Entanglement buffering with two quantum memories

Quantum networks crucially rely on the availability of high-quality entan-gled pairs of qubits, known as entangled links, distributed across distant nodes. Maintaining the quality of these links is a challenging task due to the presence of time-dependent noise, also known as decoherence. Entanglement purification protocols offer a solution by converting multiple low-quality entangled states into a smaller number of higher-quality ones. In this work, we introduce a framework to analyse the performance of entanglement buffering setups that combine entanglement consumption, decoherence, and entanglement purification. We propose two key metrics: the availability, which is the steady-state probability that an entangled link is present, and the average consumed fidelity, which quantifies the steady-state quality of consumed links. We then investigate a two-node system, where each node possesses two quantum memories: one for long-term entanglement storage, and another for entanglement generation. We model this setup as a continuous-time stochastic process and derive analytical expressions for the performance metrics. Our findings unveil a trade-off between the availability and the average consumed fidelity. We also bound these performance metrics for a buffering system that employs the well-known bilocal Clifford purification protocols. Importantly, our analysis demonstrates that, in the presence of noise, consistently purifying the buffered entanglement increases the average consumed fidelity, even when some buffered entanglement is discarded due to purification failures.


INTRODUCTION
The functionality of quantum network applications typically relies on the consumption of entangled pairs of qubits, also known as entangled links, that are shared among distant nodes [40].The performance of quantum network applications does not only depend on the rate of production of entangled links, but also on their quality.In a quantum network, it is therefore a priority for high-quality entangled states to be readily available to network users.This is a challenging task, since entangled links are typically stored in memories that are subjected to time-dependent noise, meaning that the quality of stored entangled links decreases over time.This effect is known as decoherence.
A common way of overcoming the loss in quality of entangled links is to use entanglement purification protocols [3,11,12,43].An -to- entanglement purification protocol consumes  entangled quantum states of low quality and outputs  states with a higher quality, where typically  < .The simplest form of purification schemes are 2-to-1, also known as entanglement pumping protocols.One downside of using purification is that there is typically a probability of failure, in which case the input entangled links must be discarded and nothing is produced.
In this work, we take a crucial step towards the design of high-quality entanglement buffering systems.The goal of the buffer is to make an entangled link available with a high quality, such that it can be consumed at any time for an application.We develop methods to analyse the performance of an entanglement buffering setup in a system with entanglement consumption, decoherence, and entanglement pumping.We introduce two metrics to study the performance: (i) the availability, which is the steady-state probability that a link is available, and (ii) the average consumed fidelity, Illustration of the entanglement buffering system with two quantum memories (1G1B system).Each of the nodes has two memories (G and B).Memory G is used to store the buffered link.An entangled link is generated at a rate  in memory B. If memory G is empty when the new link is generated in B, the link is immediately transferred to G. If memory G is occupied, the new link generated in B is immediately used to purify the buffered link with probability  (otherwise, the new link is discarded).The pumping protocol consumes the link in B to increase the quality of the buffered link in G, and it succeeds with probability  (otherwise, it destroys the link in G).The buffered link is consumed at a fixed rate .The quality of the entanglement stored in G decays exponentially with rate Γ.Formal definitions of the problem parameters can be found in Section 3.
which is the steady-state average quality of entangled links upon consumption.We measure the quality of quantum states with the fidelity, which is a well-known metric for this [32].
We use these metrics to study a two-node system where each of the nodes has two quantum memories, each of which can store a single qubit (see Figure 1).This system is of practical relevance since early quantum networks are expected to have a number of memories per node of this order (e.g. in [26] and [42], entanglement purification was demonstrated experimentally between two distant nodes, each with the capability of storing two qubits).We study a system where each node has one good (long-term) quantum memory, G, and one bad (short-term) memory, B, per node.We therefore refer to this entanglement buffering setup as the 1G1B system.The good memories are used to store an entangled link between the nodes that can be consumed at any time.The bad memories are used to generate a new entangled link between the nodes.The new link may be used to pump the stored link with fresh entanglement.
Calculating the temporal evolution of the fidelity of an entangled link is generally a difficult task, since the fidelity depends on the history of operations that have been applied to the link in the past.By modelling the state of the 1G1B system as a continuous-time stochastic process and using the theory of renewal processes, we are able to find analytical solutions for the availability and the average consumed fidelity of the system.We illustrate the application of these results in a simplified scenario where purification has a linear action on the quality of the buffered link.
Our main contributions are the following: • We propose two metrics to measure the performance of an entanglement buffering system: the availability and the average consumed fidelity.• We provide a simple closed-form expression for the availability in the 1G1B system.
