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

Fermionic Hamiltonians play a critical role in quantum chemistry, one of the most promising use cases

for near-term quantum computers. However, encoding nonlocal fermionic statistics using conventional qubits

results in significant computational overhead. Fermionic quantum hardware, such as fermion atom arrays,

provides a more efficient means of simulating electronic degrees of freedom. In order to better understand

the many-body entanglement structure of fermionic N -particle states we study here M -body reduced density

matrices (DMs) across various bipartitions. This naturally leads to an M -body entanglement measure by

assigning von Neumann entropy to the reduced DM. This entanglement measure generalizes the traditional

quantum chemistry concept of the 1-particle DM, which captures how a single fermion is entangled with the

rest. We also establish a connection between this framework and the mathematical structure of hypergraphs.

Specifically, we show that a special class of hypergraphs, known as t-designs, corresponds to maximally entan-

gled fermionic states. Finally, we explore fermionic many-body entanglement in random states, relating the

reduced DMs of fermionic states to the trace-fixed Wishart-Laguerre (TFWL) ensemble. In the limit of large

single-particle dimension D and a non-zero filling fraction, random states asymptotically become absolutely

maximally entangled