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 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.

Apr 23, 20260 citations

Many-body entanglement in fermion systems

N. Gigena, Martina Di Tullio, R. Rossignoli

We discuss a general bipartitelike representation and Schmidt decomposition of an arbitrary pure state of $N$ indistinguishable fermions, based on states of $M<N$ and $(N\ensuremath{-}M)$ fermions. It is directly connected with the reduced $M$ - and $(N\ensuremath{-}M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of $M$ -body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized $M$ -body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as postmeasurement states. Moreover, such entropy is an upper bound to the bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of $M$ and $N\ensuremath{-}M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of $M$ -body DMs in some strongly correlated fermionic states is also provided.We discuss a general bipartitelike representation and Schmidt decomposition of an arbitrary pure state of $N$ indistinguishable fermions, based on states of $M<N$ and $(N\ensuremath{-}M)$ fermions. It is directly connected with the reduced $M$- and $(N\ensuremath{-}M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of $M$-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized $M$-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as postmeasurement states. Moreover, such entropy is an upper bound to the bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of $M$ and $N\ensuremath{-}M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of $M$-body DMs in some strongly correlated fermionic states is also provided.

arXiv:2012.13785Physical Review AMay 20, 202115 citations

Fermionic entanglement and quantum correlation measures in molecules

J. Garcia, J. A. Cianciulli, R. Rossignoli

We analyze fermionic entanglement and correlation measures in the ground and the low temperature thermal state of the water molecule as a function of the internuclear distance in the context of the full configuration interaction approach. The aim is to obtain a general entanglement based characterization of the electronic eigenstates. We consider first the spin-up - spin-down partition and the associated Schmidt decomposition, examining the total up-down entanglement of the electronic wave function. We then consider the one- and two-body entanglement derived from the one- and two-body reduced density matrices (DMs), which measure both the deviation of the state from a Slater Determinant (SD) as well as the up-down correlation at the two-body level. All blocks of these DMs are examined. We also introduce and analyze new measures like the up-down two-body mutual information and two types of two-body negativities, the latter measuring the "inner" entanglement of the reduced two-body DMs, i.e., their deviation from a convex mixture of SDs. Finally, the dissociation limit is also analyzed, considering both the exact ground state (GS) as well as the thermal state in the zero temperature limit, representing the projector onto the "GS band" of almost degenerate lowest lying eigenstates.We analyze fermionic entanglement and correlation measures in the ground and the low temperature thermal state of the water molecule as a function of the internuclear distance in the context of the full configuration interaction approach. The aim is to obtain a general entanglement based characterization of the electronic eigenstates. We consider first the spin-up - spin-down partition and the associated Schmidt decomposition, examining the total up-down entanglement of the electronic wave function. We then consider the one- and two-body entanglement derived from the one- and two-body reduced density matrices (DMs), which measure both the deviation of the state from a Slater Determinant (SD) as well as the up-down correlation at the two-body level. All blocks of these DMs are examined. We also introduce and analyze new measures like the up-down two-body mutual information and two types of two-body negativities, the latter measuring the "inner" entanglement of the reduced two-body DMs, i.e., their deviation from a convex mixture of SDs. Finally, the dissociation limit is also analyzed, considering both the exact ground state (GS) as well as the thermal state in the zero temperature limit, representing the projector onto the "GS band" of almost degenerate lowest lying eigenstates.

quant-pharXiv:2604.07633Apr 10, 20260 citations

Statistical mechanics of multipartite entanglement in hypergraph states

Paolo Scarafile, Giorgia Trotta, Paolo Facchi +4 more

We investigate multipartite entanglement in a particular family of pure $n$ -qubit hypergraph states through a statistical-mechanics framework, where the average bipartite purity maps onto an effective Hamiltonian of $2^n$ classical binary spins. In this correspondence, each hypergraph state uniquely corresponds to a classical spin configuration, while temperature serves as a control parameter that continuously interpolates between a uniform ensemble of random hypergraph states at high temperature and maximally multipartite entangled states (MMES) at zero temperature. Remarkably, the exponential of the zero-temperature entropy directly gives the number of MMES within the set of hypergraph states. For small system sizes ( $n \leq 5$ ), we perform an exact enumeration, fully characterizing the energy landscape and associated thermodynamic observables, and validating known MMES counts. For larger systems ( $n = 6$ and $7$ ), where exact methods become computationally infeasible, we employ simulated annealing and parallel tempering algorithms to efficiently sample the exponentially large state space. Our analysis yields quantitative predictions of the number of MMES and reveals how entanglement is statistically distributed across the sets of hypergraph states. These results establish hypergraph states as an ideal platform for investigating multipartite entanglement through thermodynamic methods, offering both computational advances and physical insights into the structure of quantum entanglement in restricted families of quantum states.We investigate multipartite entanglement in a particular family of pure $n$-qubit hypergraph states through a statistical-mechanics framework, where the average bipartite purity maps onto an effective Hamiltonian of $2^n$ classical binary spins. In this correspondence, each hypergraph state uniquely corresponds to a classical spin configuration, while temperature serves as a control parameter that continuously interpolates between a uniform ensemble of random hypergraph states at high temperature and maximally multipartite entangled states (MMES) at zero temperature. Remarkably, the exponential of the zero-temperature entropy directly gives the number of MMES within the set of hypergraph states. For small system sizes ($n \leq 5$), we perform an exact enumeration, fully characterizing the energy landscape and associated thermodynamic observables, and validating known MMES counts. For larger systems ($n = 6$ and $7$), where exact methods become computationally infeasible, we employ simulated annealing and parallel tempering algorithms to efficiently sample the exponentially large state space. Our analysis yields quantitative predictions of the number of MMES and reveals how entanglement is statistically distributed across the sets of hypergraph states. These results establish hypergraph states as an ideal platform for investigating multipartite entanglement through thermodynamic methods, offering both computational advances and physical insights into the structure of quantum entanglement in restricted families of quantum states.

