  A determinant formalism in combination with \emph{ab initio} calculations
proposed recently has paved a new way for simulating and interpreting x-ray
excitation spectra. The new method systematically takes into account
many-electron effects in the Mahan-Nozi\'eres-De Dominicis (MND) theory,
including core-level excitonic effects, the Fermi-edge singularity, shakeup
excitations, and wavefunction overlap effects such as the orthogonality
catastrophe, all within a universal framework using many-electron
configurations. A heuristic search algorithm was introduced to search for the
configurations that are important for defining spectral lineshapes, instead of
enumerating them in a brute-force way. The algorithm has proven to be efficient
for calculating \ce{O} $K$ edges of transition metal oxides, which converge at
the second excitation order (denoted as $f^{(n)}$ with $n = 2$), i.e., the
final-state configurations with two \emph{e-h} pairs (with one hole being the
core hole). However, it remains unknown how the determinant calculations
converge for general cases and at which excitation order $n$ one should stop
the determinant calculation. Even with the heuristic algorithm, the number of
many-electron configurations still grows exponentially with the excitation
order $n$. In this work, we prove two theorems that can indicate the order of
magnitude of the contribution of the $f^{(n)}$ configurations, so that one can
estimate their contribution very quickly without actually calculating their
amplitudes. The two theorems are based on the singular-value decomposition
(SVD) analysis, a method that is widely used to quantify entanglement between
two quantum many-body systems. We examine the $K$ edges of several metallic
systems with the determinant formalism up to $f^{(5)}$ to illustrate the
usefulness of the theorems.
