Role of syndrome information on a one-way quantum repeater using teleportation-based error correction

We investigate a quantum repeater scheme for quantum key distribution based on the work by S. Muralidharan et al. [Phys. Rev. Lett. 112, 250501 (2014)]. Our scheme extends that work by making use of error syndrome measurement outcomes available at the repeater stations. We show how to calculate the secret key rates for the case of optimizing the syndrome information, while the known key rate is based on a scenario of coarse graining the syndrome information. We show that these key rates can surpass the Pirandola-Laurenza-Ottaviani-Banchi bound on secret key rates of direct transmission over lossy bosonic channels.

Figure 1

Encoded transmission of a logical qubit based on a block of $N_p=nm$ single photons (physical qubits). (a) Transmission through a concatenation of intermediate stations. Each station is composed of a decoder and an encoder. A unit segment is connected with the number of $nm$ optical-loss channels whose transmission is ${\eta }_0$ and qubit-error rate is $e$. The efficiency of the unit segment can be described by the success probability $P_0$ that the logical qubit is received and the average error $E_0$ of the successfully received logical qubit. (b) It is not necessary for the intermediate stations to physically execute a decoding operation to output a logical qubit. But an error correction, a syndrome measurement followed by a recovery operation, can be performed effectively by keeping the syndrome information of intermediate stations $\{i_1,i_2,\cdots ,i_N\}$. A syndrome outcome of a single station $i$ assigns conditional success probability $w_i$ and conditional logical error ${\epsilon }_i$. The set $\{w_i,{\epsilon }_i\}$ and the measured sequence $\{i_1,i_2,\cdots ,i_N\}$ determine the net performance of the transmission.

Figure 2

A quantum-teleportation-based error-correction process transfers the logical qubit of incoming photons in the block $R$ into a fresh logical qubit in the block $R^{\prime }$ and recovers the photon loss. (i) A quantum nondemolition (QND) measurement of the photon number is performed for each of the physical qubits in $R$, and the position of the photon loss in the incoming photon block $R$ is identified. (ii) A control-not (cnot) gate is applied between each of the surviving photons in block $R$ and corresponding photons in block $S$ whose logical qubit is prepared to be maximally entangled with the outgoing logical qubit in block $R^{\prime }$ as $({\left|00\right.〉}_{S{R}^{\prime }}+{\left|11\right.〉}_{S{R}^{\prime }})/\sqrt{2}$. (iii) A Bell measurement is implemented by logical $X$ and logical $Z$ measurements performed on blocks $R$ and $S$, respectively. The Pauli operators of logical qubits are decomposed into products of Pauli $X$ and Pauli $Z$ of physical qubits. (iv) Finally, the Pauli frame [16] is adjusted by a unitary operation based on the Bell measurement outcome.

Figure 3

The key rates $(R_{\mathrm{TGW}},R_{\mathrm{PLOB}},R_{\mathrm{FG}},R_{\mathrm{CG}})$ for ${\eta }_0=0.97$ as functions of the number of stations $N$. At $N\simeq 220$, the key rate of the CG scenario $R_{\mathrm{CG}}$ drops rapidly without any crossover to the PLOB bound. For the FG key rate $R_{\mathrm{FG}}$, there is a crossover with the PLOB bound $R_{\mathrm{PLOB}}$. The FG key rate shows relatively slow decay until $N\simeq 400$.

Figure 4

The key rates $(R_{\mathrm{TGW}},R_{\mathrm{PLOB}},R_{\mathrm{FG}},R_{\mathrm{CG}})$ for ${\eta }_0=0.90$ as functions of the number of stations $N$. For $N\simeq 210 (L_{\mathrm{tot}}\simeq 440\mathrm{km})$, the CG key rate drops rapidly, whereas the FG key rate holds until $N\simeq 300 (L_{\mathrm{tot}}\simeq 630\mathrm{km})$.

Figure 5

The key rates $(R_{\mathrm{TGW}},R_{\mathrm{PLOB}},R_{\mathrm{FG}},R_{\mathrm{CG}})$ for ${\eta }_0=0.81$ of the number of stations. The FG key rate $R_{\mathrm{FG}}$ marginally goes up to the PLOB bound $R_{\mathrm{PLOB}}$ around $N\simeq 190$ (which corresponds to $L_{\mathrm{tot}}\simeq 800\mathrm{km}$), while the CG scenario could not beat the PLOB bound.

