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. We show that, for any given distance $L$, the polynomial scaling of rate with distance can only be achieved if quantum memories with coherence times on the order of $L/c$ or longer, with $c$ being the speed of light in the channel, are available. The above rate degrades as a power of $\exp[-\sqrt{(L/c)/ \tau_c}]$ with distance when the coherence time $\tau_c \ll L/c$.