Measurement-Based Teleportation along Quantum Spin Chains

We examine the teleportation of an unknown spin-$1/2$ quantum state along a quantum spin chain with an even number of sites. Our protocol, using a sequence of Bell measurements, may be viewed as an iterated version of the 2-qubit protocol of C. H. Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)]. A decomposition of the Hilbert space of the spin chain into 4 vector spaces, called Bell subspaces, is given. It is established that any state from a Bell subspace may be used as a channel to perform unit fidelity teleportation. The space of all spin-0 many-body states, which includes the ground states of many known antiferromagnetic systems, belongs to a common Bell subspace. A channel-dependent teleportation parameter $\mathcal{O}$ is introduced, and a bound on the teleportation fidelity is given in terms of $\mathcal{O}$.

Figure 1

A schematic representation of the proposed teleportation protocol, showing input state $|v⟩$ and final state $X|v⟩$. (a) A Bell-basis measurement over a single pair of maximally entangled spins, as in Eq. (1), represented by a solid line between two sites. (b) A schematic representation of our protocol for a general spin quantum state, as in Eq. (2), with nontrivial entanglement, represented by the dashed line. Here we assume that the state $|\overrightarrow{j}\overrightarrow{k}\}$ lies within a Bell subspace, yielding a pure output state, $X|v⟩$. $X$ is known from the measurement outcome, as per Table .

Figure 2

Data points correspond to the fidelity of teleportation for a range of target states and channels. The fidelity is shown against the quantity $\mathcal{O}$ as defined in Eq. (6) for 2000 4-qubit channels. The dashed line is a lower bound for these points and is given by the inequality in Eq. (7). This clearly shows the tendency $\mathcal{F}\to 1$ as $\mathcal{O}\to 3$.