Secret key rates for an encoded quantum repeater

We investigate secret key rates for the quantum repeater using encoding [L. Jiang et al., Phys. Rev. A 79, 032325 (2009)] and compare them to the standard repeater scheme by Briegel, Dür, Cirac, and Zoller. The former scheme has the advantage of a minimal consumption of classical communication. We analyze the trade-off in the secret key rate between the communication time and the required resources. For this purpose we introduce an error model for the repeater using encoding which allows for input Bell states with a fidelity smaller than one, in contrast to the model given by L. Jiang et al. [Phys. Rev. A 79, 032325 (2009)]. We show that one can correct additional errors in the encoded connection procedure of this repeater and develop a suitable decoding algorithm. Furthermore, we derive the rate of producing entangled pairs for the quantum repeater using encoding and give the minimal parameter values (gate quality and initial fidelity) for establishing a nonzero secret key. We find that the generic quantum repeater is optimal regarding the secret key rate per memory per second and show that the encoded quantum repeater using the simple three-qubit repetition code can even have an advantage with respect to the resources compared to other recent quantum repeater schemes with encoding.

Figure 1

Setup of the encoded quantum repeater (adapted from [12]), BM stands for Bell measurement. In the ideal case ${\rho }_{\mathrm{enc}}=|{\tilde{\varphi }}^{+}\rangle \langle {\tilde{\varphi }}^{+}|$ with $|{\tilde{\varphi }}^{+}\rangle$ defined in Eq. (1).

Figure 2

Teleportation-based cnot, see [20].

Figure 3

Generation of the encoded state ${\rho }_{\mathrm{enc}}$ (see text).

Figure 4

Circuit for a Bell measurement.

Figure 5

Commutation rules for the cnot gates, see [21].

Figure 6

The secret fraction [Eq. (27)] plotted as a function of the gate error $\beta$ including all errors according to Eq. (12) (black circles) and those where only single bit-flip errors were correctable (red squares), $F_0=0.98$, one repeater station ($r=1$).

Figure 7

Decoding procedure: The upper three qubits are on Alice's and the lower are on Bob's side.

Figure 8

The optimal secret key rate per memory per second, see Eq. (31), as a function of the initial fidelity $F_0$ and the gate quality $p_G$, optimized over the number of repeater stations for the encoded quantum repeater ($L=600$ km). In the white region it is not possible to extract a nonzero secret key.

Figure 9

The optimal secret key rate per memory per second in Eq. (31) for the encoded (red squares) and the generic quantum repeater (black circles) as a function of the total distance $L$ (parameters: $F_0=0.98$, $p_G=0.992$).

Figure 10

The cost coefficient ($C^{\prime }=C/L$) [Eq. (40)] for the encoded (green squares), the generic quantum repeater (black circles), and the quantum repeater protocol presented in [15] (red crosses, with the effective qubit error $\varepsilon ={10}^{-4}$, for an explanation see [15]) as a function of the total distance $L$ (parameters: $F_0=0.99995$, $p_G=0.9999$, and $T_0=1$ as in [15]).

Figure 11

The optimal distance between the repeater stations $L_0$ for the cost coefficient ($C^{\prime }=C/L$), see Eq. (40), for the encoded (green squares) and the generic quantum repeater (black circles) (parameters $F_0=0.99995$ and $p_G=0.9999$, $T_0=1$ as in [15]) as a function of the total distance $L$.