Equilibrium State and Entanglement

G. B. Furman, V. Meerovich, Vladimir Sokolovsky

It was recently shown that entanglement in quantum systems being in a nonequilibrium state can appear at much higher temperatures than in an equilibrium state. However, any system is subject to the natural relaxation process establishing equilibrium. The work deals with the numerical study of entanglement dynamics in a dipolar coupled spin-1/2 system under the transition from a nonequilibrium state to an equilibrium state. The spin system is characterized by a two-temperature density matrix, and the process of the establishment of equilibrium is in the equalization of these temperatures. The method of the nonequilibrium statistical operator is used to describe the evolution of the system. The process of establishing an equilibrium state in the homonuclear spin systems at low temperature was first considered. It was shown that the time dependencies of the inverse temperatures of the spin subsystems are given by a solution of nonlinear equations in contrast to the linear equations in the well-known high-temperature approximation. We have first studied the entanglement dynamics during the equilibrium state establishment and have found that the concurrence changes nonmonotonically with time and temperatures while equilibrium is being established. Entanglement fades long before equilibrium is established in the system. It was shown that the entanglement dynamics depends strongly on the ratio of the Zeeman energy to the dipolar energy. At a high ratio, the concurrence in the system decreases quickly for a time of about 100 $\ensuremath{\mu}\text{s}$ whereas, at a low ratio, establishment of equilibrium and fading entanglement takes a prolonged time up to 1 $\text{ms}$.

Shinji Mukohyama A

Stig Stenholm

Jinshan Wu

Using the representation introduced in \cite{frame}, an artificial game in quantum strategy space is proposed and studied. Although it has well-known classical correspondence, which has classical mixture strategy Nash Equilibrium states, the equilibrium state of this quantum game is an entangled strategy (operator) state of the two players. By discovering such behavior, it partially shows the independent meaning of the new representation. The idea of entanglement of strategies, instead of quantum states, is proposed, and in some sense, such entangled strategy state can be regarded as a cooperative behavior between game players.

Paul Cummins

Ahlem Abidi

The quick progress in quantum entanglement research allows us not one to study quantum systems down to N-bodies but also to take a new look at these systems in different branches of physics; particularly the statistical thermodynamics where the application of the thermo-field dynamics (TFD) method to the investigation of entanglement is fruitful. Because the traditional methods based on the identification of a specific parameter show their limit. The process using (TFD) facilitates the understanding of entanglement because it focuses directly on the eigenstate of the system and it is useful in the equilibrium and the non-equilibrium states also. In this context, the (TFD) method is used in this paper to analyze entanglement of an electron interacting with a two-mode electromagnetic field assimilated to an electron with two harmonic oscillators. Entanglement entropies are derived between concerned, not concerned harmonic oscillator and electron compute when the system is in a thermodynamic equilibrium and non-equilibrium state. For the equilibrium case, an increase in the number of particles per unit volume increases the quantum entanglement consequently entanglement appears more important for the couple oscillator-electron than the one electron, this trend is reversed for the non-equilibrium case. By respecting the entanglement parameters, such results allow us to know the relative equilibrium state of the overall system.