Tight bound on coherent-state-based entanglement generation over lossy channels

The first stage of the hybrid quantum repeaters is entanglement generation based on transmission of pulses in coherent states over a lossy channel. Protocols to make entanglement with only one type of error are favorable for rendering subsequent entanglement distillation efficient. Here we provide the tight upper bound on performances of these protocols that is determined only by the channel loss. In addition, we show that this bound is achievable by utilizing a proposed protocol [K. Azuma, N. Sota, R. Namiki, Ş. K. Özdemir, T. Yamamoto, M. Koashi, and N. Imoto, Phys. Rev. A 80, 060303(R) (2009)] composed of a simple combination of linear optical elements and photon-number-resolving detectors.

Figure 1

The scenario of entanglement generation protocols. $|\varphi \rangle {}_A:={\sum }_{j=0,1}\sqrt{q_j}{e}^{i{\Theta }_j}|j\rangle {}_A$. Bob’s quantum operation returns qubit $B$ in a state depending on outcome $k$, and he shares the outcome with Alice by using classical communication.

Figure 2

An imaginary protocol equivalent to the real protocol in Fig. 1. $|{\varphi }^{\text{'}}\rangle {}_A:={\sum }_{j=0,1}\sqrt{q_j}{e}^{i{\Theta }_j+i(-1){}^j\phi }|j\rangle {}_A$. Channel $a\to b$ becomes ideal at the expense of the application of a phase-flip channel ${\Lambda }_A$.

Figure 3

The optimal performances of single-error-type entanglement generation for $10⩽l⩽100$ km at intervals of $10$ km, where we assume $T=e^{-l/l_0}$ and $l_0=25$ km (corresponding to $~$0.17 dB/km attenuation).