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Physical Review A

Correlation-coupling entropy as a measure of strong electron correlation and fragment-conditional density spin polarization as a measure of electron entanglement

Robert van der Meer, O. V. Gritsenko

Published Sep 27, 2019• 7 citations
Publisher

Abstract

Quantum entanglement has been one of the hottest topics in current day physics, since it is the driving force behind quantum cryptography, quantum teleportation, and quantum computing. Several measures of quantification of entanglement have been proposed, each of which can often only be applied to a few specific systems. In this paper we derive a kinematic measure of entanglement that is capable of giving a full description of Einstein-Podolsky-Rosen entanglement in molecular systems. The associated ``coupled entropy'' energy contribution generates the correct amount of strong correlation energy for the ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$ prototype systems, and is shown to be able to perform the same feat if it is explicitly used as the correlation component in the density-matrix functional context. And, finally, we propose a nonkinematic way to measure the entanglement of spins by using the conditional density of the system.

DOI
10.1103/physreva.100.032335

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A new entropy measure is proposed for the molecular electronic structure theory. The corresponding many-electron, Shannon-like measure reflects quantum entanglement between electronic subsystems with different spins. The related Renyi entropy is introduced, and its connection with the previously considered collectivity analysis is discussed. Investigation of the second-order reduced density matrix (2-RDM) allows one to define an irreducible, size-consistent component of the 2-RDM entropy and to find the irreducible component of the 2-RDM squared norm. Specific computations on several small molecules are carried out at the full configuration interaction level, providing new insights into non-classical, long-range electron correlations and quantum entanglement in these systems. The local spin analysis and the spin-correlator analysis reveal nontrivial spin structure underlying the entangled states.
One of the most intriguing concepts of quantum mechanics is quantum entanglement. Two physical systems are said to be entangled with respect to a certain variable, if their individual outcomes of the variable are undetermined before measurement, but strictly correlated. Measurement of the variable for only one system immediately determines the outcome for the other system, irrespective of the distance between the two systems. It is this strong correlation that provides a basis for perspective, powerful applications like quantum cryptography and quantum computation. The most studied entangled systems, often called EPR systems (after the pioneers Einstein, Podolsky and Rosen), are pairs of light particles or photon pairs. These photon pairs, usually generated via the optical process of a spontaneous parametric down-conversion, are, in principal, simultaneously entangled in three variables, being polarization (polarization entanglement), energy or time (time entanglement), and momentum or transverse space (spatial entanglement). The contents of this thesis cover new insights in the polarization and spatial entanglement of photons. We have investigated both the entanglement quality and the yield of entangled photons for different experimental geometries that are associated with the production and detection of the photons.
Different entropies which can characterize the inseparable quantum correlation in a two-electron entangled artificial atom proposed by Moshinsky are analyzed. The analysis is based on an exact representation for the trace of the qqq-order one-body reduced density matrix in terms of the interparticle coupling. The entanglement spectrum of the model system and the possible role of the entropy-maximum concept to practical applications of matrices are discussed as well. The ground-state energy, readily obtained from the Schr\"odinger Hamiltonian, is recalculated within a density-matrix-functional representation for its components.
This paper introduces a novel approach to quantifying quantum entanglement using entanglement entropy. Traditional entanglement measures, such as concurrence and entanglement of formation, often struggle to fully capture the intricate correlations present in entangled states. We propose a new definition of entanglement entropy based on the density matrix of the entangled system, explicitly accounting for the correlations between subsystems. This approach leverages information-theoretic entropy calculations to provide a more precise and comprehensive measure of entanglement strength and the degree of correlation. We demonstrate that this entropy-based metric offers advantages over existing measures, particularly in characterizing complex entangled states and distinguishing between different levels of entanglement. The core claim is that entanglement entropy can serve as a more accurate indicator of entanglement than existing metrics. The underlying mechanism centers around a density matrix-based entropy calculation, recognizing the interconnectedness within entangled states. The novelty lies in moving beyond purely state-based descriptions to incorporate a measure of subsystem correlations.
Einstein-Podolsky-Rosen (EPR) steering refers to the type of correlations described in the EPR paradox, where one observer seems to affect (”steer”) the state of other observer by using local measurements. There have been several works regarding characterization and quantification of EPR steering. One characteristic of this non-local correlation is that it can be asymmetric, while entanglement is symmetric. This asymmetric property is relevant for potential applications of EPR steering to quantum information, in particular to quantum cryptography and quantum teleportation. This latter refers to the process where one observer sends an unknown quantum state to Bob, who is in a different location. They communicate by classical means. Here we will show that EPR steering is a necessary resource to obtain secure continuous variable teleportation. We will also consider NOON states, which is an example of an entangled state. For this state, we will present a steering signature. This contribution reviews the work derived in Refs. [1] and [2], which was presented as an invited talk in ELAF 2017.
In order to quantify entanglement between two parts of a quantum system, one of the most used estimators is the Von Neumann entropy. Unfortunately, computing this quantity for large interacting quantum spin systems remains an open issue. Faced with this difficulty, other estimators have been proposed to measure entanglement efficiently, mostly by using simulations in the valence-bond basis. We review the different proposals and try to clarify the connections between their geometric definitions and proper observables. We illustrate this analysis with new results of entanglement properties of spin-1 chains.