Understanding quantum entanglement by thermo field dynamics

• We introduce a new method to study quantum entanglements using the TFD.
• An extended density matrix is defined in the double Hilbert space.
• Our new method with TFD can be applied to the non-equilibrium systems.
• The intrinsic quantum entanglement can be extracted.
• The extended entanglement entropy is defined using the extended density matrix.

We propose a new method to understand quantum entanglement using the thermo field dynamics (TFD) described by a double Hilbert space. The entanglement states show a quantum-mechanically complicated behavior. Our new method using TFD makes it easy to understand the entanglement states, because the states in the tilde space in TFD play a role of tracer of the initial states. For our new treatment, we define an extended density matrix on the double Hilbert space. From this study, we make a general formulation of this extended density matrix and examine some simple cases using this formulation. Consequently, we have found that we can distinguish intrinsic quantum entanglement from the thermal fluctuations included in the definition of the ordinary quantum entanglement at finite temperatures. Through the above examination, our method using TFD can be applied not only to equilibrium states but also to non-equilibrium states. This is shown using some simple finite systems in the present paper.