Classifying fermionic states via many-body correlation measures

Mykola Semenyakin; Yevheniia Cheipesh; Yaroslav Herasymenko

doi:10.22331/q-2025-04-15-1705

Classifying fermionic states via many-body correlation measures

Mykola Semenyakin^1,2, Yevheniia Cheipesh², and Yaroslav Herasymenko^3,4,5

¹Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
²Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
³QuSoft and CWI, Science Park 123, 1098 XG Amsterdam, The Netherlands
⁴QuTech, TU Delft, P.O. Box 5046, 2600 GA Delft, The Netherlands
⁵Delft Institute of Applied Mathematics, TU Delft, 2628 CD Delft, The Netherlands

Published:	2025-04-15, volume 9, page 1705
Editor:	Jens Siewert
Eprint:	arXiv:2309.07956v3
Doi:	https://doi.org/10.22331/q-2025-04-15-1705
Citation:	Quantum 9, 1705 (2025).

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Abstract

Understanding the structure of quantum correlations in a many-body system is key to its computational treatment. For fermionic systems, correlations can be defined as deviations from Slater determinant states. The link between fermionic correlations and efficient computational physics methods is actively studied but remains ambiguous. We make progress in establishing this connection mathematically. In particular, we find a rigorous classification of states relative to $k$-fermion correlations, which admits a computational physics interpretation. Correlations are captured by a measure $\omega_k$, a function of $k$-fermion reduced density matrix that we call twisted purity. A condition $\omega_k=0$ for a given $k$ puts the state in a class $G_k$ of correlated states. Sets $G_k$ are nested in $k$, and Slater determinants correspond to $k = 1$. Classes $G_{k=O(1)}$ are shown to be physically relevant, as $\omega_k$ vanishes or nearly vanishes for truncated configuration-interaction states, perturbation series around Slater determinants, and some nonperturbative eigenstates of the 1D Hubbard model. For each $k = O(1)$, we give an explicit ansatz with a polynomial number of parameters that covers all states in $G_k$. Potential applications of this ansatz and its connections to the coupled-cluster wavefunction are discussed.

Featured image: Visually explaining our classification of fermionic states of $n$ particles on $l$ modes. Classes $\mathcal{G}_k$ are defined as the states $|{v}\rangle$ for which the twisted purity vanishes ($\omega_k=0$). Twisted purity $\omega_k$ is given in terms of the k-body reduced density matrix (k-RDM) $\rho_k$ and the "twisted" (particle-hole conjugated) k-RDM $\tilde{\rho}_k$. The vanishing of $\omega_k$ is equivalent to generalized Plücker relations $\Omega^k|{v}\rangle\otimes |{v}\rangle=0$, with the standard $k=1$ Plücker relations yielding the familiar set of Slater states. Sets $\mathcal{G}_k$ are nested in $k$ and $k=n+1$ corresponds to the entire Hilbert space. The right panel shows the definitions of $\Omega$, $\rho_k$ and $\tilde{\rho}_k$.

This is the talk given by Yaroslav Herasymenko at Perimeter Institute, titled “Non-Gaussian fermionic ansatzes from many-body correlation measures“, explaining the results in this work.

Popular summary

Understanding correlations between quantum particles is one of key subjects in many-body physics and quantum information theory. The standard example of such correlations is bipartite entanglement, which has played a central role in advancing quantum communication, computational quantum physics, and beyond. However, for systems of fermions—such as electrons and some types of atoms —quantifying many-body correlations in a meaningful and rigorous way has been a significant challenge.

This paper addresses this problem through a mathematically precise framework. In particular, we introduce a new measure of $k$-body correlations called twisted purity (denoted $\omega_k$), which is derived from $k$-fermion reduced density matrices. This measure provides a natural way to classify quantum states according to their degree of $k$-body correlations.

The states belonging to each class obey a type of generalized Wick’s theorem, extending a familiar structure from uncorrelated systems. We further demonstrate that these state classes are not only of theoretical interest—they also closely align with key tools in computational physics. Specifically, we show how the classification connects to widely used ansätze such as configuration interaction and coupled-cluster states, widely used in quantum chemistry and condensed matter theory.

► BibTeX data

@article{Semenyakin2025classifying,
  doi = {10.22331/q-2025-04-15-1705},
  url = {https://doi.org/10.22331/q-2025-04-15-1705},
  title = {Classifying fermionic states via many-body correlation measures},
  author = {Semenyakin, Mykola and Cheipesh, Yevheniia and Herasymenko, Yaroslav},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1705},
  month = apr,
  year = {2025}
}

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