Entropy as a Projection Artifact: One-Body Operators are Entanglement-Inert on Sparse Lattices

Josh Loutey

Published Jul 9, 2026• 0 citations

Abstract

We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on $S^{3}$. Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm $S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}$ for He at $n_{\max}=2,3$ in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction $V_{ee}$, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling $S_n = k\log A_n$ with $A_n\propto n^{4}$, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement: $A_n=g_n^{2}$, where $g_n=2n^{2}$ is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the $(1s,1s)$ pair-state of the $V_{ee}$ graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling $S_B = A (\tilde{w}_B/\delta_B)^{\gamma}$, where $\tilde{w}_B$ is the dimensionless $V_{ee}$ mass off-diagonal in the one-body eigenbasis and $\delta_B$ the dimensionless first $H_1$ gap; the local exponent extrapolates to $\gamma_{\infty} \approx 1.96$ in the joint $Z, n_{\max}\to\infty$ limit (Richardson on $n_{\max}=2,3,4,5$), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of $2$, with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of $1/r_{12}$. GeoVac Paper 27. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

DOI: 10.5281/zenodo.21286840

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Entropy as a Projection Artifact: One-Body Operators are Entanglement-Inert on Sparse Lattices

Josh Loutey

Published Jul 9, 2026• 0 citations

Publisher

Abstract

We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on

S^{3}

. Five results are presented. First, the von Neumann entropy of any single-particle reduced density matrix derived from a non-degenerate ground state of a purely one-body fermionic Hamiltonian on a finite lattice is identically zero; numerically, we confirm

S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}

for He at

n_{\max}=2,3

in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction

V_{ee}

, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling

S_n = k\log A_n

with

A_n\propto n^{4}

, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement:

A_n=g_n^{2}

, where

g_n=2n^{2}

is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the

(1s,1s)

pair-state of the

V_{ee}

graph; entanglement is co-located with cusp mass. Fourth, on the Bargmann–Segal harmonic oscillator lattice the ground-state entropy of any two-fermion closed-shell state interacting via any central two-body potential is identically zero, a consequence of Moshinsky–Talmi total-quanta conservation; this is a rigidity statement structurally parallel to the Fock and Bargmann–Segal theorems of Paper 24. Fifth, Coulomb-lattice block entropy obeys a universal scaling

S_B = A (\tilde{w}_B/\delta_B)^{\gamma}

, where

\tilde{w}_B

is the dimensionless

V_{ee}

mass off-diagonal in the one-body eigenbasis and

\delta_B

the dimensionless first

H_1

gap; the local exponent extrapolates to

\gamma_{\infty} \approx 1.96

in the joint

Z, n_{\max}\to\infty

limit (Richardson on

n_{\max}=2,3,4,5

), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of

2

, with the residual gap attributed to multi-shell aggregation in the basis-extension limit. Entropy is slotted into the exchange-constant taxonomy as a projection-class observable: zero on the combinatorial graph, finite only after the metric embedding of

1/r_{12}

. GeoVac Paper 27. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

DOI: 10.5281/zenodo.21286840

Related papers

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Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as $O(Q^{3.36})$, and the Pauli 1-norm $\lambda = \sum_i |c_i|$—which controls both Trotter step counts and qubitization query complexity—scales as $O(Q^{1.69})$ with $R^2 = 0.997$. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H$_2$; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of $O(Q^{2.5})$, independent of molecular topology (exponent spread $0.02$ across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref], $Q^{3.9}$–$Q^{4.3}$ span fitted by us)—an advantage that grows with system size due to the $\sim 1.5$ exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH$_2$/H$_2$O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning $Z = 1$–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count: $N_{\mathrm{Pauli}} = 11.10 × Q$ (exact, non-identity terms; $d$-orbital blocks give 9.23). 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STUDIES IN MANY-BODY ENTANGLEMENT OF FERMIONS USING M-BODY DENSITY MATRIX

Irakli Giorgadze

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. 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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

Entanglement in N-harmonium: bosons and fermions

C L Benavides-Riveros, I V Toranzo, J S Dehesa

The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of $N$ particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of $N$ and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number $N_c$ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for $N > N_c$ . For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when $N$ is growing. For fixed $N$ , the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of $N$ particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of $N$ and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number $N_c$ of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for $N > N_c$. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when $N$ is growing. For fixed $N$, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.

arXiv:1404.4447Journal of Physics B Atomic Molecular and Optical PhysicsSep 24, 201435 citations

Entanglement Entropy and GUE Spectral Correlation in Quantum States Constructed from Riemann Zeta Zeros