• We develop an analytical framework to calculate the average fidelity of the links consumed in a 1G1B system.We provide a closed-form expression for pumping schemes that increase the fidelity of the entangled link linearly with the initial fidelity.
Our main findings are the following: • We confirm the intuition that, except in some edge cases, there is a trade-off between availability and average consumed fidelity: one must either consume low-quality entanglement at a higher rate, or high-quality entanglement at a lower rate.• Consider a situation where the quantum memories are subjected to decoherence and bilocal Clifford protocols are employed (this is one of the most popular and well-studied classes of purification protocols [10]).Then, pumping the stored link with fresh entanglement always increases the average consumed fidelity, even if the stored link is often discarded due to a small probability of successful pumping.
The structure of the paper is the following.In Section 2, we provide a short overview of related work and an introduction to the quantum information concepts required to understand this work.In Section 3, we explain the physical setup and provide a formal definition of the 1G1B system as a stohcastic process.In Section 4, we define the performance metrics of interest and provide analytical expressions that enable their computation.In Section 5, we analyse the system in the case where the pumping protocol produces an output state whose fidelity is a linear function of the fidelity of one of the input states.In Section 5.1, we use these results to bound the performance of the 1G1B system, in the case where bilocal Clifford protocols are employed for entanglement pumping.Lastly, in Section 6, we discuss the implications of this work and future research directions.

Related work
The performance analysis of quantum networks is unique because of the trade-off between the rate of distribution of entangled links and the quality of distributed links, both of which are important for the functionality of networking applications.This leads to interesting stochastic problems, which are important to understand the parameter regimes of a possible architecture.For example, [23,28,37] deal with the problem of generating an end-to-end entangled link across a chain of quantum repeaters, where both the rate of production and the quality of the end-to-end links are quantities of interest.Another example is the problem of generating multiple entangled links between two users with a high quality, which is treated in [9,36].In these works, the time between successfully generated entangled links is modelled by a geometric distribution.However, the time taken up by an entanglement generation attempt is generally small compared to other relevant time scales [29,34].Hence, a simplifying assumption that we make in this work is that the time between entanglement generation attempts is exponentially distributed.This is a common assumption in the quantum networking literature (see e.g.[7,31,38]), because it can enable the finding of closed-form relations between physical variables and protocol parameters.Here, we introduce and find expressions for the values of two key performance metrics in the steady state.In order to derive these solutions, we use methods from renewal theory.This means that the formulae obtained are in fact applicable to a more general case for at least one of the performance metrics; for example, the case where entanglement generation and consumption have a more general distribution (instead of being Poisson processes).
Previous work that incorporates entanglement purification schemes into the analysis of quantum network architectures typically involves numerical optimisation methods (see e.g.[39]), or only considers specific purification protocols [6].By contrast, in this work we focus on presenting the purification protocol in a general way, and finding closed-form solutions for the performance metrics of interest (albeit for a simpler architecture).This is an important step towards an indepth understanding of how one can expect purification to impact the performance of a near-term quantum network.
Other works have introduced the concept of entanglement buffering (preparing quantum links to be consumed at a later time) over a large-scale quantum network [24,35].To the best of our knowledge, the only work with a similar set-up to ours is [15], which was developed in parallel and independently of our work.There, the authors study the steady-state fidelities of a system involving two memories used for storage (good memories), and one memory used for generation (bad memory).This work differs from ours in multiple ways.For example, the analysis is done in discrete time and it is assumed that the fidelity takes a discrete set of values, whereas we do not make this assumption since we work in continuous time.Additionally, consumption of entanglement is not included in the system studied in [15], which may impact the steady-state behaviour.
Lastly, we note that previous work generally assumes a specific protocol for entanglement buffering between each pair of nodes, and does not address the following fundamental question: what is the best way to buffer entanglement between two users in a quantum network?To the best of our knowledge, we address this question for the first time.

Quantum preliminaries
Here, we introduce the quantum information concepts needed in this work.For a general reference on quantum information theory, see, e.g., ref. [32].
In classical computer science, information is generally stored in the form of bits, discrete binary variables that can take value zero or one.In quantum information, bits are generalized to qubits.Qubits describe systems that can be in a linear combination of two different states (say, state zero and state one).As a consequence, a qubit must not be described as a binary variable but as a vector | ⟩ ∈ C 2 with unit norm.We represent these vectors using the Dirac notation, and we refer to |•⟩ as a ket and to its conjugate transpose ⟨•| ≡ |•⟩ † as a bra.The state of a pure -qubit system is then described by a -dimensional vector | ⟩ ∈ C  , with  = 2  .The elements of the basis of C  are usually labeled |x⟩, with x ∈ {0, 1}  .For example, the basis of the two-qubit space C 4 can be written as {|00⟩ , |01⟩ , |10⟩ , |11⟩}.
