Entanglement Screening by Nonlinear Resonances

We show that nonlinear resonances in a classically mixed phase space allow us to define generic, strongly entangled multipartite quantum states. The robustness of their multipartite entanglement increases with the particle number, i.e., in the semiclassical limit, for those classes of diffusive noise which assist the quantum-classical transition.

Figure 1

Evolution of multipartite concurrence $C_k$ for two different numbers of qubits: $k=5$ (top), $k=8$ (bottom). Open symbols refer to unitary dynamics, while filled symbols represent the evolution under the diffusive noise described by Eq. (3). Squares correspond to an initial condition inside the nonlinear resonance island, triangles to initial conditions within the chaotic sea, in the classical phase space spanned by $p$ and $q$ (see inset). For different qubit numbers, the ratio of noise strength to the effective size of Planck’s quantum, $ϵ/{\hslash }_{\mathrm{eff}}$, is kept constant ($ϵ=0.04$ for $k=5$, and $ϵ=0.005$ for $k=8$). Black arrows indicate the value of $C_k$ for $k$-partite GHZ states.

Figure 2

Eight-partite concurrence $C_{k=8}$ as a function of initial momentum ($p=0.0,0.05,\dots ,0.95,1.0$), after 16 iterations of the Harper map, with $\chi =0.4964$. Open symbols correspond to unitary evolution, while filled symbols refer to unitary evolution amended by diffusive noise, Eq. (3). Squares represent the initial position $q=0.5$, and triangles $q=0.25$. Peaks and dips of $C_{k=8}(t)$ are located exactly at the center of the nonlinear resonance island in classical phase space.

Figure 3

Evolution of multipartite concurrence $C_{k=8}$ in the presence of phase damping (triangles, Eq. (4)) and depolarizing noise (squares, Eq. (5)), with $ϵ=0.04$ in Eqs. (4, 5). Filled and open symbols refer to initial conditions in the chaotic domain and within the resonance island, respectively, precisely as in Fig. 1.