Entanglement production by the magnetic dipolar interaction dynamics

Abstract

We consider two qubits prepared in a product state and evolved under the magnetic dipolar interaction (MDI). We describe the dependence of the entanglement generated by the MDI with time, with the interaction parameters, and with the system’s initial state, identifying the symmetry and coherence aspects of those initial configurations that yield the maximal entanglement. We also show how one can obtain maximum entanglement from the MDI applied to some families of partially entangled initial states.

Appendix: Dynamics of local quantum coherence for initial product-pure states

Here, we use the $l_{1}$-norm coherence [31], $C(\rho )=\sum _{j\ne k}|\langle j|\rho |k\rangle |$, to quantify quantum coherence. By taking the partial trace [38] over one of the two dipoles, whose composite state is (8), we obtain the reduced density operator $\rho _{r}=\mathrm {Tr}_{p}(|\Psi _{t}\rangle \langle \Psi _{t}|)$. The quantum coherence of this state reads

$$\begin{aligned} 2^{-2}C^{2}(\rho _{r})= & {} \alpha _{a}^{2}\beta _{a}^{2}(\alpha _{b}^{4}+\beta _{b}^{4})\cos ^{2}t+\alpha _{b}^{2}\beta _{b}^{2}(\alpha _{a}^{4}+\beta _{a}^{4})\sin ^{2}t+2\alpha _{a}^{2}\beta _{a}^{2}\alpha _{b}^{2}\beta _{b}^{2}\cos 2t\cos 4t\nonumber \nonumber \\&\quad -\,\alpha _{a}\beta _{a}\alpha _{b}\beta _{b}(\alpha _{a}^{2}\beta _{b}^{2}+\beta _{a}^{2}\alpha _{b}^{2})\sin 2t\sin 4t. \end{aligned}$$

(A.1)

In Fig. 4, we plot this quantity as a function of time and of the dipole a initial state for some initial states of dipole b. Comparison with Fig. 1 confirms the non-existence of a general temporal correlation between the values of coherence and entanglement.