Let us now consider a two-qubit system | ⟩ ∈ C 4 .When one of the qubits is in state | ⟩ 1 ∈ C 2 and the other qubit is in state | ⟩ 2 ∈ C 2 , we can write the joint state of the system as the tensor product of the individual qubits: | ⟩ = | ⟩ 1 ⊗ | ⟩ 2 .This is called a product state.When the two qubits are entangled, it is not possible to describe their joint state as a tensor product.One of the intuitive effects of the entanglement is that, if both qubits are measured, the measurement outcomes will be correlated.This is the basis for many quantum networking applications, such as quantum key distribution [14] and quantum secret sharing [8,20].
Noise and quantum operations, such as gates and measurements, modify the state of a quantum system.When dealing with noisy systems, it is generally more convenient to represent quantum states using the density matrix formalism instead of kets and bras.The density matrix of a pure state | ⟩ can be written as the outer product  = | ⟩⟨ |.When the system is not pure, it is called mixed, and it is written as a mixture of pure states:  =    |  ⟩⟨  |, with   ∈ [0, 1] and    = 1.Intuitively, this corresponds to a system with some degree of uncertainty from a classical point of view.For example, consider a device that prepares the state | 1 ⟩ with probability  and the state | 2 ⟩ with probability 1 − .We can write the output state of the device as a mixed state We refer to a two-qubit state  as an entangled link (in our setup, each qubit is located at a distant node, and the entanglement can be regarded as "link" connecting them).Entangled links can be entangled to different degrees.The Bell states are examples of two-qubit maximally entangled (pure) states: Measurements on the individual qubits of any of these states yield maximally correlated outcomes.All two-qubit maximally entangled states are equivalent, since they can be mapped via single-qubit operations to each other.Therefore, we can measure the quality of the entanglement of a two-qubit state by measuring how close the state is to one of the Bell states, say | + ⟩.Formally, we do this using the fidelity of the state: where  is the density matrix of an arbitrary two-qubit state.
Lastly, an important type of mixed entangled state is the Bell-diagonal state: The fidelity of this state is  BD .Bell-diagonal states are relevant because any two-qubit state can be transformed into Bell-diagonal form while preserving the fidelity by applying extra noise, a process known as twirling [4,22].A specific instance of Bell-diagonal state is the Werner state [41]: with fidelity  ∈ [0, 1].The Werner state corresponds to a maximally entangled state that has been subjected to isotropic noise.

THE 1G1B (ONE GOOD, ONE BAD) SYSTEM
We now define the 1G1B system.In Section 3.1, we describe and motivate the model of the system.In Section 3.2, we define the variables of interest precisely.This facilitates the definition of the performance metrics in Section 4.

System description
Below we provide a list of assumptions that model the 1G1B system, and provide motivation for each assumptions.An illustration of the system is given in Figure 1.
( i.e., the attempt can fail with some probability and, when this occurs, a failure flag is raised.Therefore, the generation of a single link may take multiple attempts.The time taken by an attempt is typically fixed (this is both the case in present-day quantum networks [34] and an assumption that is commonly made in the theoretical analysis of quantum networks [9,23,24]).Then, the time between attempts follows a geometric distribution.Since the probability of successful generation and the length of the time step is often small compared to other relevant time scales [29,34], we use a continuous approximation.
(3) When a link is newly generated in memory B, if memory G is empty (no link present), the new link is immediately placed there.If memory G is not empty, the nodes immediately either (i) attempt pumping with probability , or (ii) discard the new link from memory B (probability 1 − ).This step is included because it may not always be a good idea to carry out pumping, due to there being a possibility of this failing.(4) Links stored in memory G are Werner states.
The state in the good memory is therefore fully described by one parameter: the fidelity  .Any state can be transformed into a Werner state with the same fidelity by applying extra noise, a process known as twirling [4,22].Hence, this assumption constitutes a worst-case model.(5) While in memory G, states are subject to depolarising noise with memory lifetime 1/Γ.Depolarising noise can also be seen as a worst-case noise model [13].After a time  in memory, this maps the state fidelity  to .
(6) Consumption requests arrive according to a Poisson process with rate .When a consumption request arrives, if there is a stored link in memory G, it is immediately used for an application (and therefore removed from the memory).If there is no link available, the request is ignored.This means that a consumption request is equally likely to arrive in any small interval of time, and is one of the most simple ways to model arrivals.(7) Assumptions about pumping: (a) Pumping is carried out instantaneously.This is because the execution time is generally much lower than the other timescales involved in the problem.For example, in state-of-the-art setups, an entangled link is generated approximately every 0.5 s [34], while entanglement pumping may take around 0.5 • 10 −3 s [26].If the nodes are far apart, classical communication between them would only add a negligible contribution to the purification protocol (e.g.classical information takes less than 10 −4 s to travel over 10 km of optical fiber).(b) Pumping succeeds with constant probability .
