Characterizing maximally many-body entangled fermionic states by using $M$-body density matrix

Irakli Giorgadze; Haixuan Huang; Jordan Gaines; Elio J. König; Jukka I. Väyrynen

doi:10.22331/q-2025-06-24-1778

Characterizing maximally many-body entangled fermionic states by using $M$-body density matrix

Irakli Giorgadze¹, Haixuan Huang¹, Jordan Gaines¹, Elio J. König^2,3, and Jukka I. Väyrynen¹

¹Department of Physics and Astronomy, Purdue University, West Lafayette, Indiana 47907 USA
²Department of Physics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA
³Max-Planck Institute for Solid State Research, 70569 Stuttgart, Germany

Published:	2025-06-24, volume 9, page 1778
Editor:	Himadri Shekhar Dhar
Eprint:	arXiv:2412.09576v2
Doi:	https://doi.org/10.22331/q-2025-06-24-1778
Citation:	Quantum 9, 1778 (2025).

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Abstract

Fermionic Hamiltonians play a critical role in quantum chemistry, one of the most promising use cases for near-term quantum computers. However, since encoding nonlocal fermionic statistics using conventional qubits results in significant computational overhead, fermionic quantum hardware, such as fermion atom arrays, were proposed as a more efficient platform. In this context, we here study the many-body entanglement structure of fermionic $N$-particle states by concentrating on $M$-body reduced density matrices (DMs) across various bipartitions in Fock space. The von Neumann entropy of the reduced DM is a basis independent entanglement measure which generalizes the traditional quantum chemistry concept of the one-particle DM entanglement, which characterizes how a single fermion is entangled with the rest. We carefully examine upper bounds on the $M$-body entanglement, which are analogous to the volume law of conventional entanglement measures. To this end we establish a connection between $M$-body reduced DM and the mathematical structure of hypergraphs. Specifically, we show that a special class of hypergraphs, known as $t$-designs, corresponds to maximally entangled fermionic states. Finally, we explore fermionic many-body entanglement in random states. We semianalytically demonstrate that the distribution of reduced DMs associated with random fermionic states corresponds to the trace-fixed Wishart-Laguerre random matrix ensemble. In the limit of large single-particle dimension $D$ and a non-zero filling fraction, random states asymptotically become absolutely maximally entangled.

Featured image: A maximally $2$-body entangled $4$-fermion state in $13$ available orbitals, represented with its associated hypergraph. The hypergraph is a $2 – (13, 4, 1)$ combinatorial design.

Popular summary

In this work, we ask two questions:
1. In what state of $N$ indistinguishable fermions can we have $M$ fermions being maximally entangled with the rest $N-M$ fermions? In other words, what does a maximally $M$-body entangled $N$-fermion state look like?
2. How much $M$-body entangled a random $N$-fermion state is?

We find that each $N$-fermion state can be mapped to an $N$-uniform hypergraph, and a special type of hypergraphs, called $M$-designs, correspond to maximally $M$-body entangled states. We also find that a random $N$-fermion state is maximally $M$-body entangled in the limit when the number of available orbitals is large, for all $M$ from $1$ up to $N/2$.

► BibTeX data

@article{Giorgadze2025characterizing,
  doi = {10.22331/q-2025-06-24-1778},
  url = {https://doi.org/10.22331/q-2025-06-24-1778},
  title = {Characterizing maximally many-body entangled fermionic states by using {$M$}-body density matrix},
  author = {Giorgadze, Irakli and Huang, Haixuan and Gaines, Jordan and K{\"{o}}nig, Elio J. and V{\"{a}}yrynen, Jukka I.},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1778},
  month = jun,
  year = {2025}
}

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