.