In general this is not true, because the probability of purification success depends on the input states, which may vary.However, this does allow one to find bounds on the operating regimes of the system by considering the best-case and worst-case values of  (see Section 5.1).(c) Suppose that the link in memory G has fidelity  and the link in memory B is in state  new .If pumping succeeds, the output link has fidelity  (,  new ), and remains in the good memory.If pumping fails, all links are discarded from the system.
Here, the jump function  (,  new ) ∈ [0, 1] is dependent on the choice of the purification protocol.
Implicit in the above is that the process of entanglement generation, pumping and consumption ((2),(3),( 6) and (7b)) are independent.We provide a summary of the parameters involved in the 1G1B system in Table 1.

System definition
In this subsection, we define the state of the system mathematically, which will be the main object of study in the rest of this work.We view the state of the system as the number of rounds of pumping that the link in memory has undergone.From now on, when we refer to 1G1B, we refer to the stochastic process that evolves according to the following definition.
Definition 3.1 (1G1B system).Let  () be the state of the 1G1B system at time .This takes values ∅ if there is no link in memory,  ≥ 0 if there is a link in memory which is the result of  successful pumping rounds, (4) where  = 0 corresponds to a link in memory that has not undergone any pumping.Assume that the system starts with no link, i.e.  (0) = ∅.The system transitions from state ∅ to state 0 when a new link is generated and placed in the good memory, which was previously empty.The rate of transition from ∅ to 0 is then given by the entanglement generation rate .Pumping success occurs when a new link is produced (rate ), pumping is attempted (probability ), and pumping succeeds (probability ).Therefore, the transition from state  to  + 1 occurs with rate .The final allowed transition is from  to ∅ which occurs due to consumption or purification failure, which occurs with rate  + (1 − ).
We also refer to the state  ≥ 0 as the th purification level.Since the transitions between each state in 1G1B occur according to an exponential distribution with rate that is only dependent on the current state of the system, this is a continuous-time Markov chain (CTMC) on the state space {∅, 0, 1, ...}.The resulting CTMC and the rate of transitions is depicted in Figure 2.This is the main object of study in our work.
Recall that we are also interested in the fidelity of the link in memory.This is dependent not only on the state  () ∈ {∅, 0, 1, ...}, but also on the time spent in the states leading up to the current purification level.This motivates the following definition.Definition 3.2.Suppose that  () = .Then, we define random variable ì  () to be the length-( +1) vector storing the times spent in the recent purification levels 0, 1, . . .,  leading up to the current one, where time   () was spent in the most recent visit to state  ( ≤  − 1), and time   () is the time spent so far in state .See Figure 3 for a depiction of this.
We also need a framework with which to compute the fidelity at time .Recalling assumption (5) of Section 3.1, we denote decoherence by the following.
denote the action of depolarising noise on the state fidelity  .This has action .
We now formally define the jump function.
Definition 3.4.After successfully applying purification to a Werner state with fidelity  and a general two-qubit state  new , the output state has fidelity  (,  new ).We refer to  as the jump function of the protocol.
We note that every purification protocol has a corresponding jump function.The exact form of  is dependent on the choice of pumping protocol, but in general is a continuous rational function of  , taking values in [0, 1].
We also need to compute the fidelity after many rounds of decoherence and pumping.This essentially means composing   and  .Definition 3.5.Let  ( ) ( 0 , ...,   ) denote the fidelity after spending time  0 , ...,   in each purification level 0, 1, ..., .This may be defined recursively as with where  new is the fidelity of  new .
Note that  ( ) is a continuous and bounded function of its inputs, since the same is true for   and  .We are now equipped to define the fidelity of the system.Definition 3.6.The fidelity of the 1G1B system at  is given by Example of the evolution of the fidelity of the buffered entanglement over time.The fidelity experiences a sudden boost every time a pumping protocol is successful.Then, it decays exponentially due to decoherence.Each state in the CTMC is identified by the number of times the current buffered link has been purified.If  () = , the random variables {  () :  = 0, 1, ...,  − 1} are the times spent in each state of the CTMC immediately leading up to the current state, and   () is the time so far spent in state .
Note that this formulation can also be adapted to incorporate a system where we apply a different pumping protocol in each state of the CTMC.In that case, we would employ a more general recurrence relation: where the  ( ) is the jump function corresponding to the pumping protocol applied in state  of the CTMC.For simplicity, however, we study recurrence relations of the form ( 5).This may be used to model the situation where the same pumping protocol is applied every time, or provide bounds for using multiple protocols, as we do in Section 5.1.