quant-pharXiv:2609.24508Sep 22, 20260 citations

Distribution of the entanglement entropy of a non-ergodic quantum state

Devanshu Shekhar, Pragya Shukla

We theoretically derive the probability densities of the entanglement measures of a pure non-ergodic many-body state, represented in a bipartite product basis and with its reduced density matrix described by a generalized, multi-parametric Wishart ensemble with unit trace. Our results indicate significant fluctuations of the measures around their average behavior (specifically for the states away from separability and maximum entanglement limits). The information is relevant not only for hierarchical arrangement of entangled states (e.g., revealing the flaws in their characterization based on average behavior) but also for phase transition studies of many body systems.We theoretically derive the probability densities of the entanglement measures of a pure non-ergodic many-body state, represented in a bipartite product basis and with its reduced density matrix described by a generalized, multi-parametric Wishart ensemble with unit trace. Our results indicate significant fluctuations of the measures around their average behavior (specifically for the states away from separability and maximum entanglement limits). The information is relevant not only for hierarchical arrangement of entangled states (e.g., revealing the flaws in their characterization based on average behavior) but also for phase transition studies of many body systems.

arXiv:2402.01102Feb 2, 20240 citations

Many-Body Entanglement Properties of Finite Interacting Fermionic Hamiltonians

Irakli Giorgadze, Grayson Welch, Haixuan Huang +2 more

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.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.

quant-pharXiv:2603.11300Mar 13, 20260 citations

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.

We discuss a general bipartitelike representation and Schmidt decomposition of an arbitrary pure state of

N

indistinguishable fermions, based on states of

M<N

and $(N\ensuremath{-}M)$ fermions. It is directly connected with the reduced

M

- and $(N\ensuremath{-}M)$-body density matrices (DMs), which have the same spectrum in such states. The concept of

M

-body entanglement emerges naturally in this scenario, generalizing that of one-body entanglement. Rigorous majorization relations satisfied by the normalized

M

-body DM are then derived, which imply that the associated entropy will not increase, on average, under a class of operations which have these DMs as postmeasurement states. Moreover, such entropy is an upper bound to the bipartite entanglement entropy generated by a class of operations which map the original state to a bipartite state of

M

and $N\ensuremath{-}M$ effectively distinguishable fermions. Analytic evaluation of the spectrum of

M

-body DMs in some strongly correlated fermionic states is also provided.

We investigate multipartite entanglement in a particular family of pure

n

-qubit hypergraph states through a statistical-mechanics framework, where the average bipartite purity maps onto an effective Hamiltonian of

2^n

classical binary spins. In this correspondence, each hypergraph state uniquely corresponds to a classical spin configuration, while temperature serves as a control parameter that continuously interpolates between a uniform ensemble of random hypergraph states at high temperature and maximally multipartite entangled states (MMES) at zero temperature. Remarkably, the exponential of the zero-temperature entropy directly gives the number of MMES within the set of hypergraph states. For small system sizes (

n \leq 5

), we perform an exact enumeration, fully characterizing the energy landscape and associated thermodynamic observables, and validating known MMES counts. For larger systems (

n = 6

and

7

), where exact methods become computationally infeasible, we employ simulated annealing and parallel tempering algorithms to efficiently sample the exponentially large state space. Our analysis yields quantitative predictions of the number of MMES and reveals how entanglement is statistically distributed across the sets of hypergraph states. These results establish hypergraph states as an ideal platform for investigating multipartite entanglement through thermodynamic methods, offering both computational advances and physical insights into the structure of quantum entanglement in restricted families of quantum states.

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

M

-body entanglement in the presence of an Abelian symmetry.