J. Garcia, J.A. Cianciulli and R. Rossignoli of Instituto de Física La Plata and Comisión de Investigaciones Científicas (CIC) have developed new methods to quantify entanglement within the water molecule, analyzing how this quantum property changes as the distance between atoms varies. Their work, grounded in the full configuration interaction approach, introduces novel measures, including “up-down two-body mutual information” and “two-body negativities”, to characterize the electronic eigenstates of the molecule.

The researchers examined entanglement not only in the water molecule’s ground state but also in a thermal state approaching absolute zero, representing a projection onto the lowest lying eigenstates. This analysis aims to provide a general entanglement-based characterization of electronic structure.

Fermionic Entanglement Measures in Molecular Systems

Researchers extended existing entanglement formalisms to encompass higher-order interactions and bosonic systems, creating a hierarchy of measures based on the spectral properties of multi-body reduced density matrices (DMs). These matrices, when applied to quantum chemistry, traditionally focus on the von Neumann entropy of one-particle RDMs and higher-order RDMs, initially tested on small molecular systems. Other quantum information theory techniques include Shannon entropy calculations of Configuration Interaction (CI) expansion coefficients.

Cumulants of the RDMs, representing portions of higher-order RDMs independent of lower orders, have also been employed to describe electron correlation, alongside analyses of static and dynamic correlations. The team introduced new correlation measures for fermionic systems with fixed numbers of up and down particles, beginning with total up-down mutual information and negativity, quantifying correlations between spin-up and spin-down subsystems.

The mutual information measures both classical and quantum correlations, while negativity isolates quantum correlations, differing in how they assess entanglement in mixed states like those found in thermal systems. In pure states, the negativity is directly linked to the up-down entanglement entropy, calculated from singular values resulting from the up-down Schmidt decomposition of the state. A positive value for negativity indicates up-down entanglement even when the mutual information is zero.

Analysis of the one-body reduced density matrix, denoted as $ρ^{1}$ , provides a measure of deviation from a single Slater Determinant (SD), as the matrix is idempotent only when the state is a SD. The elements of $ρ^{1}$ are defined by the expectation value of creation and annihilation operators, offering a pathway to quantify entanglement independent of specific spin modes.

Up-Down Entanglement via Schmidt Decomposition of Water

The research uses the Schmidt decomposition to examine total up-down entanglement within the electronic wave function, providing a granular view of electron correlation. This approach allows for quantifying the interplay between spin-up and spin-down electrons as the molecule stretches and compresses, offering insights into bond behavior at the quantum level. Beyond a global measure of entanglement, the work also focuses on one- and two-body reduced density matrices, which assess deviation from a simple independent-particle picture.

The exact eigenstates of the water molecule can be written as a sum over the Schmidt decomposition, as detailed in the paper. This entropy, derived from the partial traces of the density matrix, quantifies the entanglement between electrons with opposite spins. The authors introduce and analyze 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.

One- and Two-Body Reduced Density Matrices for Correlation

Beyond quantifying total entanglement, recent work focuses on characterizing correlations within the electronic structure of molecules using one- and two-body reduced density matrices (RDMs). These matrices provide a detailed picture of how electrons interact, moving beyond a simple assessment of overall entanglement to pinpoint correlations at specific levels. The analysis examines not only how much entanglement exists, but also how it manifests in the deviation of a molecular state from a single Slater determinant, a foundational concept in quantum chemistry. Any departure from this single determinant indicates the presence of entanglement and correlation effects.

The approach allows for a basis-independent quantification of correlation effects, meaning the results are not tied to a specific mathematical representation of the electronic wave function. As the paper states, “Unlike systems of distinguishable components, where entanglement is defined via the tensor product structure of the Hilbert space, fermionic systems require a formulation that explicitly respects the antisymmetry principle,” highlighting the unique challenges of applying entanglement concepts to systems governed by Fermi-Dirac statistics.