PERFORMANCE METRICS
In this Section, we define two metrics to evaluate the performance of an entanglement buffering system: the availability and the average consumed fidelity.We also provide analytical expressions for both metrics in the 1G1B system.

Availability
A natural measure for the quality of service provided to users is the probability that a consumption request may be served at any given time.If there is a link stored in the good memories, the consumption request is immediately served.However, if there is no entanglement available, the request is ignored.Letting  ( () = ) be the probability that the system is in state  at time , we define the steady-state distribution as Then, we define our first performance metric as follows.
Definition 4.1 (Availability).The availability  is defined as which is the probability that there is a link in memory in the limit  → ∞.
This definition can be applied to any entanglement buffering setup.In the 1G1B system, the availability is well-defined, as shown in Appendix A. Moreover, it is possible to derive a closed-form expression for the availability, as stated in the proposition below.Proposition 4.2.Consider the 1G1B system (Definition 3.1).The availability is given by and the rest of the steady-state distribution is given by See Appendix A for a proof of this proposition.We note that this can be derived in a straightforward manner using the balance equations for a CTMC.Instead, we use renewal theory, for two reasons.Firstly, this approach ties in neatly with the proof of the formula for the average fidelity (see the next subsection).Secondly, this approach provides a formula for the availability that is more general, as it also applies to the case where entanglement generation is described by a general random variable instead of being exponentially distributed.

Average consumed fidelity
The quality of service of an entanglement buffering system can also be measured in terms of the quality of the entanglement provided to the users.Therefore, the average fidelity of the entangled links upon consumption can be used as an additional metric to assess the performance of the system.

Definition 4.3 (Average consumed fidelity).
The average consumed fidelity is the average fidelity of the entangled link upon consumption, in the steady state.More specifically, In the definition of  , we condition on not being in ∅ since consumption events do not happen when there is no link present.As before, this performance metric can be applied to any entanglement buffering setup.In the case of the 1G1B system, it is possible to derive an analytical expression for  which explicitly depends on the steady-state distribution.The formula is given in the following theorem.
Sketch proof of Theorem 4.4.A first step is to expand by conditioning on the value of  (), Fig. 4.An example timeline of the 1G1B process.Black dashes are link generation and removal.Purple dashes are pumping rounds.If there is a link present at time , the random variable  () is the total time spent so far in ¬∅ (link present).Pumping rounds occur within the time  () as a Poisson process with rate .This may be used to characterise the distribution of ì  () in the limit  → ∞.
In Proposition A.18, it is shown that when  → ∞, the limit can be brought inside of the sum, and so where we have used the definition of the steady-state distribution and the availability (see ( 8) and ( 9)).The values   may be computed using Proposition 4.2.The remaining work is then to show that lim which essentially requires the characterisation of the limiting distribution of ì  (), since from Definition 3.6 we recall that E  ()| () =  = E  ( ) ì  ()  () =  .Since  ( ) is a continuous function of its inputs, this is achieved with the following result: conditional on  () = , ì  () → ( 0 , . . .,   ) in distribution as  → ∞, where the   are i.i.d.random variables with  0 ∼ Exp(+).There are two main steps to show this (see Figure 4 for graphical intuition): (1) Let  () be the total time spent so far in ¬∅ (link in memory G) at the time .The first step is to show that  () →  in distribution as  → ∞, where  ∼ Exp( + (1 − )).This is shown with renewal theory.For more details, see the results of Appendix A.1 (2) Characterise the limiting distribution of the time spent in each purification level within the time  ().These are the   ().We use the fact that pumping rounds occur as a Poisson process within the time  ().For more details, see the results of Appendix A.2.
Finally, since  ( ) is a continuous function of its inputs, (15) follows.□ The particularly simple form of ( 14) can be attributed to the fact that in a CTMC, the time spent in a state is not influenced by the state to which the system transitions.As an example, in the CTMC from Figure 5, the time spent in state B before a transition does not depend on the transition itself, and this time is exponentially distributed with rate   +   .In the 1G1B system, the times spent in the states  = 0, 1, . . .,  − 1 leading up to state  are all exponentially distributed with rate  +  + (1 − ) =  + .As a consequence, the average fidelity after  successful purifications,   , does not depend on the probability of successful purification .
Having systematic closed-form expressions for the functions  ( ) enables the efficient computation of   and, therefore,  .The calculation of  ( ) in closed-form for a general  is quite involved, since (5) becomes a rational difference equation with arbitrary coefficients.However, in A B C Fig. 5.In a CTMC, the time spent in a state is independent of the transition that happens next.In this example, the time spent in state B before leaving is exponentially distributed with rate   +   .
the following sections we consider a jump function which is linear, for which it is possible to find a closed-form solution for  .

