Entanglement and off-diagonal long-range order of an η-pairing state

Off-diagonal long-range order (ODLRO) which is believed to be one characteristic of superconductivity is a quantum phenomenon not describable in classical mechanical terms. The quantum state constructed by eta-pairing demonstrates ODLRO. Entanglement is a key concept of the quantum information processing and has no classical counterpart. We study the entanglement property of the eta-pairing quantum state by concurrence and entropy which are two measures of the entanglement. We show that the concurrence of entanglement between one-site and the rest sites is exactly the correlation function of the ODLRO for the eta-pairing state in the thermodynamic limit. So, when the eta-pairing state is entangled, it demonstrates ODLRO and is thus in the superconducting phase, if it is a separable state, there is no ODLRO. In the thermodynamical limit, the entanglement between the M-site and other sites of the eta-pairing state does not vanish. Other types of ODLRO of the eta-pairing state are presented. We show that the behaviour of the ODLRO correlation functions is equivalent to that of the entanglement of the eta-pairing state. The scaling of the entropy of the entanglement for the eta-pairing state is studied.