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Europhysics Letters (EPL)

Correlation energy as a measure of non-locality: Quantum entanglement of helium-like systems

Rodolfo O. Esquivel, Sheila López‐Rosa, J. S. Dehesa

Published Aug 1, 2015• 24 citations
Publisher

Abstract

In this work we discuss the essential quantum origin of correlation energy as measured through the wave function regardless of any extrinsic Hamiltonian. The ambiguous physical meaning of correlation energy is clarified by identifying the non-dynamical correlations inherent in the system state of Slater rank different than one with the quantum phenomenon of nonlocality. This is achieved by relating correlation energy to entanglement as measured through the von Neumann and the linear entropies. Indeed, for helium-like systems with varying Z we observe one-to-one correspondence between entanglement and correlation energy. We present numerical evidence of the linear relation between the correlation energy and the quantum entanglement for various members of the helium isoelectronic series by use of highly correlated wave functions of configuration-interaction type.

DOI
10.1209/0295-5075/111/40009

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In this work we discuss the essential quantum origin of correlation energy as measured through the wave function regardless of any extrinsic Hamiltonian. The ambiguous physical meaning of correlation energy is clarified by identifying the non-dynamical correlations inherent in the system state of Slater rank different than one with the quantum phenomenon of nonlocality. This is achieved by relating correlation energy to entanglement as measured through the von Neumann and the linear entropies. Indeed, for helium-like systems with varying Z we observe one-to-one correspondence between entanglement and correlation energy. We present numerical evidence of the linear relation between the correlation energy and the quantum entanglement for various members of the helium isoelectronic series by use of highly correlated wave functions of configuration-interaction type.
Abstract In this work we have performed state-of-the-art configuration-interaction (CI) calculations to determine the linear and von Neumann entanglement entropies for the helium-like systems with varying nuclear charge Z in the range . The focus of the work resides on determining accurate entanglement values for 2-electron systems with the lowest computational cost through compact CI-wave functions. Our entanglement results for the helium atom fully agree with the results obtained with higher quality wave functions of the Kinoshita type (Dehesa [5]). We find that the correlation energy is linearly related to the entanglement measures associated with the linear and von Neumann entropies of the single-particle reduced density matrizes, which sheds new light on the physical implications of entanglement in helium-like systems. Moreover, we report CI-wave-function-based benchmark results for the entanglement values for all members of the helium isoelectronic series with an accuracy similar to that of Kinoshita-type wave functions. Finally, we give parametric expressions of the linear and von Neumann entanglement measures for two-electron systems as Z varies from 1 to 10.
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.
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.
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.
Abstract We perform a detailed numerical study of the very peculiar thermodynamic properties of the localized high-energy phase of the discrete non-linear Schrödinger equation (DNLSE). A numerical sampling of the microcanonical ensemble done by means of Hamiltonian dynamics reveals a new and subtle relation between the presence of the localized phase and a property of the system that we have called classical entanglement . Our main finding is that a quantity defined for our classical system in perfect analogy with the entanglement entropy of quantum ones, and that we have therefore called S ent , grows with the system size N in the localized phase as S ent ( N ) ∼ log ⁡ ( N ) , therefore revealing the presence of subtle non-local correlations between any finite portion of the system and the rest of it. This manifestation of classical entanglement beautifully captures the lack of system separability in the DNLSE localized phase, revealing how statistical correlations specific to the microcanonical ensemble and non-reproducible in the canonical one, may concur to determine a property totally analogous to the one produced by non-local quantum correlations.
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.