Universal entanglement timescale for Rényi entropies

Recently it was shown that the growth of entanglement in an initially separable state, as measured by the purity of subsystems, can be characterized by a timescale that takes a universal form for any Hamiltonian. We show that the same timescale governs the growth of entanglement for all Rényi entropies. Since the family of Rényi entropies completely characterizes the entanglement of a pure bipartite state, our timescale is a universal feature of bipartite entanglement. The timescale depends only on the interaction Hamiltonian and the initial state.

Figure 1

(a) $S_2\left({\rho }_A\right)$ for the Fock state with $N=3$ and $C_e=1$ is sinusoidal and $C^{\infty }$ smooth. $S_2$ is compared to the quadratic approximation with timescale $\lambda T_{\mathrm{ent},e}=1/4$ (dashed red line). (b) $S\left({\rho }_A\right)$ for the same state is differentiable, but $d^{2}S/d{t}^2$ is discontinuous at $t=0$ (inset, dashed line). Units of $\ln \left(2\right)$ are used in all figures.

Figure 2

(a) $S_2\left({\rho }_A\right)$ for the coherent state with $\nu =3$ and $C_e=1$. The small-$t$ behavior is independent of $\nu$ and described by the quadratic timescale $\lambda T_{\mathrm{ent},e}=1$ (dashed red line). (b) $S\left({\rho }_A\right)$ for the same state is differentiable, but $d^{2}S/d{t}^2$ is discontinuous at $t=0$ (inset, dashed line).

Figure 3

(a) $S_2\left({\rho }_A\right)$ for the coherent state with $\nu =3$ and $C_g=1$, where $T_{\mathrm{ent},g}^{-1}=0$ indicates that the state remains effectively separable for a significant time. The leading behavior around $t=0$ is sixth order in $t$. (b) $S\left({\rho }_A\right)$ for the same state is $C^5$ smooth, with $d^{2}S/d{t}^2{|}_{t=0}=0$ (inset, solid line) and $d^{6}S/d{t}^6$ discontinuous at $t=0$ (inset, dashed line).