Propagation of quantum information through a spin system

It has been recently suggested that the dynamics of a quantum spin system may provide a natural mechanism for transporting quantum information. We show that one-dimensional rings of qubits with fixed (time-independent) interactions, constant around the ring, allow high-fidelity communication of quantum states. We show that the problem of maximizing the fidelity of the quantum communication is related to a classical problem in Fourier wave analysis. By making use of this observation we find that if both communicating parties have access to limited numbers of qubits in the ring (a fraction that vanishes in the limit of large rings) it is possible to make the communication arbitrarily good.

Figure 1

Propagation of the state $\mid N∕2⟩$ in a $100$-site Heisenberg spin ring.

Figure 2

Propagation of a truncated Gaussian-modulated $W$ state of width $10$ and wave number $N∕4$ in a $100$-site Heisenberg spin ring.

Figure 3

Dots: size of the final wave packet in the Heisenberg spin ring for $N=50$ to $N=5000$. (The initial wave packet is a Gaussian-modulated wave of variance $N^{1∕3}$ truncated after $2N^{1∕3}$ sites.) Final width $F$ is defined to be when more than $95\%$ of the wavepacket lies between $F$ sites. Solid line: graph of $2.8N^{1∕3}$ for comparison. The apparent jumps of integer amounts greater than $1$ in the numerical results is an artifact of the way the width of the final wave packet is calculated. The final wave packet has a sequence trailing oscillations. In order to get an area of more than some arbitrary amount (like $0.95$) it is sometimes necessary to include several more sites in a go.