ENTANGLEMENT BUFFERING WITH A LINEAR JUMP FUNCTION
In a purification protocol with a linear jump function, the output fidelity is a linear function of the fidelity of one of the input entangled links.In this Section, we compute a closed-form solution for the average consumed fidelity in a 1G1B system assuming a linear jump function.Then, we analyse the performance of the system using the performance metrics defined in Section 4 (availability and average consumed fidelity).In Section 5.1, we focus on bilocal Clifford protocols, an important type of purification scheme.For a given value of target availability, we provide upper and lower bounds on the average consumed fidelity that can be achieved by any bilocal Clifford protocol in the 1G1B system.Purification protocols with linear jump functions are relevant for two main reasons: (i) Purification protocols are generally more effective within some range of input fidelities (the increase in fidelity is larger when the input fidelities are within some interval).If the system operates within a small range of fidelities, one may approximate the true jump function with a linear jump function.(ii) One can find linear jump functions that upper and lower bound a set of jump functions of interest.These may then be used to upper and lower bound a fidelity-based performance metric (such as the average consumed fidelity) of a system that has the freedom to employ any of these jump functions.
In Section C.1, we demonstrate (ii) in the case where bilocal Clifford protocols are employed in the 1G1B system.The output fidelity of a bilocal Clifford protocol can be upper and lower bounded by nontrivial linear functions when one of the input states is a Werner state, as shown in Appendix C (using some additional minor assumptions).Consider a pumping scheme that takes as input a Werner state with fidelity  and an arbitrary state  new .In the 1G1B system, these are the states in the good and the bad memories, respectively.A linear jump function can be written as with 0 ≤ ( new ) ≤ 1 and In what follows, we implicitly assume that  and  depend on  new .We now derive a closed-form solution for the average consumed fidelity of 1G1B when the jump function is linear, using Theorem 4.4.The formula requires knowledge of the steady state distribution {  :  = ∅, 0, 1, ...}, and the expected fidelities   , as defined in (14).Recall that we assume a constant , and therefore the steady-state distribution is independent of the jump function.Hence, we can use the formula for   from Proposition 4.2.The work then lies in computing the   , which are dependent on the choice of jump function.The first step to compute   is to find an explicit solution for the function  ( ) .The linear jump function (16) allows us to do this by solving the recurrence relation (5).The explicit form of  ( ) is provided in the following proposition (see Appendix B.2 for a proof).Proposition 5.1.Consider a 1G1B system with  (,  new ) =  +  and  (0) ( 0 ) =   0 ( new ), where  new is the fidelity of the state  new .Then, where the constants  ( )  are given by  (0) 0 =  new − 1 4 , and for  > 0.
In the following Lemma, we use the formulae for  ( ) to derive a closed-form expression for   and for the average consumed fidelity.Lemma 5.2.Consider a 1G1B system with  (,  new ) =  +  and  (0) ( 0 ) =   0 ( new ), where  new is the fidelity of the state  new .Then, the average fidelity after  ≥ 0 purification rounds is given by where  =  +  and  = /( + Γ).Moreover, the average consumed fidelity is given by The closed-form solution (20) is obtainable since  = 1  ∞ =0     is a geometric series with the linear jump function.In the following proposition, we see how this varies with  and .We provide a proof of Lemma 5.2 and Proposition 5.3 in Appendix B.2.We now have closed-form expression for  and  linear , which allows for a thorough analysis of the performance of the 1G1B system with the linear jump function.In particular, the following conclusions may already be drawn.
• Result () from Proposition 5.3 implies that the average consumed fidelity is maximized for  = 0 or  = 1.Consider a 1G1B system with a fixed set of parameters and a pumping scheme with a linear jump function.If the pumping protocol is good enough (e.g. when  ≥  new (1 − ), as explained in Appendix B.2), then pumping every time a link is generated ( = 1) maximises the average consumed fidelity.Sometimes, the pumping protocol chosen may impact the average consumed fidelity negatively and in that case one should never pump entanglement ( = 0) to increase the average consumed fidelity.• Result () from Proposition 5.3 provides similar insights: a pumping protocol with a good jump function always benefits from a larger probability of success, i.e.  linear is maximized for  = 1.When the protocol is detrimental, failure ( = 0) benefits the overall procedure, since it frees the good memory and allows for a fresh entangled link to be allocated there.Other parameters used in this example (times and rates in the same arbitrary units): . This jump function corresponds to a linear approximation of a specific bilocal Clifford protocol (the DEJMPS protocol) in the high-fidelity regime [11].