Quantifying Deviation from Slater Determinants

Central to this work is the analysis of reduced density matrices, which describe the state of a subset of electrons within the molecule. The team’s approach quantifies both the overall deviation from a Slater determinant and the up-down correlation at the two-body level, examining all blocks of these matrices to build a detailed picture of electron interactions.

They introduced “up-down two-body mutual information” and two types of “two-body negativities” as novel measures, designed to isolate and quantify the “inner” entanglement within the reduced two-body density matrices, essentially, how far these matrices are from being representable as a mixture of single determinants. The researchers observed that, for large internuclear distances, this value approaches 13/6, with 15 negative eigenvalues.

Up-Down Two-Body Mutual Information as a Correlation Tool

Beyond quantifying overall entanglement, researchers have focused on measures sensitive to correlations between pairs of electrons, specifically examining the “inner” entanglement within reduced two-body density matrices. This approach introduces new tools for understanding how electrons interact within molecules, going beyond simple assessments of whether entanglement exists to characterizing its specific nature at the two-body level. These two-body negativities vanish when describing a state as a convex mixture of Slater determinants, providing a sensitive indicator of non-classical correlations within the two-electron sector.

This convergence occurs because the electronic wave function, in that limit, can be expressed as linear combinations of products of electron pairs, simplifying the correlation structure. The study also compared these measures for both the ground state and a thermal state at extremely low temperatures, represented as a projector onto the “GS band” of almost degenerate lowest lying eigenstates.

Two-Body Negativities Reveal “Inner” Entanglement

Researchers have introduced and analyzed two types of “two-body negativities” as measures of this internal entanglement, quantifying the deviation of the reduced DMs from what would be expected in a simple mixture of Slater determinants. These new metrics provide a sensitive probe for non-classical correlations within the two-body sector of the electronic structure, offering a more detailed picture of electron interactions. This approach allows for the investigation of entanglement characteristics in conditions approaching energetic instability, revealing how these correlations persist even as the molecule begins to dissociate.

The study found that both full and two-body up-down negativities approach a constant value as internuclear distances increase, suggesting a fundamental limit to how much entanglement can be lost during molecular separation. The paper states, referencing the behavior of the negativity measures as the distance between atoms increases.

Full Configuration Interaction Approach to Electronic Eigenstates

The full configuration interaction approach, used to analyze the water molecule, reveals that entanglement characteristics shift predictably as the distance between oxygen and hydrogen atoms changes. These calculations, performed with the STO-3G basis set, demonstrate a smooth evolution from a near single-determinant state at equilibrium to a correlated state with increasing distance.

The weighting factors, q_n, are determined by the Boltzmann distribution, dependent on the energy difference between each eigenstate and the ground state, and a parameter β. This approach moves beyond static snapshots of molecular structure to explore how entanglement behaves in a dynamic, albeit low-temperature, environment. The team’s work introduces new tools for characterizing these quantum correlations.

Dissociation Limit Analysis of Ground and Thermal States

Eigenvalues of these RDMs, alongside those of the total density matrix, were calculated for both the ground state and the thermal state in the dissociation limit, providing a detailed picture of electronic correlation. Analytical limits for these values as the internuclear distance approaches infinity are 2(2/3 + log_23) for the ground state and differing values for the thermal state, indicating a shift in electronic structure.

Spectral Properties of M-Body Reduced Density Matrices

These RDMs reveal not only how much a system deviates from a simple Slater Determinant, a standard reference point in quantum chemistry, but also the degree of up-down correlation at the two-body level, providing a more subtle picture of electronic interactions. All blocks of these matrices were examined to fully map these correlations.

In the dissociation limit, these eigenvalues converge towards rational numbers distinct from those of the ground state, with specific degeneracy numbers readily derived from prior asymptotic expressions. This approach allows for a detailed understanding of entanglement characteristics under varying conditions, extending beyond static descriptions of molecular bonding.

👉 More information
🗞 Fermionic entanglement and quantum correlation measures in molecules
✍️ J. Garcia, J. A. Cianciulli and R. Rossignoli
🧠 DOI: http://link.aps.org/doi/10.1103/7l1l-1czq

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