Temperature dependence of interaction-induced entanglement

Both direct and indirect weak nonresonant interactions are shown to produce entanglement between two initially disentangled systems prepared as a tensor product of thermal states, provided the initial temperature is sufficiently low. Entanglement is determined by the Peres-Horodecki criterion, which establishes that a composite state is entangled if its partial transpose is not positive. If the initial temperature of the thermal states is higher than an upper critical value $T_{\mathit{uc}}$ the minimal eigenvalue of the partially transposed density matrix of the composite state remains positive in the course of the evolution. If the initial temperature of the thermal states is lower than a lower critical value $T_{\mathit{lc}}⩽T_{\mathit{uc}}$ the minimal eigenvalue of the partially transposed density matrix of the composite state becomes negative, which means that entanglement develops. We calculate the lower bound $T_{\mathit{lb}}$ for $T_{\mathit{lc}}$ and show that the negativity of the composite state is negligibly small in the interval $T_{\mathit{lb}}<T<T_{\mathit{uc}}$. Therefore the lower-bound temperature $T_{\mathit{lb}}$ can be considered as the critical temperature for the generation of entanglement. It is conjectured that above this critical temperature a composite quantum system could be simulated using classical computers.

Figure 1

The coupling scheme for two directly interacting systems.

Figure 2

(Color online) The shaded area in the parameter space of the inverse initial temperature $T$ of two spins and the logarithm of the inverse coupling strength $\gamma$, represents values of $T$ and $\gamma$, where entanglement does not develop in the course of the evolution. The composite system of two spins evolves from the initial product of thermal states under the Hamiltonian $\widehat{\mathbf{H}}=(1∕2)\omega [{\widehat{\mathbf{\sigma }}}_z^a\otimes \widehat{\mathbf{1}}+(\sqrt{2}-1)\widehat{\mathbf{1}}\otimes {\widehat{\mathbf{\sigma }}}_z^b]+\gamma ({\widehat{\mathbf{\sigma }}}_x^a\otimes {\widehat{\mathbf{\sigma }}}_x^b-{\widehat{\mathbf{\sigma }}}_y^a\otimes {\widehat{\mathbf{\sigma }}}_y^b)$. The evolution is calculated numerically for $\omega =1$. The border of the shaded area represents $T_{\mathit{uc}}$ calculated numerically. The dashed line represents $T_{\mathit{lb}}$ according to Eq. (22). Up to the coupling $\gamma =0.1T_{\mathit{lb}}$ approximates $T_{\mathit{lc}}$ very well. Temperature is measured in units of $\omega$.

Figure 3

(Color online) The time-averaged negativity as a function of initial temperature. The composite system is constructed from two interacting four-level subsystems. The initial state is a product of thermal states. The evolution is generated numerically by the Hamiltonian (2) (for details of the Hamiltonian see the text) with $\gamma =0.05$. The dashed lines correspond to the lower-bound temperature $T_{\mathit{lb}}$, Eq. (22), and to the upper critical temperature $T_{\mathit{uc}}$, found numerically. It can be seen that the entanglement is vanishingly small in the interval $T_{\mathit{lb}}<T<T_{\mathit{uc}}$. Temperature is measured in units of $\omega$.}

Figure 4

(Color online) The time-averaged negativity as a function of initial temperature of the composite system. The composite system is constructed from two interacting four-level subsystems. The initial state is a product of thermal states. The evolution is generated numerically by the Hamiltonian (2) (for details of the Hamiltonian see the text) with $\gamma =0.05$. The dashed lines correspond to the lower-bound temperature $T_{\mathit{lb}}$, Eq. (22), to the numerical value of the upper critical temperature $T_{\mathit{uc}}$ and to the value $T_{uc}^{*}$, corresponding to the largest spectrum spacing $\Delta E_{max}$. We see that entanglement is vanishingly small at $T_{\mathit{lb}}<T<T_{\mathit{uc}}$, as expected, and $T_{uc}^{*}$ is a good approximation to the upper critical temperature $T_{\mathit{uc}}$. Temperature is measured in units of $\omega$.

Figure 5

Scheme of interaction for two noninteraction systems in contact with a common third party.

Figure 6

(Color online) The shaded area in the parameter space of the inverse initial temperature $T$ of the slow spin and the logarithm of the inverse coupling strength $\gamma$, represents values of $T$ and $\gamma$ where entanglement does not develop in the course of the evolution. The composite system of two slow spins interacting with a fast four-level system evolves from the initial product of thermal states under the Hamiltonian (47). The dashed line is the plot of $T_{\mathit{lb}}$, Eq. (46). Up to the coupling $\gamma =1$ its correspondence with the border of the shaded area is very good. Temperature is measured in units of $\omega$.

Figure 7

(Color online) The shaded area in the parameter space of the inverse initial temperature $T$ of the fast spins and the logarithm of the inverse coupling strength $\gamma$ represents values of $T$ and $\gamma$ where entanglement does not develop in the course of the evolution. The composite system of two fast spins interacting with the slow four-level system evolves from the initial product of thermal states under the Hamiltonian (52). The dashed line is the plot of $T_{\mathit{lb}}$, Eq. (51). Up to the coupling $\gamma =1$ its correspondence with the border of the shaded area is good. Temperature is measured in units of $\omega$.