Tiago Barbosa Dias Maciel

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Josh Loutey

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Jul 9, 20260 citations

We characterize the operator-theoretic origin of entanglement entropy in the GeoVac Coulomb lattice, a finite spectral graph realization of the hydrogenic angular momentum eigenbasis on

S^{3}

S_{\mathrm{kin}}/S_{\mathrm{full}} \sim 10^{-14}

for He at

n_{\max}=2,3

in the angular momentum eigenbasis. All ground-state entanglement is generated by the two-body interaction

V_{ee}

, with the non-degeneracy qualifier essential: open-shell and dissociated-bond states escape the floor. Second, the holographic area-law scaling

S_n = k\log A_n

with

A_n\propto n^{4}

, asserted on one-particle grounds in earlier framework papers, is rigorously recovered as a pair-counting statement:

A_n=g_n^{2}

, where

g_n=2n^{2}

is the shell degeneracy. The factor of four is the two-body signature. Third, in the Coulomb lattice the cusp concentrates as a single ``hot node'' on the

(1s,1s)

pair-state of the

V_{ee}

S_B = A (\tilde{w}_B/\delta_B)^{\gamma}

, where

\tilde{w}_B

is the dimensionless

V_{ee}

mass off-diagonal in the one-body eigenbasis and

\delta_B

the dimensionless first

H_1

gap; the local exponent extrapolates to

\gamma_{\infty} \approx 1.96

in the joint

Z, n_{\max}\to\infty

limit (Richardson on

n_{\max}=2,3,4,5

), slightly below the non-degenerate second-order Rayleigh–Schrödinger value of

2

1/r_{12}

(n, l, m)

. Four computational results are presented. First, energy-entanglement decoupling: the off-diagonal one-body Hamiltonian (graph Laplacian topology, coupling constant

\kappa = -1/16

) contributes

10

–

39%

of the correlation energy but less than

0.2%

of the entanglement entropy; the off-diagonal electron repulsion contributes

100%

of the entanglement. Entanglement entropy scales as

S \sim Z^{-2.56}

across He-like ions. Second, basis-intrinsic sparsity: among orbital bases for systems with spherical symmetry, the angular momentum eigenbasis is the unique basis producing sparse electron repulsion integrals (ERIs). Any unitary rotation of orbital indices, no matter how small, fills the ERI tensor from

42%

99%

density in a step-function transition at the identity. The nonzero ERI count is

Z

-independent, confirming that sparsity is purely geometric. Third, entanglement network topology: orbital mutual information maps reveal a hub migration pattern—

1s \to 2s \to 2p

—that tracks the partially filled shell across the first row of the periodic table. Fourth, core-valence decoupling: the core-valence mutual information falls to

\approx 2×10^{-3}

Z = 4

and below

10^{-3}

for

Z \geq 5

, quantitatively justifying the composed fiber bundle factorization used in the GeoVac molecular framework. In the composed molecular architecture, per-block entanglement is

R

-independent, with core/bond entropy ratios of

50×

at matched qubit count. These results establish that the angular momentum eigenbasis decouples one-body energy transport from two-body entanglement, and that the

O(Q^{2.5})

Pauli scaling of the GeoVac qubit Hamiltonians is a direct consequence of this basis choice. This is Paper 26 in the GeoVac series. GeoVac Paper 26. Part of the GeoVac research corpus (precision observations group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

O(Q^{3.15})

Pauli string terms in the number of qubits

Q

, compared to

O(Q^{4+})

for conventional Gaussian-basis Hamiltonians (our own log-log fit of the published Pauli counts of Trenev et al. [ref] gives

Q^{4.25}

for LiH and

Q^{3.92}

for H

_2

O, spanning

Q^{3.9}

–

Q^{4.3}

). The advantage originates in angular momentum selection rules—enforced by Gaunt integrals at the two-electron integral level—which cause the electron repulsion integral (ERI) tensor density to decay as

\sim 1/M

(empirically

\approx M^{-0.85}

–

M^{-1}

) with the number of spatial orbitals

M

, while Gaussian ERI tensors remain near-dense at

O(1)

. Beyond term counts, we quantify two operationally relevant cost metrics: the number of qubitwise-commuting (QWC) measurement groups scales as

O(Q^{3.36})

, and the Pauli 1-norm

\lambda = \sum_i |c_i|

—which controls both Trotter step counts and qubitization query complexity—scales as

O(Q^{1.69})

with

R^2 = 0.997

. The sub-quadratic 1-norm scaling is the key result for fault-tolerant simulation: it implies that Trotter circuit depth grows far more slowly with system size than term counts alone would suggest. All Gaussian baselines are validated against published integral values (Szabo and Ostlund for H