When the jump function is good (i.e. when  linear is monotonically increasing in ), we observe a trade-off between  linear and the availability , which is a decreasing function of , as can be seen from (10).This behaviour is shown in Figure 6.If we rarely purify (small ), a low-quality entangled state (small  linear ) will be available most of the time (large ).In that case, the average consumed fidelity can be lower than the fidelity of newly generated links, since the entanglement is not being purified often enough to compensate the noise introduced by the memory over time (in Figure 6, the average consumed fidelity is below the dashed line for small ).When purification is performed more often (larger ), the quality of the stored entanglement will be higher (larger  linear ), at the expense of a more limited availability (smaller ), since purification can fail and destroy the entanglement stored in the long-term memory.This trade-off disappears when the pumping scheme is deterministic ( = 1): the availability remains constant when varying  since purification will always succeed and the stored entanglement will not be destroyed.Note that, if the system is dominated by decoherence (Γ ≫ , ), the average consumed fidelity will always be smaller than  0 .
As a validation check, we also implemented a Monte Carlo simulation of the 1G1B system, which provided the same availability and average consumed fidelity that we obtained analytically.

Operating regimes of bilocal Clifford protocols
In this Subsection, we study the operating regimes of the 1G1B system, under the assumption that the pumping protocol employed is a bilocal Clifford protocol [10,25].Firstly, we find upper and lower bounds for the availability.Then, for a desired value of the availability within these bounds, we find lower and upper bounds for the average consumed fidelity that can be provided by bilocal Clifford protocols.This analysis finds limits to the performance of the 1G1B buffering system.
Bilocal Clifford protocols are one of the most well-studied types of protocol [10,16,25].One of their main advantages is that they are relatively simple to execute, since they involve a basic set of gates.To the best of our knowledge, bilocal Clifford circuits have been the only purification protocols implemented experimentally so far (see, e.g.[26,42]).In Appendix C we provide further details on bilocal Clifford protocols.
Let us start our performance analysis by discussing the availability.The maximum value that can be achieved by any protocol (bilocal Clifford or not) is /( + ), as can be seen from (10).This maximum value is obtained when there is no pumping or the pumping protocol succeeds deterministically, i.e. when  = 0 or  = 1.The availability is lower bounded by /(2 + ), and the minimum value is attained when a pumping protocol is always applied and it never succeeds, i.e., when  = 1 and  = 0.
To find bounds for the average consumed fidelity, we first need to bound the jump functions of all bilocal Clifford protocols, which we do in the following Lemma.We only consider nontrivial protocols, i.e. we do not consider bilocal Clifford protocols with  (,  new ) =  or  (,  new ) =  new , where  new is the fidelity of  new .The former trivial jump function corresponds to a protocol that leaves the buffered link untouched, while the second trivial jump function corresponds to a protocol that replaces the buffered link by the newly generated link.Lemma 5.4.Let  (,  new ) be the jump function of a nontrivial bilocal Clifford protocol ( (,  new ) ≠  and  (,  new ) ≠  new , where  new is the fidelity of  new ).Assume  new is a Bell-diagonal state: with where  min = min( 1 ,  2 ,  3 ).Then, for all  ∈ [ 1 4 ,  * ], the jump function is lower bounded as follows: where with , the jump function is upper bounded as with , and  u = Moreover, the success probability of the protocol is bounded by  l ≤  ≤  u , where , and A proof of Lemma 5.4 can be found in Appendix C. Note that, despite the fact that we assume that newly generated entangled links are Bell-diagonal, other forms of the density matrix are also valid in practice, since they can be brought to Bell-diagonal form by adding extra noise [4,22].
Note also that the lower bound for the jump function (23) only applies when the fidelity of the buffered link is below  * , but this is always the case in the 1G1B system, as shown in Appendix C.
If we regard  new as a fixed parameter, the upper and lower bounds to the jump function ( 23) and ( 25) are linear in  , and the bounds to the success probability (27) are constant.It is now possible to find an upper and lower bound for the average consumed fidelity by combining Lemmas 5.2 and 5.4, as we do in the following corollary.
Corollary 5.5.The average consumed fidelity of the 1G1B system when using any (nontrivial) bilocal Clifford protocol is lower bounded by and upper bounded by where  l ,  l ,  l ,  u ,  u , and  u , are given by ( 24), (26), and (27).