Figure 6

The gray area shows the regime where the FG key rate surpasses the PLOB bound $R_{\mathrm{FG}}\ge R_{\mathrm{PLOB}}$. The dashed area shows the CG key rate surpasses the PLOB bound $R_{\mathrm{CG}}\ge R_{\mathrm{PLOB}}$. For the dotted level lines ${\eta }_0=0.97$ and ${\eta }_0=0.81$, we can see that the CG key rate $R_{\mathrm{CG}}$ cannot surpasses the PLOB bound, while the FG key rate can (see Figs. 3 and 5). On the other hand, the middle of these two level lines shows a wide regime in which both $R_{\mathrm{FG}}$ and $R_{\mathrm{CG}}$ can surpass the PLOB bound (see the actual behavior of the key rates for the dotted level ${\eta }_0=0.90$ in Fig. 4). Blue dashed, red dotted, and green solid lines show the boundaries where $R_{\mathrm{FG}}$ becomes 2, 4, and 10 times larger than $R_{\mathrm{CG}}$, respectively, $(r:=R_{\mathrm{FG}}/R_{\mathrm{CG}})$. The code size is $(n,m)=(2,2)$, and the physical error rate is $e=5\times {10}^{-4}$. The areas will shrink when the physical error becomes larger (see Fig. 7).

Figure 7

Areas beating the PLOB bound for the smallest block size $(n,m)=(2,2)$. The two largest areas for the physical error rate $e=5.0\times {10}^{-4}$ (the ones in Fig. 6) shrink as $e$ becomes higher. For $e=1.0\times {10}^{-3}$ in the case of the CG scenario $R_{\mathrm{CG}}$, there is no area where the PLOB bound can be beaten, while there still exists a substantial area to beat the PLOB bound for the FG scenario $R_{\mathrm{FG}}$. The right end of the loops for the FG scenario are slightly ragged due to a numerical instability in estimating $R_{\mathrm{FG}}$.

Figure 8

Areas where the FG key $R_{\mathrm{FG}}$ beats the PLOB bound in the presence of a coupling loss for the smallest code $(n,m)=(2,2)$ with a physical error rate of $e=5\times {10}^{-4}$. The largest loop shows the case without coupling loss $({\eta }_c=1)$. The area shrinks with the existence of coupling loss. For a 3% coupling loss $({\eta }_c=0.97)$ we could not find the area.

Figure 9

The areas where the CG key $R_{\mathrm{CG}}$ beats the PLOB bound in the presence of a coupling loss for the smallest code $(n,m)=(2,2)$ with a physical error rate of $e=5\times {10}^{-4}$. The largest loop shows the case without coupling loss $({\eta }_c=1)$. The area shrinks due to coupling loss. We could not find the area when the coupling loss is $2.4\% ({\eta }_c=0.976)$.

Figure 10

Areas beating the PLOB bound for a block size of $(n,m)=(3,2)$ with physical error rates of $e=7.5\times {10}^{-4},e=1.0\times {10}^{-3}$, and $e=1.5\times {10}^{-3}$. For this block size, we can observe wider areas to beat the PLOB bound compared with Fig. 7. In particular, we find a relatively wide area for the CG scenario with $e=1.0\times {10}^{-3}$ beating the PLOB bound (the middle green dashed loop), and there exists such a regime even for a higher error rate of $e=1.5\times {10}^{-3}$ (the smallest yellow dashed loop). The rightmost boundaries for the FG scenario are jagged due to a numeric instability. Again, for each of the physical error rates, the FG scenario always extends the longest distance to beat the PLOB bound compared with the CG scenario.

Figure 11

Areas beating the PLOB bound for a block size of $(n,m)=(3,3)$. The solid loops are for the FG key rate, and the dashed loops are for the CG rates. The physical error rates are, respectively, $e=7.5\times {10}^{-4},e=1.0\times {10}^{-3}$, and $e=1.5\times {10}^{-3}$ from the largest solid loop to the smallest solid loop for the FG scenario, while there is no area for the CG scenario with $e=1.5\times {10}^{-3}$. Compared with the block size of $(3,2)$, the areas for FG key rates grow wider, while the areas for the CG scenario rather shrink. The rightmost boundaries for the FG scenario are jagged due to the numeric instability.

Figure 12

Areas beating the PLOB bound for a block size of $(n,m)=(4,3)$. Solid loops are for the FG scenario, whereas the dashed loops are for the CG scenario. The physical error rates are, respectively, $e=1.0\times {10}^{-3},e=1.5\times {10}^{-3}$, and $e=2.5\times {10}^{-3}$ from the largest loops to the smallest loops. A distinguishing feature is the growth of the areas toward the distance direction around ${\eta }_0\sim 0.97$. This suggests that one can obtain better performance by using many intermediate stations with a short unit distance. The rightmost boundaries are jagged due to the numeric instability.

Figure 13

Classification of patterns. The white circles indicate the position of a lost photon. For this example, the block size is $(n,m)=(7,6)$, and the classifying index is determined to be $\vec{u}={(u_0,u_1,u_2,u_3,u_4,u_5,u_6)}^T={(3,1,2,1,0,0,0)}^T$. This implies $n_{LP}={\sum }_{k=0}^{m-1}k{u}_k=8$.