Entanglement Entropies of Non-Equilibrium Finite-Spin Systems

For the purpose of clarifying a new approach to understanding quantum entanglement using thermofield dynamics (TFD), entanglement entropies of non-equilibrium finite-spin systems are examined for both traditional and extended cases. The extended entanglement entropy, $\hat{S}$, is derived, and it is found that the conditions for the maximum entangled state can be obtained through this approach. The capacity of the TFD-based method to distinguish between states in quantum systems is confirmed.

Recently, a new approach to understanding quantum entanglement using thermofield dynamics (TFD) [1][2][3] has been proposed in Ref. [4]. In this new treatment of quantum entanglement with TFD, extended density matrices have been formulated on the double Hilbert space (ordinary and tilde Hilbert spaces), and the entanglement states show a quantummechanically complicated behavior. The new TFD-based method allows the entanglement states to be easily understood, because the states in the TFD tilde space play the role of tracers of the initial states. In the new analysis, a general formulation of the extended density matrices has been constructed and applied to some simple cases. Consequently, it has been found that the intrinsic quantum entanglement can be distinguished from the thermal fluctuations included in the definition of the ordinary quantum entanglement at finite temperatures. Based on the analysis presented in Ref. [4], it has been argued that the new TFD-based method is applicable not only to equilibrium states but also to non-equilibrium states. However, analysis of the entanglement entropies of non-equilibrium systems was not conducted in Ref. [4] and, therefore, examination of the entanglement entropies of nonequilibrium systems with the use of TFD is of current interest. In the present communication, therefore, the "extended" entanglement entropies of non-equilibrium spin systems are intensively investigated in both the dissipative and non-dissipative cases, based upon a TFD algorithm.
Let us consider the spin system described by the Hamiltonian incorporating the spin operators, , of the subsystems, A and B, respectively. The state, |s , of the total system is then denoted by the direct product, |s = |s A , s B = |s A |s B . Using the base, the matrix form of the Hamiltonian (1) is then expressed as For the equilibrium states in terms of Hamiltonian expressed in Eq.(2) , the ordinary density matrix, ρ eq , of this system can be obtained as where β is the inverse temperature and the partition function, Z(β), is defined as Let us turn our attention to a non-equilibrium system with dissipation, which is described by the Hamiltonian of Eq.(2). The time dependence of the ordinary density matrix, ρ(t), of this system is given by the dissipative von Neumann equation [5,6], where with ǫ being a dissipation parameter. The solution of Eq. (5) is expressed as for any initial value, ρ 0 , where the unitary operator, U(t), denotes and ω := J/h. Because the explicit expression of ρ(t) in Eq. (6) is complicated for any initial condition, hereafter, let us confine ourselves to the initial condition, The ordinary entanglement entropy, S, is defined by As can be seen from Eq. (10), S converges to the constant, k B log 2, at t → ∞.
We are now in a position to investigate the extended density matrix,ρ, in the TFD double Hilbert space. Note thatρ has been defined in Ref. [4] as follows: ρ := |Ψ Ψ|, |Ψ := ρ(t) 1/2 s |s,s = ρ(t) 1/2 s |s |s , using the ordinary density matrix, ρ(t), in Eq. (6) where the matrix elements b d1 , b d2 , b cf and b qe are respectively obtained as analytic functions of t, β, ǫ and ω, and correspond to the two diagonal components (d1 and d2), the classical fluctuations (cf) and the quantum entanglements (qe) ofρ A , respectively. The parameter, b qe , in Eq.(12) expresses the quantum entanglement effect. This quantum fluctuation appears only in quantum systems, and it has been used as an order parameter of quantum systems at zero temperature [9][10][11][12][13][14]. The time dependences of the parameter, b qe , in several cases are shown in Figs.1. As can be seen from Eq. (12), only the intrinsic quantum entanglement is extracted clearly in the TFD formulation. In particular, it can be understood that the entangled state of the system emerges through a single product, such as |+ +||− − |, in The "extended" entanglement entropy is defined aŝ using the renormalizedρ A in Eq. (12) [4]. The insertion of Eq.(12) into Eq.(13) and subsequent simplification eventually yield Hence, theŜ expression of this system also incorporates analytic functions of t, β, ǫ and ω, however, the full calculation is quite tedious. So, we show the numerical behaviour of S,Ŝ and b qe for a few cases in Figs.1(a)−(c) (in units of k B = 1). As can be seen from these figures, S,Ŝ and b qe converge to finite values at t → ∞, respectively. As a consequence, the traditional entanglement entropy, S, becomes larger than the extended entanglement entropy,Ŝ, at t → ∞.
In this communication, we have examined the extended entanglement entropies of nonequilibrium spin systems in both the dissipative and non-dissipative cases, based upon the TFD formulation. The extended entanglement entropy,Ŝ, is derived using the extended density matrix,ρ A , and is compared to the traditional entanglement entropy, S, and the pareameter, b qe inρ A . These results are summarized in Figs.1 and show that the conditions yielding the maximum entangled state can be obtained using these three quantities.
However, the amplitudes ofŜ, S and b qe differ, and it seems that b qe inρ A may be a useful tool for studying the quantum entanglement, because it does not require the non-linear calculations such as logarithms.
It is apparent that the new TFD-based method enables us to clearly distinguish between the various states of quantum systems. The parameters b d1 and b d2 correspond to the