Controlled generation of genuine multipartite entanglement in Floquet Ising spin models

We propose a method for generation of genuine multipartite entangled states in a short-range Ising spin chain with periodic global pulses of magnetic field. We consider an integrable and a nonintegrable Floquet system that is periodic in time and has constant quasienergy gaps with degeneracies. We start with all spins polarized along one direction and show that they evolve into states with high entanglement by calculating the average entanglement entropy and geometric measure of entanglement. We show that some of these states have a high number of parties involved in the entanglement by calculating the quantum Fisher information. Such controlled generation of multipartite entanglement has potential applications in quantum information processing.

Figure 1

Degeneracy of quasienergies in the (a) ${\mathcal{U}}_0$ and (b) ${\mathcal{U}}_x$ systems for system size $L=10$. Both systems have equally spaced quasienergies. The ${\mathcal{U}}_0$ system has spacings of $\pi /\left(2L\right)$. The spacings in the ${\mathcal{U}}_x$ system is not a simple function of the system size.

Figure 2

Variation of the normalized average entanglement entropy (AEE) $S\left(l\right)/l$ for the ${\mathcal{U}}_0$ and ${\mathcal{U}}_x$ system with successive Floquet periods in a system size $L=10$ with the $z$ initial state.

Figure 3

Variation of the geometric measure of entanglement $E_g$ with consecutive Floquet periods for the ${\mathcal{U}}_0$ system of sizes (a), (b) $L=10$ and (c), (d) $L=8$. In the case of the $z$ initial state in panels (a) and (c), we see here that this entanglement measure does not distinguish between the state with $L/2$ Bell pairs at $n=L/2$ and the other intermediary states. In the case of the $y$ initial state in panels (b) and (d), a drop in $E_g$ is seen at $n=L,L+1$. This is due to the system reaching states with high number of particles involved in the entanglement (the GHZ states).

Figure 4

Variation of the geometric measure of entanglement $E_g$ with consecutive Floquet periods in the ${\mathcal{U}}_x$ system of sizes (a), (b) $L=10$ and (c), (d) $L=8$ starting from the $z$ initial state in panels (a) and (c) and the $y$ initial state in panels (b) and (d). For system size $L=10$, the $E_g$ values are high at most points. The points with low values of $E_g$ ($n=10,11,...$) are points where there is a high number of particles involved in the entanglement. A similar plot is seen for other even system sizes (up to $L\le 12$) except for $L=8$. For $L=8$ in panels (c) and (d), the plot is similar to that seen in the ${\mathcal{U}}_0$ system [Figs. 3 and 3].

Figure 5

Variation of the maximum value of $F_Q$ with consecutive Floquet periods in the ${\mathcal{U}}_0$ system of sizes (a), (b) $L=10$ and (c), (d) $L=8$ starting from the $z$ initial state in panels (a) and (c) and the $y$ initial state in panels (b) and (d). $\kappa \left(k\right)$ is the maximum value of QFI for $k$-producible states. A value of $F_Q$ greater than $\kappa \left(k\right)$ indicates that the state has at least $(k+1)$-particle entanglement. In panels (a) and (c), the system reaches states with two-particle entanglement, while in panels (b) and (d) system reaches states with $L$-particle entanglement.

Figure 6

Variation of the maximum value of $F_Q$ with consecutive Floquet periods in the ${\mathcal{U}}_x$ system of sizes (a), (b) $L=10$ and (c), (d) $L=8$ starting from the $z$ initial state in panels (a) and (c) and the $y$ initial state in panels (b) and (d). $\kappa \left(k\right)$ is the maximum value of QFI for $k$-producible states. The system with even system size is seen to have at least $L/2$-particle entanglement (checked for $L\le 12$). However, $L=8$ in panels (c) and (d) is an exception as the $F_Q$ plot suggests only at least two-particle entanglement.