One of the main challenges of realizing quantum networks is providing entangled links between end nodes over larger distances. Due to noise, distributing high-quality entangled qubit pairs is demanding. Entanglement-purification protocols have been introduced to maintain the quality of such entangled links by transforming multiple low-quality entangled states into fewer entangled links of higher quality [1,2]. The disadvantage of using entanglement purification is that, with some probability, the purification fails, and the input links must be discarded.

In their recent work [3], Davies, Iñesta, and Wehner introduce a high-quality entanglement-buffering system with the goal of guaranteeing the permanent availability of high-quality entangled links that can be used for applications at any time. Specifically, the authors consider a two-node system with two quantum memories each: one good quantum memory for long-term storage and one bad (short-term) quantum memory for  entanglement generation. Buffered links are stored in the good memories. In the meantime, entangled links are generated in the bad memory with some rate $\lambda$. In case a new link is successfully generated, it is either transferred to the long-term memory for storage or it is used to purify the existing buffered link. If the purification fails, the new link as well as the buffered link are destroyed. By purification, the quality of the buffered link can be increased, and the exponential decay of the stored link in the good memory due to depolarization is counteracted. The resulting output link has fidelity $J(F, \rho_{new})$ where $F$ is the fidelity of the previous buffered link and $\rho_{new}$ is given by the generated link in the bad memory. The quantity $J(F, \rho_{new}) \in  [0,1]$ is called the jump function and depends on the choice of the underlying purification protocol.

A main contribution of [3] is the development of a framework for analyzing buffering systems based on two different metrics: availability, which is the probability that an entangled link exists, and average consumed fidelity, which measures the quality of consumed links in the steady state. The paper derives analytical expressions for these metrics by modeling the two-node system as a continuous-time stochastic process. A closed-form expression for the availability in the one-buffering system is provided and an analytical framework to calculate the average fidelity of the links consumed in the setup is developed.

A main finding is to show the trade-off between availability and fidelity: Consuming high-quality entanglement happens at a slower rate, while lower-quality entanglement can be consumed more frequently. Furthermore, it is shown that in a practical scenario (with a noisy good memory and the pumping protocol being bilocal Clifford), regularly refreshing stored links with new entanglement enhances the average consumed fidelity, even though some links may be discarded due to the low success rate of the refreshing process.

Unlike previous research, which often relies on numerical optimization [4] or specific purification protocols [5], this work presents a more general approach to entanglement purification and finds closed-form solutions for performance metrics.  In [6], a similar setup is developed independently. In contrast to [3], the system considered in [6] uses two good memories for storage. Also, the analysis is done in discrete time, and the consumption of entanglement is not taken into account. Therefore, the steady-state behavior can differ from the results in [3].

The approach in [3] is a significant step towards understanding the impact of purification on near-term quantum-network performance since the results do not rely on a specific protocol for entanglement purification. Consequently, the developed mathematical framework can also be used to analyze the performance of more complex systems, such as advanced purification protocols.

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