Entanglement flow in multipartite systems

We investigate entanglement dynamics in multipartite systems, establishing a quantitative concept of entanglement flow: both flow through individual particles and flow along general networks of interacting particles. In the former case, the rate at which a particle can transmit entanglement is shown to depend on that particle’s entanglement with the rest of the system. In the latter, we derive a set of entanglement rate equations, relating the rate of entanglement generation between two subsets of particles to the entanglement already present further back along the network. We use the rate equations to derive a lower bound on entanglement generation in qubit chains, and compare this to existing entanglement creation protocols.

Figure 1

A network of interacting particles showing interactions and sets defined in the entanglement rate equations. Interactions “crossing the boundaries” of $A$ or $B$ are indicated by thicker lines.

Figure 2

Numerically calculated generalized singlet fraction curves $f_k\left(t\right)$ saturating the inequalities in the entanglement rate equations (2a, 2b). The final solid curve is for $k=50$, i.e., the singlet fraction of the end qubits is separated by an interaction distance $d=100$. [The dashed curves show the corresponding upper and lower bounds $u_k\left(t\right)$ and $l_k\left(t\right)$ for $k=50$, from Sec. 4B.]

Figure 3

Entanglement dynamics in the entanglement generation protocol based on Khaneja and Glaser’s state transfer scheme [26], for a chain of ten qubits. Successive curves show the evolution of generalized singlet fractions $F_1$ through $F_5$, numbered as in Sec. 3C.

Figure 4

Entanglement dynamics in the entanglement generation scheme for odd chain lengths from Ref. 28, here for nine qubits. Successive curves show the evolution of generalized singlet fractions $F_1$ through $F_4$, numbered as in Sec. 3C. (Note that times are not comparable to those in Fig. 3, since interaction strengths in [28] are not normalized.)

Figure 5

Scaling with chain-length $L$ of the time $T_{\mathrm{ent}}$ required to create a maximally entangled state between the ends. The points show numerical results obtained by saturating the rate equations (2a, 2b). The solid and dashed curves show the analytic lower and upper bounds, $T_{\mathrm{ent}}⩾\sqrt{\lfloor L∕2\rfloor ∕2}$ and $T_{\mathrm{ent}}⩽\sqrt{\lfloor L∕2\rfloor }$, respectively.