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arxiv: 2506.18709 · v3 · submitted 2025-06-23 · ❄️ cond-mat.str-el · quant-ph

Local classical correlations between physical electrons in Hubbard systems

Gabriele Bellomia , Adriano Amaricci , Massimo Capone This is my paper

Pith reviewed 2026-05-19 08:08 UTC · model grok-4.3

classification ❄️ cond-mat.str-el quant-ph
keywords Hubbard modellocal correlationsclassical correlationsreduced density matrixnatural orbitalsmutual informationentanglement
0
0 comments X

The pith

In Hubbard-type models conserving orbital- and spin-resolved electron numbers, local electron correlations are fully classical.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper computes local nonfreeness, an unbiased measure of electron correlation at one lattice site, as the mutual information between local natural spin orbitals. It then proves that in Hubbard-type models preserving the electron count per orbital and per spin, the local reduced density matrix factors completely into independent parts in the natural basis. This separability implies that the correlations remain classical, containing no quantum features beyond any entanglement that may be present. Comparisons across different treatments of magnetic and nonmagnetic states further show that the strength of these local classical correlations is controlled by processes occurring between distant sites.

Core claim

Local electron correlations in Hubbard-type models that conserve the orbital- and spin-resolved electron number are fully classical, since the local reduced density matrix is separable in the natural basis and no quantum correlations beyond entanglement are present.

What carries the argument

Separability of the local reduced density matrix in the natural spin-orbital basis, which converts local nonfreeness into mutual information between those orbitals.

If this is right

  • Local nonfreeness equals classical mutual information between natural spin orbitals under the stated conservation laws.
  • Different approximate theories of magnetic and nonmagnetic states produce markedly different values of these local classical correlations because of their treatment of inter-site processes.
  • The link between local classical correlations and nonlocal entanglement supplies a route to relate conventional quantum resources to nonfreeness using quantities that are experimentally accessible.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • In any lattice model obeying the same per-orbital and per-spin number conservation, local sites should behave as effectively classical objects for purposes of correlation measures.
  • Experiments that simulate Hubbard physics in ultracold atoms or quantum-dot arrays could test the predicted separability by reconstructing the local density matrix.
  • The result suggests a broader pattern: conservation laws that fix local particle numbers may systematically suppress local quantum resources other than entanglement.

Load-bearing premise

Hubbard-type models conserve the number of electrons separately for each orbital and each spin direction at every site.

What would settle it

A direct calculation or measurement that finds irreducible quantum correlations beyond entanglement inside the local reduced density matrix of a Hubbard model while the orbital- and spin-resolved electron numbers remain strictly conserved.

Figures

Figures reproduced from arXiv: 2506.18709 by Adriano Amaricci, Gabriele Bellomia, Massimo Capone.

Figure 1
Figure 1. Figure 1: FIG. 1. Local quasiparticle weight (top) and local mutual [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Local magnetization (top), local double occupancy [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Local magnetization (top), local double occupancy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

We demonstrate that the local nonfreeness, an unbiased measure of correlation between electrons at a single lattice site, can be computed as the mutual information between local natural spin orbitals. This leads us to prove a general result: local electron correlations in Hubbard-type models that conserve the orbital- and spin-resolved electron number are fully classical, since the local reduced density matrix is separable in the natural basis and no quantum correlations beyond entanglement are present. Finally, we compare different theoretical descriptions of magnetic and nonmagnetic states, showing that local classical correlations are drastically influenced by nonlocal processes. These results confirm the relation between local classical correlations within an open system and nonlocal entanglement and they provide a clear path for the study of the relationship between traditional quantum resources and the nonfreeness in terms of experimentally accessible quantities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that local nonfreeness (an unbiased correlation measure) between electrons on a single site in Hubbard-type models equals the mutual information between local natural spin orbitals. It proves that when the Hamiltonian conserves orbital- and spin-resolved electron number, the local reduced density matrix is separable (diagonal) in the natural basis, so all local correlations are classical with no quantum contributions beyond entanglement. Numerical comparisons across magnetic and nonmagnetic states then illustrate that these local classical correlations are strongly affected by nonlocal processes.

