Transfer of arbitrary two-qubit states via a spin chain

We investigate the fidelity of the quantum state transfer (QST) of two qubits by means of an arbitrary spin-$\frac{1}{2}$ network, on a lattice of any dimensionality. Under the assumptions that the network Hamiltonian preserves the magnetization and that a fully polarized initial state is taken for the lattice, we obtain a general formula for the average fidelity of the two qubits QST, linking it to the one- and two-particle transfer amplitudes of the spin excitations among the sites of the lattice. We then apply this formalism to a 1D spin chain with $XX$-Heisenberg type nearest-neighbour interactions adopting a protocol that is a generalization of the single qubit one proposed in Paganelli et al. [Phys. Rev. A 87, 062309 (2013)]. We find that a high-quality two qubit QST can be achieved provided one can control the local fields at sites near the sender and receiver. Under such conditions, we obtain an almost perfect transfer in a time that scales either linearly or, depending on the spin number, quadratically with the length of the chain.

Figure 1

The spin graph $\mathcal{K}$ by means of which we aim at achieving high-quality 2-QST via a Rabi-like mechanism between the two ends of the spin chain. The quantum state to be sent is encoded on the spins $\mathcal{S}=\{1,2\}$ (blue), whereas the receivers are located at $\mathcal{R}=\{N-1,N\}$ (red). Barrier qubits, residing at sites $n=\{3,N-2\}$, on which a strong magnetic field $h$ is applied are colored in green.

Figure 2

Density plot of $|a_{kn}|$ versus $k$ and $n$ for $N=46$ (left) and $N=50$ (right) with $h=100J$. In the case of $N\ne 3n-1$ (top panel) it is clearly shown that the spatial distribution of the eigenvectors exhibits the presence of 4 quadri-localized energy eigenstates; whereas, in the case of $N=3n-1$ (bottom panel), two more eigenstates appear with non negligible amplitudes also on sites $\mathcal{S}$ and $\mathcal{R}$. Their presence gives rise to a more complicated dynamical behavior of the fidelity because of the larger number of degrees of freedom (and frequencies) effectively entering the time evolution of $\left|\Psi \right(0)\rangle$.

Figure 3

Top panel: Approximated average fidelity ${\overline{F}}_a\left(t\right)$, and, for comparison, $|sin{\omega }_1^{-}t|$ shown as a function of time. Notice that $F_a$ is so rapidly oscillating that it appears to fill the entire red region in the plot. The sinusoidal function approximately gives an envelop of the fidelity with the same periodicity, which easily allows us to identify several reading windows for the 2-QST. Bottom panel: Full expression for the average fidelity ${\overline{F}}_a\left(t\right)$ (red), $sin{\omega }_0^{-}tcos{\omega }_0^{+}t$ (blue), and $sin{\omega }_1^{-}t$ (green) shown as a function of time around the optimal transfer time $t_1$, denoted by the vertical dashed line. Notice that $|sin{\omega }_1^{-}t|\simeq 1$ and that the fidelity attains very high values near $t_1$ which recur several times, with a frequency of order of $J^{-1}$, allowing again for several reading windows.

Figure 4

Top panel: Exact average fidelity $\overline{F}\left(t\right)$ (yellow) and approximate fidelity ${\overline{F}}_a\left(t\right)$ (red) versus $t$ around $t_1$. Notice that $\overline{F}\left(t\right)>{\overline{F}}_a\left(t\right)$ is always fulfilled and that, at the optimal time $t^{*}$ almost perfect quantum state transfer is achieved, with $\overline{F}\left(t\right)\simeq 0.999$. Bottom panel: Transfer time $t^{*}$ versus $h$ for $N=30$: The function $f\left(h\right)=\frac{\pi }{2}{h}^2$ (dashed line) fits perfectly with the numerical data for $t^{*}$. The range of values of the magnetic field $h$ is such that ${\overline{F}}_a\left(t\right)>0.95$.

Figure 5

Top panel: Transfer time $t^{*}$ for the case of spin chain's of length $N=3n-1$ as a function of $h$. The time $t^{*}$ increases linearly with $h$, although with slightly different coefficients depending on whether $N$ is divisible by 4. In the figure the two cases are exemplified by $N=32$ and $N=38$, respectively. Bottom panel: Transfer time $t^{*}$ for the case of spin chain's of length $N=3n-1$ with magnetic field $h=4000$ as a function of $N$. The time increases linearly with $N$, with clearly distinct coefficients depending on the divisibility by 4 of $N$.

Figure 6

Maximum of the approximate fidelity ${\overline{F}}_a\left(t^{*}\right)$ obtained by choosing $h=4000$. The red points are the Rabi-like 2-QST and the maximum does not depend on $N$; the blue points are for $N=3n-1$ and even $N$ performs better than odd $N$. For $N\gg 1$, both of the curves converge to the $N\ne 3n-1$ case.