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

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.

Symmetric hypergraph states: Entanglement quantification and robust Bell nonlocality

“…Despite of its simple definition, it is very hard to compute due to a large number of optimization parameters. The geometric measure has been analytically estimated only for a few classes of states [4,6,7], upper and lower bounds have been derived [8,9,10,11,12], and numerical methods have been considered to find states with high entanglement [13,14,15]. In order to ease the complexity of the problem, symmetries of a quantum state have been utilized.…”

Symmetric hypergraph states: entanglement quantification and robust Bell nonlocality

“…Despite its simple definition, it is very hard to compute due to a large number of optimization parameters. The geometric measure has been analytically estimated only for a few classes of states [4,6,7], upper and lower bounds have been derived [8][9][10][11][12], and numerical methods have been considered to find states with high entanglement [13][14][15]. In order to ease the complexity of the problem, symmetries of a quantum state have been utilized.…”

Determining quantum correlations in bipartite systems - from qubit to qutrit and beyond

“…An interesting aspect of finding the value of correlation measure is a possibility of relating it to the mean values of a selected observables of a system. One example of such relation is given for qubits, where entanglement measure can be expressed in terms of the mean value of spin [4].…”