Entanglement of eta-pairing state with off-diagonal long-range order
  Off-diagonal long-range order (ODLRO) is a quantum phenomenon not describable in classical mechanical terms. It is believed to be one characteristic of superconductivity. The quantum state constructed by eta-pairing which demonstrates ODLRO is an eigenstate of the three-dimensional Hubbard model. Entanglement is a key concept of the quantum information processing and has no classical counterpart. We study the entanglement property of eta-pairing quantum state. The concurrence is a well-known measure of quantum 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 superconducting phase, if it is a separable state, there is no ODLRO. In the thermodynamic limit, the entanglement between M-site and other sites of the eta-pairing state does not vanish. Other types of ODLRO of eta-pairing state are presented. We show that the behavior 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. 
Heng Fan, Seth Lloyd