Thermal and ground-state entanglement in Heisenberg XX qubit rings

Quantum entanglement is an important prediction of quantum mechanics and constitutes indeed a valuable resource in quantum information processing [1]. Much efforts are devoted to the study and characterization of it. Very recently one kind of natural entanglement, the thermal entanglement [2–7], is proposed and investigated. The investigations of this type of entanglement, groundstate entanglement [8,9] in quantum spin models, and relations [10,11] between quantum phase transition [12] and entanglement provide a bridge between the quantum information theory and condensed matter physics. Consider a thermal equilibrium state in a canonical ensemble. In this situation the system state is described by the Gibb’s density operator ρT = exp (−H/kT )/Z, where Z =tr[exp (−H/kT )] is the partition function, H the system Hamiltonian, k is Boltzmann’s constant which we henceforth will take equal to 1, and T the temperature. As ρT represents a thermal state, the entanglement in the state is called thermal entanglement [2]. In a recent paper [6] we showed that in the isotropic Heisenberg XXX model the thermal entanglement is completely determined by the partition function and directly related to the internal energy. In general the entanglement can not be determined only by the partition function. In this paper, we consider a physical Heisenberg XX N -qubit ring with a magnetic field and aim to obtain some general results about the thermal and ground-state entanglement. We consider a four-qubit model as an example and examine in detail the properties of entanglement. Both the pairwise and genuinely many-body entanglement are considered. In our model the qubits interact via the following Hamiltonian [13]