We analyze a general bipartite-like representation of arbitrary pure states of N indistinguishable partcles, valid for both bosons and fermions, based on
M- and $(N\ensuremath{-}M)$-particle states. It leads to exact $(M,N\ensuremath{-}M)$ Schmidt-like expansions of the state for any
M<N and is directly related to the isospectral reduced
M- and $(N\ensuremath{-}M)$-body density matrices ${\ensuremath{\rho}}^{(M)}$ and ${\ensuremath{\rho}}^{(N\ensuremath{-}M)}$. The formalism also allows for reduced yet still exact Schmidt-like decompositions associated with blocks of these densities, in systems having a fixed fraction of the particles in some single-particle subspace. Monotonicity of the ensuing
M-body entanglement under a certain set of quantum operations is also discussed. Illustrative examples in fermionic and bosonic systems with pairing correlations are provided, which show that in the presence of dominant eigenvalues in ${\ensuremath{\rho}}^{(M)}$, approximations based on a few terms of the pertinent Schmidt expansion can provide a reliable description of the state. The associated one- and two-body entanglement spectrum and entropies are also analyzed.
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.
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.
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.
We analyze the problem of quantifying entanglement in pure and mixed states of fermionic systems with a fixed number parity yet not necessarily a fixed particle number. The mode entanglement between one single-particle level and its orthogonal complement is first considered, and an entanglement entropy for such a partition of a particular basis of the single-particle Hilbert space
H is defined. The sum over all single-particle modes of this entropy is introduced as a measure of the total entanglement of the system with respect to the chosen basis and it is shown that its minimum over all bases of
H is a function of the one-body density matrix. Furthermore, we show that if minimization is extended to all bases related through a Bogoliubov transformation, then the entanglement entropy is a function of the generalized one-body density matrix. These results are then used to quantify entanglement in fermion systems with four single-particle levels. For general pure states of such a system a closed expression for the fermionic concurrence is derived, which generalizes the Slater correlation measure defined by J. Schliemann et al. [Phys. Rev. A 64, 022303 (2001)], implying that particle entanglement may be seen as minimum mode entanglement. It is also shown that the entanglement entropy defined before is related to this concurrence by an expression analogous to that in the two-qubit case. For mixed states of this system the convex roof extension of the previous concurrence and entanglement entropy is evaluated analytically, extending the results in previous reference to general states.
We investigate in detail the behavior of the bipartite fluctuations of particle number $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{N}$ and spin ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{S}}^{z}$ in many-body quantum systems, focusing on systems where such U(1) charges are both conserved and fluctuate within subsystems due to exchange of charges between subsystems. We propose that the bipartite fluctuations are an effective tool for studying many-body physics, particularly its entanglement properties, in the same way that noise and full counting statistics have been used in mesoscopic transport and cold-atomic gases. For systems that can be mapped to a problem of noninteracting fermions, we show that the fluctuations and higher-order cumulants fully encode the information needed to determine the entanglement entropy as well as the full entanglement spectrum through the R\'enyi entropies. In this connection, we derive a simple formula that explicitly relates the eigenvalues of the reduced density matrix to the R\'enyi entropies of integer order for any finite density matrix. In other systems, particularly in one dimension, the fluctuations are in many ways similar but not equivalent to the entanglement entropy. Fluctuations are tractable analytically, computable numerically in both density matrix renormalization-group and quantum Monte Carlo calculations, and in principle accessible in condensed-matter and cold-atom experiments. In the context of quantum point contacts, measurement of the second charge cumulant showing a logarithmic dependence on time would constitute a strong indication of many-body entanglement.