Entanglement dynamics and quantum-state transport in spin chains

We study the dynamics of a Heisenberg $XY$ spin chain with an unknown state coded into one qubit or a pair of entangled qubits, with the rest of the spins being in a polarized state. The time evolution involves magnon excitations, and through them the entanglement is transported across the channel. For a large number of qubits, explicit formulas for the concurrences, measures for two-qubit entanglements, and the fidelity for recovering the state some distance away are calculated as functions of time. Initial states with an entangled pair of qubits show better fidelity, which takes its first maximum value at earlier times, compared to initial states with no entangled pair. In particular initial states with a pair of qubits in an unknown state $\alpha \uparrow \uparrow +\beta \downarrow \downarrow$ are best suited for quantum-state transport.

Figure 1

The average fidelity $F_r$ as a function of $T$, for $r=100$, for the three types of states discussed. For the entangled states, $s=l-m=25$ has been used. The first maximum for the unentangled is at $T=r=100$, and for the $B1$ and $B2$ states, at $T=r-s$.

Figure 2

The concurrence $C_{ij}$ as a function of $T$, using ${\mid \beta \mid }^2=1∕2$. It can be seen how concurrence builds up between the initial sites $l$, $l+1$, and $l+r$ for the unentangled initial state, and how the concurrence decreases between the initial sites $l$ and $m$ for $B1$ and $B2$ Bell states.

Figure 3

The average fidelity $G_r$ as a function of $T$ for $r=50,100$ using $s=l-m=10$. A constant 0.5 has been added on for $B1$ state to show both on the same graph.