_2

; analytical STO for He). For composed multi-center systems, the pseudopotential approximation of the composed natural geometry framework [ref] yields block-diagonal electron repulsion integrals with a universal Pauli scaling of

O(Q^{2.5})

, independent of molecular topology (exponent spread

0.02

across three molecules). At equal qubit counts, this represents two or more orders of magnitude fewer Pauli terms than Gaussian baselines (published counts of Trenev et al. [ref],

Q^{3.9}

–

Q^{4.3}

span fitted by us)—an advantage that grows with system size due to the

\sim 1.5

exponent gap. These comparisons are at matched qubit counts, not matched accuracy; the composed basis carries classical equilibrium-geometry errors of 5–26% (LiH/BeH

_2

_2

O; Sec. (ref)), making the framework optimal for fixed-geometry quantum simulation rather than geometry optimisation. Across the composed molecular library (35 molecules spanning

Z = 1

–56, H through Ba, including the first transition series ScH–ZnH), the composed Pauli count is exactly linear in the qubit count:

N_{\mathrm{Pauli}} = 11.10 × Q

(exact, non-identity terms;

d

-orbital blocks give 9.23). These counts are realized under a pair-diagonal angular approximation (

m_a=m_c

m_b=m_d

) adopted deliberately for sparsity; the exact global-

M_L

Coulomb rule (used in the framework's precision-physics paths, and considered but left on the table here) gives

\approx 2.5×

more terms with the same scaling, under which the

d

-block coefficient is in fact higher, not lower (Sec. (ref)). GeoVac Paper 14. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.

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.

The ground-state entanglement of a single particle of the N-harmonium system (i.e., a completely-integrable model of

N

particles where both the confinement and the two-particle interaction are harmonic) is shown to be analytically determined in terms of

N

and the relative interaction strength. For bosons, we compute the von Neumann entropy of the one-body reduced density matrix by using the corresponding natural occupation numbers. There exists a critical number

N_c

of particles so that below it, for positive values of the coupling constant, the entanglement grows when the number of particles is increasing; the opposite occurs for

N > N_c

. For fermions, we compute the one-body reduced density matrix for the closed-shell spinned case. In the strong coupling regime, the linear entropy of the system decreases when

N

is growing. For fixed

N

, the entanglement is found (a) to decrease (increase) for negatively (positively) increasing values of the coupling constant, and (b) to grow when the energy is increasing. Moreover, the spatial and spin contributions to the total entanglement are found to be of comparable size.

2, 8, 20, 40, 70, 112

emerge directly from graph state counting, and the physical magic numbers

2, 8, 20, 28, 50, 82, 126

are recovered with a spin–orbit strength

v_{ls}/\hbar\omega \approx 0.17

and a Nilsson

\ell(\ell+1)

correction

d_{\ell\ell}/\hbar\omega \approx 0.02

. We build the first two-species qubit encoding for the deuteron (

1p+1n

16

qubits,

592

non-identity Pauli terms,

1

-norm

\approx 342

MeV) using the Minnesota nucleon–nucleon potential and Moshinsky–Talmi brackets, and extend the architecture to helium-4 (

2p+2n

16

qubits,

712

Pauli terms) with intra-species antisymmetrization and proton–proton Coulomb repulsion. Despite a Hilbert-space dimension

12.25×

larger than the deuteron, He-4 requires only

1.20×

more Pauli terms — a direct consequence of the angular selection rules that carry through to multi-particle systems. A composed nuclear–electronic deuterium encoding (

26

qubits) validates the multi-block architecture and quantifies the

\sim 10^{13}

coefficient ratio between nuclear, electronic, and hyperfine scales. We state and apply the Fock projection rigidity theorem: the

S^3

conformal projection that makes electronic Hamiltonians equivalent to a graph Laplacian is unique to the Coulomb potential and does not extend to HO or Woods–Saxon systems. For nuclear Hamiltonians, the GeoVac contribution is therefore the angular machinery and qubit encoding, not the conformal projection. All nuclear physics inputs (Minnesota parameters, spin–orbit strength,

\hbar\omega

) follow the standard shell-model literature; the quantitative binding energies at

N_{\mathrm{shells}}=2

are far from experiment and should be read as encoding-validation benchmarks, not as a nuclear-structure calculation. GeoVac Paper 23. Part of the GeoVac research corpus (quantum computing group). Code, tests, and the full paper series: https://github.com/jloutey-hash/geovac. Corpus archive DOI: 10.5281/zenodo.20482394. Corpus version at deposit: v4.75.0.