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 the 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. We also 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. We also analyze many-body entanglement in interacting fermionic systems by using the
M-body DM. We demonstrate that if a particle number conserving fermionic Hamiltonian contains only up to
M-body interaction terms, then its
N-particle ground state cannot be maximally
M-body entangled. As a key step in the proof, we show that the energy expectation value of a maximally
M-body mixed state is equal to the spectral mean of the Hamiltonian on the corresponding
N-particle subspace. We further demonstrate that the many-body entanglement structure of a ground state can place quantitative constraint on the interaction strength of its parent Hamiltonian. We illustrate the theorem and its implications in Hubbard and extended SYK models. Going beyond ground states, we analyze entanglement generation under unitary dynamics from Slater-determinant initial states in these models. We determine early-time growth and estimate entanglement saturation times. Finally, we derive explicit symmetry-refined saturation upper bounds for
M-body entanglement in the presence of an Abelian symmetry.
Learning compact, interpretable descriptions of quantum many-body states is an important task in quantum science. We study the task of learning the Slater determinant with maximum fidelity to an arbitrary fermionic many-body state, with motivation from both Hartree-Fock methods and agnostic tomography. Given an
n-fermion wavefunction built from
m fermionic modes, we provide classical and quantum algorithms returning a Slater determinant with fidelity within
ε of maximal in time
mpoly(n,1/ε). We prove matching hardness lower bounds, assuming standard complexity conjectures, along some parameter axes. Given access to quantum copies, we prove this can be accomplished with
poly(m,n,1/ε) copies of
ρ. We also show that above a fidelity of
2/3 any stationary point is the unique global maximum while below
2/3 the optimization landscape can have spurious stationary points, and hence
2/3 marks a transition point in the optimization landscape for this problem. We apply the algorithm to the Fermi-Hubbard model, extracting the closest Slater determinant from neural quantum state solutions. Together, our results provide algorithmic tools with provable guarantees in understanding fermionic many-body systems with classical or quantum simulation.
The concept of entanglement in many‑body physics has traditionally focused on spin or particle‑number degrees of freedom. Here we introduce a novel class of fermionic quantum phases in which orbital degrees of freedom of identical particles become entangled across spatially distinct sites. We construct the general form of the many‑body wavefunction for a system of N fermions occupying L orbitals per site, and show that the ground state of certain interacting Hamiltonians can be written as |\Psi> = \sum_{i,j} c_{ij} | \phi_i \phi_j >, where | \phi_i > and | \phi_j > denote single‑particle orbital states localized at different lattice sites. This entangled orbital structure leads to emergent properties distinct from conventional band insulators, Mott insulators, or spin liquids. We discuss the symmetry constraints, the role of fermionic statistics, and the topological aspects that arise when the orbital entanglement is global. Theoretical analysis predicts that FOEPs can host protected edge modes, exotic quasiparticles, and non‑Abelian statistics in two‑dimensional lattices. We also outline experimental signatures that could be observed in engineered cold‑atom systems, photonic lattices, or layered transition‑metal dichalcogenides, and argue that the discovery of FOEPs would open a new frontier in the search for quantum materials with high‑temperature superconductivity or quantum information processing capabilities.
We analyze many-body entanglement in interacting fermionic systems by using the
M-body reduced density matrix. We demonstrate that if a particle number conserving fermionic Hamiltonian contains only up to
M-body interaction terms, then its
N-particle ground state cannot be maximally
M-body entangled. As a key step in the proof, we show that the energy expectation value of a maximally
M-body mixed state is equal to the spectral mean of the Hamiltonian on the corresponding
N-particle subspace. We further demonstrate that the many-body entanglement structure of a ground state can place quantitative constraint on the interaction strength of its parent Hamiltonian. We illustrate the theorem and its implications in Hubbard and extended SYK models. Going beyond ground states, we analyze entanglement generation under unitary dynamics from Slater-determinant initial states in these models. We determine early-time growth and estimate entanglement saturation times. Finally, we derive explicit symmetry-refined saturation upper bounds for
M-body entanglement in the presence of an Abelian symmetry.
Entanglement plays a central role in numerous fields of quantum science. However, as one departs from the typical "Alice versus Bob" setting into the world of indistinguishable fermions, it is not immediately clear how the concept of entanglement is defined among these identical particles. Our endeavor to recover the notion of subsystems, or mathematically speaking, the tensor product structure of the Hilbert space, lead to two natural pictures of defining fermionic entanglement: the particle picture and the mode picture. In the particle picture, entanglement characterizes the deviation of a fermionic quantum state from the non-interacting ones, e.g., single Slater determinants. In the mode picture, we recover the notion of subsystems, by referring to the partitioning of the orbital/mode that the fermions occupy, which allows us to naturally adopt the formalism of entanglement between distinguishable constituents. Both pictures reveal essential and interconnected aspects of fermionic entanglement, and thus offer precise tools for studying electron entanglement in highly relevant systems such as atoms and molecules. We showcase here two applications: i) resolving the correlation paradox in the molecular dissociation limit, ii) quantitative electronic structure analysis with orbital entanglement.
We show that one-body entanglement, which is a measure of the deviation of a pure fermionic state from a Slater determinant (SD) and is determined by the mixedness of the single-particle density matrix (SPDM), can be considered as a quantum resource. The associated theory has SDs and their convex hull as free states, and number conserving fermion linear optics operations (FLO), which include one-body unitary transformations and measurements of the occupancy of single-particle modes, as the basic free operations. We first provide a bipartitelike formulation of one-body entanglement, based on a Schmidt-like decomposition of a pure
N-fermion state, from which the SPDM [together with the $(N\ensuremath{-}1)$-body density matrix] can be derived. It is then proved that under FLO operations the initial and postmeasurement SPDMs always satisfy a majorization relation, which ensures that these operations cannot increase, on average, the one-body entanglement. It is finally shown that this resource is consistent with a model of fermionic quantum computation which requires correlations beyond antisymmetrization. More general free measurements and the relation with mode entanglement are also discussed.