Geometric measure of entanglement for pure states and mean value of spin

TL;DR

This paper derives an explicit geometric measure of entanglement for pure states, relates it to mean spin values, and applies it to various quantum states, enabling experimental entanglement estimation and analysis of bipartite entanglement in complex systems.

Contribution

It provides a new explicit formula linking entanglement with mean spin, applicable to various quantum states, and demonstrates its usefulness in calculating entanglement in spin systems.

Findings

Explicit formula for geometric entanglement in spin systems

Entanglement relates to mean spin measurement, facilitating experiments

Maximal bipartite entanglement in Dicke and trigonometric states

Abstract

We derive an explicit expression for geometric measure of entanglement for spin and other quantum system. A relation of entanglement in pure state with the mean value of spin is given, thus, at the experimental level the local measurement of spin may allow to find the value of entanglement. The obtained form of the measure is applied to the explicit characterization of bipartite entanglement for $n$-qubit systems in the Werner state, Dicke state, GHZ state and trigonometric states. In particular for Werner-like states the rule of sums is found and it is shown that deviations from the symmetricity of such states diminishes the amount of entanglement. For Dicke states the maximal value of bipartite entanglement is achieved when number of excitations is half of the total number of qubits in these states. For trigonometric states the bipartite entanglement is maximal and does not depend on the number of qubits. We also consider entanglement of discrete-continuous systems on the example of entanglement of spin with continuous variables of electron. The relation of entanglement with the mean value of spin is very useful for calculation of entanglement. With the help of this relation we find in explicit form the entanglement of the one spin with the rest in a spin chain during the evolution determined by the Ising Hamiltonian.