Time Evolution of the 3-Tangle of a System of 3-Qubit Interacting through a XY Hamiltonian

Abstract

We consider a pure 3-qubits system interacting through a XY-Hamiltonian with antiferromagnetic constant J. We employ the 3-tangle as an efficient measure of the entanglement between such a 3-qubit system. The time evolution of such a 3-tangle is studied. In order to do the above, the 3-tangle associated to the pure 3-qubits state|ψ(t)⟩ = c0(t)|000⟩+c1(t)|001⟩+c2(t)|010⟩+c3(t)|011⟩+ c4(t)|100⟩+c5(t)|101⟩+c6(t)|110⟩+c7(t)|111⟩ is calculated as a function of the initial coefficients {ci(t = 0)} (i = 0,1,...,7), the time t and the antiferromagnetic constant J. We find that the 3-tangle of the 3-qubits system is periodic with period t = 4π/J. Furthermore, we also find that the 3-tangle as a function of the time t and J has maximal and minimum values. The maximal values of the 3-tangle can be employed in Quantum Information Protocols (QIP) that use entanglement as a basic resource. The pattern found for the 3-tangle of the system of three qubits interacting through a XY Hamiltonian as a function of J and the time t resembles to a quantized physical quantity.