Correlation Energy and Entanglement Gap in Continuous Models

TL;DR

This paper explores the relationship between entanglement and correlation energy in a bipartite harmonic oscillator model, revealing how correlation energy acts as an entanglement witness and discussing its physical interpretation.

Contribution

It provides a detailed analysis of the entanglement-energy relationship in the Moshinsky model and introduces the concept of correlation energy as an entanglement gap.

Findings

Correlation energy is nonlinearly related to entanglement for small couplings.

A set of separable states can have larger overlap with the ground state but higher energy.

Correlation energy acts as an energy scale to distinguish entangled states from separable states.

Abstract

Our goal is to clarify the relation between entanglement and correlation energy in a bipartite system with infinite dimensional Hilbert space. To this aim we consider the completely solvable Moshinsky's model of two linearly coupled harmonic oscillators. Also for small values of the couplings the entanglement of the ground state is nonlinearly related to the correlation energy, involving logarithmic or algebraic corrections. Then, looking for witness observables of the entanglement, we show how to give a physical interpretation of the correlation energy. In particular, we have proven that there exists a set of separable states, continuously connected with the Hartree-Fock state, which may have a larger overlap with the exact ground state, but also a larger energy expectation value. In this sense, the correlation energy provides an entanglement gap, i.e. an energy scale, under which measurements performed on the 1-particle harmonic sub-system can discriminate the ground state from any other separated state of the system. However, in order to verify the generality of the procedure, we have compared the energy distribution cumulants for the 1-particle harmonic sub-system of the Moshinsky's model with the case of a coupling with a damping Ohmic bath at 0 temperature.