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Israel Journal of Chemistry

Quantum Entanglement and Electron Correlation in Molecular Systems

Hefeng Wang, Sabre Kais

Published Dec 1, 2007• 28 citations
Publisher

Abstract

Abstract We study the relation between quantum entanglement and electron correlation in quantum chemistry calculations. We prove that the Hartree–Fock (HF) wave function does not violate Bell's inequality, and thus is not entangled, whereas the configuration interaction (CI) wave function is entangled since it violates Bell's inequality. Entanglement is related to electron correlation and might be used as an alternative measure of the electron correlation in quantum chemistry calculations. As an example we show the calculations of entanglement for the H2 molecule and how it correlates with the traditional electron correlation, which is the difference between the exact and the HF energies.

DOI
10.1560/ijc.47.1.59

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Abstract We study the relation between quantum entanglement and electron correlation in quantum chemistry calculations. We prove that the Hartree–Fock (HF) wave function does not violate Bell's inequality, and thus is not entangled, whereas the configuration interaction (CI) wave function is entangled since it violates Bell's inequality. Entanglement is related to electron correlation and might be used as an alternative measure of the electron correlation in quantum chemistry calculations. As an example we show the calculations of entanglement for the H2 molecule and how it correlates with the traditional electron correlation, which is the difference between the exact and the HF energies.
We study the relation between quantum entanglement and electron correlation in quantum chemistry calculations. We prove that the Hartree-Fock (HF) wave function does not violate Bell's inequality, thus is not entangled while the configuration interaction (CI) wave function is entangled since it violates Bell's inequality. Entanglement is related to electron correlation and might be used as an alternative measure of the electron correlation in quantum chemistry calculations. As an example we show the calculations of entanglement for the H2_22​ molecule and how it is related to electron correlation of the system, which is the difference between the exact and the HF energies.
Electron–electron correlation in quantum chemistry calculations can be analysed in terms of entanglement between electrons. Two exactly solvable models: two fixed spin-1/2 particles and two-electron two-site Hubbard model are used to define and discuss the entanglement as a function of the system parameters. Ab initio configuration interaction calculation for entanglement is presented for the H2 molecule. Qualitatively, entanglement and electron–electron correlation have similar behaviour. Thus, entanglement might be used as an alternative measure of electron correlation in quantum chemistry calculations.
We use the concept of quantum entanglement to give a physical meaning to the electron correlation energy in systems of interacting electrons. The electron correlation is not directly observable, being defined as the difference between the exact ground state energy of the many-electron Schr ̈odinger equation and the Hartree–Fock energy. Using the configuration interaction method for the hydrogen molecule, we calculate the correlation energy and compare it with the entanglement as a function of the nucleus– nucleus separation. In the same spirit, we analyze a dimer of ethylene, which represents the simplest organic conjugate system, changing the relative orientation and distance of the molecules to obtain the configuration corresponding to maximum entanglement.
This thesis contains four projects: entanglement and quantum phase transition in one-dimensional quantum dots system; entanglement and electron correlation in molecular systems; quantum teleportation in one-dimensional quantum dots system and simulation of molecular systems on a quantum computer. Chapter 1 presents a general introduction to quantum computing. A brief history and motivation for quantum computing is introduced. In Chapter 2, we studied the entanglement of an array of quantum dots system modeled by one-dimensional Hubbard Hamiltonian and the relation between quantum entanglement and quantum phase transition. In Chapter 3, entanglement in molecular systems is studied. It is proved that the full configuration interaction (FCI) wave function violates Bell’s inequality, thus it is entangled; while the Hartree-Fock (HF) wave function does not violate Bell’s inequality, so it is unentangled. Based on this, we studied the relation between quantum entanglement and electron correlation in molecular systems. We suggested that quantum entanglement might be used as an alternative way to measure electron correlation in molecular systems. The application of entanglement in an array of quantum dots system in quantum teleportation is studied in Chapter 4. We showed that the space-spin entanglement in the quantum dots system can be filtered to obtain pure spin (space) entanglement, and it can be used in quantum teleportation. We also showed that by selecting appropriate Hamiltonian for the C-NOT gate, the spin-based information can be transformed into charge-based information. The pulse sequence for realizing the C-NOT operation is given. In Chapter 5, we studied the simulation of molecular systems on a quantum computer. We proposed a quantum algorithm to solve the Schrödinger equation using the basis set method based on the multi-configuration self-consistent (MCSCF) wave function. By using this algorithm, the potential energy surfaces of molecules, even some complicated regions, can be studied more efficiently than if the simpler HF wave function was employed. This algorithm can give high success probability, almost deterministic results for ground state and excited states. A small increase in the number of configuration state functions in the MCSCF wave function can dramatically increase the success probability of the quantum algorithm. A multi-reference configuration interaction approach is suggested for the treatment of larger systems. We generalized this method to the finite element method and simulated a quantum system of one particle in Yukawa potential. We showed that numerical methods for solving differential equations can be implemented efficiently on a quantum computer.
Recent proposals of measuring bipartite Renyi entropy experimentally involve techniques that hold exactly for non-interacting quantum particles. Here we consider the difference between such measurements and the actual Renyi entropy for ground state fermion ab initio molecular systems. To calculate various entanglement measures we extended several different techniques for use with variational Monte Carlo, Hartree-Fock and quantum chemistry methods. Our results show that in systems with strong electron correlations the Renyi entropy may not be accurately determined with the proposed measurements. In addition, we find significant physical insight can be gained by calculating entanglement properties in molecular systems. We see that the Renyi entropy and the entanglement Hamiltonian encode information about the character of the covalent bonds in a molecular system and that such information may lead to better descriptions of bonded systems that have been traditionally hard to describe with standard techniques. In particular our results for the C2_{2}2​ dimer suggests all eight valence electrons play a significant role in covalent bonding.
The concept of correlation is central to all approaches that attempt the description of many-body effects in electronic systems. Multipartite correlation is a quantum information theoretical property that is attributed to quantum states independent of the underlying physics. In quantum chemistry, however, the correlation energy (the energy not seized by the Hartree-Fock ansatz) plays a more prominent role. We show that these two different viewpoints on electron correlation are closely related. The key ingredient turns out to be the energy gap within the symmetry-adapted subspace. We then use a few-site Hubbard model and the stretched H2{\mathrm{H}}_{2}H2​ to illustrate this connection and to show how the corresponding measures of correlation compare.