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-qubit state | ψ t = c 0 t | 000 + c 1 t | 001 + c 2 t | 010 + c 3 t | 011 + c 4 t | 100 + c 5 t | 101 + c 6 t | 110 + c 7 t | 111 is calculated as a function of the initial coefficients c i t = 0 i = 0,1 , … , 7, the time t and the antiferromagnetic constant J. We find that the 3-tangle of the 3-qubit system is periodic with period t = 4 π / 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.