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Journal of Nanjing University of Posts and Telecommunications

The Formula of Entanglement Evaluation for Two-Electron Systems

Longyan Gong

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

The formula of entanglement evaluation for singleelectron systems is extended to that for two-electron systems.The accurate formula is derived for the concerrence between two arbitrary sites.It is pointed out that the effect of the interaction between particles on extanglement can be studied with the help of the evaluation of the site-averaged concurrence and spectrum-avaraged concurrence,and the localization properties of two-particle states can be studied with the help of entanglement.

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The formula of entanglement evaluation for singleelectron systems is extended to that for two-electron systems.The accurate formula is derived for the concerrence between two arbitrary sites.It is pointed out that the effect of the interaction between particles on extanglement can be studied with the help of the evaluation of the site-averaged concurrence and spectrum-avaraged concurrence,and the localization properties of two-particle states can be studied with the help of entanglement.
Several convenient formulas for the entanglement of two indistinguishable delocalized spin-1∕21∕21∕2 particles are introduced. These generalize the standard formula for concurrence, valid only in the limit of localized or distinguishable particles. Several illustrative examples are given.
We introduce a new measure of quantum entanglement, the orbital concurrence, designed to quantify orbital correlations in interacting multi-orbital electron systems. This quantity extends the concept of concurrence, originally developed for charge and spin degrees of freedom, to the orbital sector, providing a compact and physically transparent characterization of orbital entanglement. To test the validity and usefulness of this measure, we apply it to the single-site Anderson model, a paradigmatic and exactly solvable system that captures the essential interplay between localized correlated electrons and itinerant ones. In this framework, we derive exact, fully analytical expressions for the orbital concurrence and investigate its behavior in different regimes of interaction strength and hybridization. Our results demonstrate the diagnostic power of the proposed measure and open the way to its use in more complex multi-orbital impurities and lattice models.
Practical realizations of qubits and systems of qubits are implement by particles with states from infinite-dimensional space, such as space of wave functions for coordinate representation of quantum mechanics. Properties of the entangled states of continuous variables differ from such for discrete variables. That occurs due to existence of unlimited operators of observables and their uncertainties. In this work we used and extended idea of entanglement estimation for the case of continuous variables with the help of covariance matrix norm. Well-known concurrence was calculated with the help of covariance of observables. The goal of this paper is to check if it is possible to apply this idea to continuous variables. We consider coherent electron pair as the object of our investigation. We also consider coordinate and momentum of coherent electron pair as basis of observables. In this work, a new characteristic of entanglement is proposed. We evaluated that using correlation function instead of covariance. For the small distances between particles, it coincides with covariance matrix norm but for the big distances, it remains restricted.
Entanglement of spins is analyzed for two electrons extracted from a mixed many-electron state by projecting onto the two-electron subspace. The concurrence formulae are expressed in a compact form for states with a well-defined square of the total spin projection. As an example, the thermal entanglement for a qubit pair with an anisotropic Heisenberg and the Dzyaloshinskii-Moriya interactions in an inhomogeneous magnetic field is given analytically. Remarkably, the concurrence of a pair of electrons with antiparallel spins and in a delocalised orbital state is given by the scalar product of the state with its spin-flipped state and not with the time-reversed state.
We investigate the Hubbard model, and the extended Hubbard model with and without correlated hopping, as an approximation to the average single-site (or local) entanglement of two-electron one-dimensional quantum dot chains with long range interactions. We focus on the effect of the chain length on the approximation accuracy. We also consider the particle-particle spatial entanglement of the three Hubbard models using the technique of Ref. [3] and compare the accuracy of this approximation to the numerically exact results of the nanostructure system.
By mapping the Fock space of many local fermionic modes isomorphically onto a many-qubit space and using the measure of concurrence, this paper studies numerically the mode entanglement of two spinless electrons with on-site interaction U moving in the one-dimensional Harper model. Generally speaking, for electrons in extended regimes (potential parameter λ < 2), the spectrum-averaged concurrence N〈C〉 first decreases slowly as λ increases until its local minimum, then increases with λ until its peak at λ = 2, while for electrons in localized regimes (λ > 2), N〈C〉 decreases drastically as λ increases. The functions of N 〈C〉 versus λ are different for electrons in extended and localized regimes. The maximum of N〈C〉 occurs at the point λ = 2, which is the critical value in the one-dimensional single-particle Harper model. From these studies it can distinguish extended, localized and critical regimes for the two-particle system. It is also found for the same λ that the interaction U always induce the decreases of concurrence, i.e., the concurrence can reflect the localization effect due to the interaction. All these provide us a new quantity to understand the localization properties of eigenstates of two interacting particles.