Quantum repeaters and computation by a single module: Remote nondestructive parity measurement

We introduce a simple photonic probing scheme of remote nondestructive parity measurement (RNPM) on a pair of matter qubits. The protocol works as a single module for key operations such as entanglement generation, Bell measurement, and parity check measurement, which are sufficient not only for building up a quantum repeater but also for equipping it with entanglement distillation. Moreover, the RNPM protocol can also be used for generating cluster states toward measurement-based quantum computation.

Figure 1

The RNPM protocol and its applications. (a) The RNPM protocol. (b) A circuit equivalent to the successful RNPM protocol, where a phase-flip channel ${\Lambda }_{\epsilon }\left(\widehat{\rho }\right):=(1-\epsilon )\widehat{\rho }+\epsilon \widehat{Z}\widehat{\rho }\widehat{Z}$ with phase error probability $\epsilon$ is applied as the penalty of photon losses. $\epsilon$ may depend on the outcome returned by photon detectors. In the lossless limit, the RNPM protocol works as the ideal NP measurement. (c) Quantum repeaters based on the RNPM protocols. (d)–(f) Applications of the RNPM protocol: (d) Bell measurement (BM), (e) parity check measurement, and (f) a gate for extending one-dimensional cluster states, where the measurement instrument means $Z$-basis measurement and the dashed arrow implies the transmission of the measurement outcome.

Figure 2

The minimum time $T$ needed to generate entanglement with $F=0.9$, $0.7$ over distance $L$ under the use of threshold detectors (TDs) and single-photon detectors (SPDs): (a) $\tau =0.95$ and $\eta =0.9$; (b) $\tau =0.98$ and $\eta =0.95$. $c=2\times {10}^8$ m/s and $L_{\mathrm{att}}=22$ km. The direct transmission time ${\left(f\eta T_L\right)}^{-1}$ of the photon from 10 GHz ($f={10}^{10}$) single-photon source (SPS) is also shown as a reference.

Figure 3

For ${\beta }^2=0.04,0.08,0.12$ with $\tau =0.98$ and $\eta =0.95$, the efficiencies of the recurrence method based on the RNPM protocols with single-photon detectors as a function of fidelity $F$ of the initial Werner states to a Bell state. (a) The success probability $P_s=P_{\mathrm{rec}}\left(F\right){\left[P_{2\eta {\beta }^2}\left(1\right)\right]}^2$. (b) The fidelity $F^{\prime }$ of the final pair.

Figure 4

A modified RNPM protocol. This protocol is executed only by Alice and Bob through an optical channel between them. The performance of this protocol is equivalent to that of the original RNPM protocol with $L_A=L_0$, but the stabilization of the relative phase between pulses $b_1$ and $b_2$ is easier.

Figure 5

For $L_A=l_0$ and $L_B=0$, the minimum time $T$ needed to generate entanglement with $F=0.9$, $0.7$ over distance $L$ under the use of threshold detectors (TDs) and single-photon detectors (SPDs): (a) $\tau =0.95$ and $\eta =0.9$; (b) $\tau =0.98$ and $\eta =0.95$. $c=2\times {10}^8$ m/s and $L_{\mathrm{att}}=22$ km. The direct transmission time ${\left(f\eta T_L\right)}^{-1}$ of the photon from a 10-GHz ($f={10}^{10}$) single-photon source (SPS) is also described as a reference.