Hybrid quantum repeater protocol with fast local processing

We propose a hybrid quantum repeater protocol combining the advantages of continuous and discrete variables. The repeater is based on the previous work of Brask et al. [Phys. Rev. Lett. 105, 160501 (2010)] but we present two ways of improving this protocol. In the previous protocol entangled single-photon states are produced and grown into superpositions of coherent states, known as two-mode cat states. The entanglement is then distributed using homodyne detection. To improve the protocol, we replace the time-consuming nonlocal growth of cat states with local growth of single-mode cat states, eliminating the need for classical communication during growth. Entanglement is generated in subsequent connection processes. Furthermore the growth procedure is optimized. We review the main elements of the original protocol and present the two modifications. Finally the two protocols are compared and the modified protocol is shown to perform significantly better than the original protocol.

Figure 1

Steps of the protocol in Ref. [27]. (a) Entanglement is generated using two sources of two-mode squeezed vacuum. One mode from each source is transmitted to a balanced beam splitter and the outputs are measured. Detection of a single-photon heralds entanglement between the remaining modes stored in quantum memories (QM). (b) Growth of cat states. Two entangled states are combined locally on balanced beam splitters and the $\widehat{X}$ quadrature is measured. Success is conditioned on the sum of the outcomes taking a value close to zero. (c) Entanglement swapping. One mode from each state is combined on a balanced beam splitter and the $\widehat{X}$ and $\widehat{P}$ quadrature of the outputs are measured. Success is conditioned on a value of the $\widehat{X}$ outcome close to zero.

Figure 2

Steps of the modified repeater. (a) Local growth of cat states. The output modes from two sources of two-mode squeezed vacuum states are combined on a balanced beam splitter and the $\widehat{X}$ quadrature of one of the outputs is measured conditioned on a click in both SPD detectors. Conditioned on the measurement outcome the resulting state is kept for further processing. (b) Entanglement generation from single-mode cat states. Small parts are tapped off from two input cat states at two separate locations, and the remaining parts are stored in quantum memories (black dots). The fraction tapped off is controlled by the reflectivity $r$ of the local beam splitters. The reflected signals are transmitted to a central, balanced beam splitter, and the output ports are measured. Conditioned on a click in either of the detectors, the memories are prepared in an entangled state.

Figure 3

Optimized production rate of approximate squeezed cat states after (a) $m$ $=$ 2,3 iterations and (b) $m$ $=$ 4,5 iterations. The dashed lines are the optimal curves and the solid curves are obtained for identical acceptance intervals in all iterations. The fidelity was calculated with the target state in Eq. (7) and the rate is given in units of the source repetition rate.

Figure 4

The fidelity of the connected state with respect to the state $|{\Psi }_m\rangle$ plotted against the rescaled probability of a successful connection. We have assumed the input states to be of the form in Eq. (9).

Figure 5

Influence of finite acceptance intervals in the growth on the state after connection. The fidelity of the connected state is with respect to the state $|{\Psi }_m\rangle$ and $\vec{\Delta }$ is represented through $R_{\mathrm{growth}}$. Here we have neglected losses in the optical fibers and $R_{\mathrm{growth}}$ is in units of the rate at which the one-photon input states for the growth can be provided.

Figure 6

[Upper curves (left axis)] The fidelity after entanglement swapping as a function of the $\widehat{P}$ outcome for (a) $m=1$, (b) $m=2$, and (c) $m=3$, and various values of the $\widehat{X}$ outcome. The fidelity is $F\left({\widehat{\rho }}_{\Psi }\right)$ where ${\widehat{\rho }}_{\Psi }$ results from swapping two copies of $|{\Psi }_m\rangle$. [Lower curves (right axis)] The corresponding probability distributions of $p$ for each $\widehat{X}$ outcome.

Figure 7

The optimal rates of the present and previous repeater protocols for different values of $r_{\mathrm{rep}}$. The protocols are optimized over the parameters listed in Table , under the constraint $F\ge 80\%$. The altered repeater performs significantly better than the previous protocol even for $r_{\mathrm{rep}}=1$ MHz.