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Journal of Chemical Theory and Computation

Entanglement and Mutual Information in Molecules: Comparing Localized and Delocalized Orbitals

Lorenzo Tenti, Stefan Peeters, Emmanuel Giner, Celestino Angeli

Published Dec 4, 2024• 4 citations
Publisher

Abstract

The use of the mutual information (MI) as a measure of the entanglement in quantum systems has gained a consensus in recent years, even if there is an ongoing effort to distinguish the classical and quantum contributions contained therein. This quantity has been first introduced in condensed matter physics, in particular, in studies based on the density matrix renormalization group method. This method has been successfully adapted to quantum chemistry problems, opening the way to compute MI also in molecular systems. A key aspect of this quantity is its dependence on the one-electron (orbital) basis set, even for wave functions that are invariant under unitary transformation of the orbitals. In this work, we investigate the role of the orbital basis set (delocalized or localized, following different strategies) for wave functions expressed as linear combinations of Slater determinants and we give the analytic expression for the MI for a few special cases. This study aims to improve the knowledge of the relationship between the characteristics of the chemical bond (considering a few paradigmatic molecules, H 2, F 2, N 2, and short linear polyenes) and the properties of interest in the field of quantum information theory.

DOI
10.1021/acs.jctc.4c01101

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•Feb 26, 2013•0 citations
The use of the mutual information (MI) as a measure of the entanglement in quantum systems has gained a consensus in recent years, even if there is an ongoing effort to distinguish the classical and quantum contributions contained therein. This quantity has been first introduced in condensed matter physics, in particular, in studies based on the density matrix renormalization group method. This method has been successfully adapted to quantum chemistry problems, opening the way to compute MI also in molecular systems. A key aspect of this quantity is its dependence on the one-electron (orbital) basis set, even for wave functions that are invariant under unitary transformation of the orbitals. In this work, we investigate the role of the orbital basis set (delocalized or localized, following different strategies) for wave functions expressed as linear combinations of Slater determinants and we give the analytic expression for the MI for a few special cases. This study aims to improve the knowledge of the relationship between the characteristics of the chemical bond (considering a few paradigmatic molecules, H 2, F 2, N 2, and short linear polyenes) and the properties of interest in the field of quantum information theory.
A recent development in quantum chemistry has established the quantum mutual information between orbitals as a major descriptor of electronic structure. This has already facilitated remarkable improvements in numerical methods and may lead to a more comprehensive foundation for chemical bonding theory. Building on this promising development, our work provides a refined discussion of quantum information theoretical concepts by introducing the physical correlation and its separation into classical and quantum parts as distinctive quantifiers of electronic structure. In particular, we succeed in quantifying the entanglement. Intriguingly, our results for different molecules reveal that the total correlation between orbitals is mainly classical, raising questions about the general significance of entanglement in chemical bonding. Our work also shows that implementing the fundamental particle number superselection rule, so far not accounted for in quantum chemistry, removes a major part of correlation and entanglement seen previously. In that respect, realizing quantum information processing tasks with molecular systems might be more challenging than anticipated.
Natural orbitals, defined in electronic structure and quantum chemistry as the molecular orbitals diagonalizing the one-particle reduced density matrix of the ground state, have been conjectured for decades to be the perfect reference orbitals to describe electron correlation. In the present work we applied the Wave function-Adapted Hamiltonian Through Orbital Rotation (WAHTOR) method to study correlated empirical ansätze for quantum computing. In all representative molecules considered, we show that the converged orbitals are coinciding with natural orbitals. Interestingly, the resulting quantum mutual information matrix built on such orbitals is also maximally sparse, providing a clear picture that such orbital choice is indeed able to provide the optimal basis to describe electron correlation. The correlation is therefore encoded in a smaller number of qubit pairs contributing to the quantum mutual information matrix.
Abstract The second quantum revolution is all about exploiting the quantum nature of atoms and molecules to execute quantum information processing tasks. To boost this growing endeavor and by anticipating the key role of quantum chemistry therein, our work establishes a framework for systematically exploring, quantifying and dissecting correlation effects in molecules. By utilizing the geometric picture of quantum states we compare—on a unified basis and in an operationally meaningful way—total, quantum and classical correlation and entanglement in molecular ground states. To unlock and maximize the quantum informational resourcefulness of molecules an orbital optimization scheme is developed, leading to a paradigm-shifting insight: a single covalent bond equates to the entanglement 2 ln ( 2 ) . This novel and more versatile perspective on electronic structure suggests a generalization of valence bond theory, overcoming deficiencies of modern chemical bonding theories.
The chemical bond is an important local concept to understand chemical compounds and processes. Unfortunately, like most local concepts, the chemical bond and the bond order do not correspond to any physical observable and thus cannot be determined as an expectation value of a quantum chemical operator. We recently demonstrated [Boguslawski et al., J. Chem. Theory Comput., 2013, 9, 2959-2973] that one- and two-orbital-based entanglement measures can be applied to interpret electronic wave functions in terms of orbital correlation. Orbital entanglement emerged as a powerful tool to provide a qualitative understanding of bond-forming and bond-breaking processes, and allowed for an estimation of bond orders of simple diatomic molecules beyond the classical bonding models. In this article we demonstrate that the orbital entanglement analysis can be extended to polyatomic molecules to understand chemical bonding.
Orbital entropies, pair entropies, and mutual information have become popular tools for analysis of strongly correlated wave functions. They can quantitatively measure how strongly an orbital participates in the electron correlation and reveal the correlation pattern between different orbitals. However, this pattern can become rather complicated and sometimes difficult to interpret for large active spaces and is not invariant with respect to the spin projection (MsM_sMs​) component of the spin multiplet state. We introduce a modified spin-free orbital entropy, pair entropy, and mutual information, which simplify the correlation analysis and are invariant with respect to MsM_sMs​. By comparison of these quantities with their "original" spin-including counterparts one can distinguish static correlation due to spin couplings from the "genuine" strong correlation due to a multiconfigurational character of the wave function. Recently, Evangelista introduced mutual correlation computed from the two-body cumulant as an alternative correlation measure, which is easier to compute for some methods. We present here a spin-free analogue of this quantity, which is MsM_sMs​ invariant, and perform a comparison of the entropy-based and cumulant-based correlation measures in both spin-free and original variants. We illustrate the approaches on iron-sulfur bound complexes with one and two iron atoms.
The thesis aims to study the applicability of techniques of Quantum Information theory to complex systems, like molecules and biomolecules. Recently, concepts of Quantum Information theory have been applied to the study of phenomena involving molecules, for instance the interaction light-biomolecules. In particular, the entanglement has been proposed to quantify the non-classicality of interactions. The computation of molecular structures and interactions, on the other hand, has been studied for decades mainly to understand reactions? kinetic and energetic characteristics. It is not straightforward to apply these methods to the computation of quantum correlations. We have therefore explored the computability of such effects using Quantum Information techniques. We started with the study of the electronic correlations in the Helium atom, that is the simplest non trivial case, in order to evaluate the computational problems and to develop approximate methods suitable to more complex systems. We start recalling the definition and basic properties of entanglement, as it is considered in Quantum Information theory. We illustrate how it can be measured and the difficulties one encounters when identical particles are involved, then we concentrate on the case of Helium in particular. In order to study the entanglement in Helium, we treat separately the singlet and the triplet configurations. The method we use consists in computing the reduced density matrix and then the entropy. The possible choices of coordinates, basis functions and computation strategies are discussed, together with their impact on the algorithms. The reduced, single-electron von Neumann and linear entropy for several low-energy eigenstates of Helium are computed by means of a simple configuration-interaction variational method.