Quantum repeaters with imperfect memories: Cost and scalability

Memory dephasing and its impact on the rate of entanglement generation in quantum repeaters is addressed. For systems that rely on probabilistic schemes for entanglement distribution and connection, we estimate the maximum achievable rate per employed memory for our optimized partial nesting protocol, when a large number of memories are being used in each node. The above rate scales polynomially with distance, $L$, if quantum memories with infinitely long coherence times are available or if we employ a fully fault-tolerant scheme. For memories with finite coherence times and no fault-tolerant protection, the above rate optimistically degrades exponentially in $\sqrt{L}$, regardless of the employed purification scheme. It decays, at best, exponentially in $L$ if no purification is used.

Figure 1

(Color online) A quantum repeater with multiple quantum memories per node. At each round, we first employ an entanglement distribution protocol to entangle any unentangled memory pairs over $D_0$ links. It takes at least $T_{\text{ED}}=L_0/c$ to learn about the success or failure of these attempts, which succeed with probability $P_S$, hence $T_{\text{ED}}$ is the shortest period at which any protocol can run. At any such cycle, we also match up entangled pairs at different stations, according to our partial nesting protocol (see the text), to perform Bell-state measurements (BSMs), which succeed with probability $P_M$.

Figure 2

(Color online) (a) A quantum repeater link with nesting level $n=2$. Here, we have only shown memories $A_2,\dots ,A_7$ that contribute to entanglement generation between memories $X$ and $Y$. In the 2-PNP, all measurements are informed so if measurements labeled by BSM1 occur at time $t_1$, BSM2 will occur at time $t_1+T_1$. In the 1-PNP, however, all measurements happen at time $t_1$. In such a case, BSM2 will be a blind BSM. (b) A snapshot of memories involved in creating entanglement between $X$ and $Y$ in the general case of $m\text{-PNP}$. The assumption here is that we know that $X$ and $A_{2^m}$, $A_{2^m+1}$ and $A_{2^{m+1}}$, and so on are entangled because all previous measurements have been informed. The remaining measurements in $2^n/2^{m-1}-1$ stations shown above all occur at the same time. Among these BSMs, those measurements that correspond to nesting level $m$, labeled by $\text{BSM}m$ are informed, and the rest are blind.

Figure 3

(Color online) An optimistic estimate to the generation rate of maximally entangled states per employed memory, $R_m^{\left(n\right)}$, for the $m\text{th}$ order partial nesting protocol, in the presence of memory dephasing. The implicit assumption is that we employ a sufficiently large number of memories to minimize the waiting times due to classical communication. In (a), we have plotted $R_m^{\left(n\right)}$, optimized over $m$, versus the nesting level, for three different values of coherence time. In (b), we have plotted $R_m^{\left(n\right)}$ versus coherence time, optimized over $m$ and $n$. For fixed values of ${\tau }_c$, the rate degrades exponentially in $\sqrt{L}$. The pairs $\left(n,m\right)$, on several points on the graph, show the optimum nesting levels, $n$, and the optimum numbers of informed BSMs, $m$, at those points. In both (a) and (b), we assume that our entanglement distribution succeeds with probability $P_S=0.2\times {10}^{-0.01L_0}$ and the total distance is $L=1000\text{km}$, thus $L/c=5\text{ms}$ in optical fibers.

Figure 4

(Color online) Estimates for the generation rate of maximally entangled states per employed memory, $R_m^{\left(n\right)}$, versus coherence time, optimized over $m$ and $n$, for $P_M=0.75$, $P_S=0.2\times {10}^{-0.01L_0}$, $L=1000\text{km}$, and $L/c=5\text{ms}$. The employed entanglement measures in these graphs are entanglement cost for the solid lines, and entanglement of distillation for the dashed lines. For fixed values of ${\tau }_c$, the curves scale exponentially in $L$ if no purification is used and exponentially in $\sqrt{L}$ with purification; see dash-dotted line for comparison. The two-tuples $\left(n,m\right)$, on several points on the graph, show the optimum nesting levels, $n$, and the optimum numbers of informed BSMs, $m$, at those points.

Figure 5

(Color online) Estimates for the generation rate of maximally entangled states per employed memory, $R_m^{\left(n\right)}$, versus total distance $L$, optimized over $m$ and $n$, for different values of coherence time at $P_M=0.5$ and $P_S=0.2\times {10}^{-0.01L_0}$. The employed entanglement measure in these graphs is the entanglement cost and a proper purification protocol is used. It can be seen that, for ${\tau }_c⪢L/c$, the curves scale polynomially with $L$, whereas, for ${\tau }_c⪡L/c$, they scale exponentially with $\sqrt{L}$. The two-tuples $\left(n,m\right)$, on several points on the graph, show the optimum nesting levels, $n$, and the optimum numbers of informed BSMs, $m$, at those points. They, respectively, scale with $\text{log}L$ and $\text{log}\sqrt{L}$, for large distances.