Significance. If the separability result holds, the work supplies a parameter-free link between local classical correlations in an open system and nonlocal entanglement, together with a concrete route to express traditional quantum resources in terms of experimentally accessible natural-orbital quantities. The reliance on conservation laws rather than fitted parameters or self-referential equations is a clear strength.

major comments (2)
  1. [Main proof (following abstract)] The central claim rests on a general proof that conservation of n_{i l σ} forces the local RDM to be diagonal in the natural basis, yet the manuscript provides no explicit derivation steps showing how [H, n_{i l σ}] = 0 eliminates all off-diagonal matrix elements between distinct occupation sectors. This step is load-bearing for the separability and classicality assertions.
  2. [Section deriving nonfreeness as mutual information] The reduction of the mutual-information expression for nonfreeness to a purely classical Shannon quantity is asserted once the RDM is diagonal, but no intermediate algebra is shown that confirms the quantum discord or entanglement terms vanish identically under the stated conservation condition.
minor comments (2)
  1. [Notation and definitions] Define the natural spin-orbital basis explicitly at first use and state whether it is obtained from the one-body reduced density matrix or from a different procedure.
  2. [Numerical results] Add a brief error analysis or convergence check for the numerical comparisons of magnetic versus nonmagnetic states.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. The positive assessment of the significance is appreciated. We address each major comment below and indicate the revisions we will make to strengthen the presentation of the central proof.

read point-by-point responses

  1. Referee: [Main proof (following abstract)] The central claim rests on a general proof that conservation of n_{i l σ} forces the local RDM to be diagonal in the natural basis, yet the manuscript provides no explicit derivation steps showing how [H, n_{i l σ}] = 0 eliminates all off-diagonal matrix elements between distinct occupation sectors. This step is load-bearing for the separability and classicality assertions.

    Authors: We agree that the link between the commutation relation and the vanishing off-diagonal elements of the local RDM merits an expanded derivation. The manuscript states the result but condenses the intermediate algebra. In the revised version we will insert a dedicated paragraph immediately after the statement of the conservation condition. The added steps will show that [H, n_{ilσ}] = 0 implies the many-body eigenstates lie in definite occupation-number sectors for each orbital and spin; the local reduced density matrix is obtained by tracing out all other sites, which preserves the block-diagonal structure in the occupation basis; and the natural spin orbitals, being the eigenbasis of this local RDM, therefore render it strictly diagonal. This establishes separability and the purely classical character of the local correlations. revision: yes

  2. Referee: [Section deriving nonfreeness as mutual information] The reduction of the mutual-information expression for nonfreeness to a purely classical Shannon quantity is asserted once the RDM is diagonal, but no intermediate algebra is shown that confirms the quantum discord or entanglement terms vanish identically under the stated conservation condition.

    Authors: We accept that the explicit cancellation of the quantum-discord and entanglement contributions should be written out. Once the local RDM is diagonal in the natural-orbital basis, the von Neumann entropy of the joint state equals the sum of the individual entropies, so the quantum mutual information reduces exactly to the classical Shannon mutual information. In the revision we will add the short algebraic sequence: I(A:B) = S(ρ_A) + S(ρ_B) − S(ρ_AB) with ρ_AB diagonal implies the off-diagonal coherences (which generate discord) are absent, confirming that only classical correlations remain. This will be placed immediately after the definition of nonfreeness. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central derivation establishes that local electron correlations are classical in Hubbard-type models by showing that conservation of orbital- and spin-resolved electron number (via [H, n_{i l σ}] = 0) forces the local reduced density matrix to be diagonal in the natural spin-orbital basis, hence separable with no coherences between occupation sectors. The mutual information then reduces to a classical Shannon quantity as a direct mathematical consequence of this separability. This follows from the model definition and standard properties of natural orbitals without any fitted parameters, self-referential equations, or load-bearing self-citations. The result is self-contained and externally verifiable from the commutation relations alone.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption of number conservation in Hubbard-type models and on the mathematical property that the local reduced density matrix becomes separable in the natural orbital basis; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption Hubbard-type models conserve the orbital- and spin-resolved electron number
    This conservation is the explicit condition used to prove separability of the local reduced density matrix in the natural basis.

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Forward citations

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