Now, we analyse the limits of the performance of the 1G1B system using the bounds on  from Corollary 5.5.Let us start with a 1G1B system with perfect memories, i.e. with Γ = 0.This corresponds to an ideal situation that we can use as a benchmark: once we introduce noise, the average consumed fidelity will be lower than in this ideal case.Figure 7(a) shows the achievable combinations of average consumed fidelity and availability for  new = 0.8, generation rate  = 1, and consumption rate  = 0.1.Below, we list some important observations that may be drawn from this Figure: • The regions shaded in grey correspond to unattainable values of average fidelity and availability, and they apply to any pumping scheme (bilocal Clifford or not).The average consumed fidelity cannot be larger than the one provided by a hypothetical protocol with jump function  (,  new ) = 1 and probability of success  = 1, which is applied with probability  = 1 (however, such a protocol does not exist).• The performance of a 1G1B system that uses any bilocal Clifford protocol is contained within the region shaded in blue and yellow.The yellow/blue line corresponds to a hypothetical protocol with jump function and success probability saturating the lower/upper bounds from ( 23) and (27).For a fixed target availability, the blue line provides an upper bound on the maximum average consumed fidelity that can be achieved by using bilocal Clifford protocols.Here, we observe again the tradeoff between both performance metrics: if our target availability is very close to the maximum value, we cannot increase the average consumed fidelity beyond  new (dotted line); but as we decrease the desired availability, we can achieve a higher consumed fidelity until we reach a maximum value.• As a reference, we show the performance of the replacement protocol (red star): in such a protocol, every time a new link is generated in the bad memory, the link in the good memory is replaced by the new one, without any form of purification.The replacement protocol is not bilocal Clifford because success is always declared (in bilocal Clifford circuits, success depends on some measurement outcomes [10]).This simple protocol achieves maximum availability, given by  = /( + ).However, since no purification is performed, this protocol cannot increase the fidelity above the initial value  new .In the absence of decoherence, the replacement protocol is equivalent to applying no purification at all ( = 0).
In Figure 7(b), we perform a similar analysis for a 1G1B system in which the good memory has a finite lifetime, i.e.Γ > 0. This is a more realistic scenario.The following observations may be drawn from this Figure: • Imperfect memories decrease the average consumed fidelity but do not affect the availability.The availability is unaffected by the decoherence experienced by the entangled links, and therefore can take the same range of values as in Figure 7(a).• The replacement protocol no longer provides an average fidelity  new .Instead, the average fidelity is lower than  new since the quality of the state stored in the good memory decreases over time and is never increased beyond  new due to the absence of purification.However, the replacement protocol performs better than no pumping at all ( = 0).This is because the system can improve its fidelity every time a new link is produced, instead of waiting for a consumption event.• In the presence of noise, the lower and upper bounds for bilocal Clifford protocols also shift towards lower values of average fidelity.Both the upper and lower bounds take their minimum value at  = 0.This means that, in the presence of noise, any pumping protocol will increase the average consumed fidelity, i.e. any pumping is better than no pumping, even if it succeeds with the lowest-possible probability.This is in contrast to when there is no noise (Figure 7(a)), where the lower bound takes its minimum at  = 1 and no such conclusion can be drawn.In fact, this conclusion (any pumping is better than no pumping) always applies when the amount of noise, Γ, is above a certain threshold.In Appendix B.2 we compute this threshold analytically.

CONCLUSIONS AND OUTLOOK
Our work sheds light on how to buffer high-quality entanglement shared among remote nodes in a quantum network.We have proposed two metrics to measure the performance of an entanglement buffering system: the availability and the average consumed fidelity.The availability corresponds to the fraction of time in which entanglement is available for consumption.The average consumed fidelity measures the quality of the entanglement upon consumption.We have used these performance metrics to analyse the 1G1B system, an entanglement buffering setup that uses two quantum memories per node.One of these memories has a finite lifetime and is used to buffer the entanglement, while the other memory is only used for entanglement generation.Entanglement generated in the bad memory can be used to pump the entanglement stored in the good memory.
We have modelled the system as a continuous-time stochastic process and used renewal theory to derive analytical expressions for both performance metrics.Our results confirm the intuition that, except in some edge cases, there is a trade-off between consuming entanglement at a high rate (high availability) and consuming high-quality entanglement (high average consumed fidelity).Remarkably, we found that, in a practical scenario (i.e. when the pumping protocol is bilocal Clifford and there is noise in the good memory), pumping the buffered entanglement is better than no pumping in terms of average consumed fidelity, even if the pumping has some probability of failure.
An assumption that allows us to find analytical solutions for our performance metrics is that the success probability of purification is constant over time.The model would be more realistic if the probability of successful purification was dependent on the state fidelity at that time, since this is the case for most protocols (in particular, the probability of successful purification is typically lower for input states with lower fidelity).This may mean that, realistically, the computation of the average fidelity when conditioning on successful purification may bias the system towards higher fidelity.However, we believe the comparison of our model (constant success probability) with a more realistic one incorporating this effect (success probability dependent on  ()) to be beyond the scope of this work, since we expect this to greatly complicate the analysis of the problem.
Our proposed metrics can be used to evaluate the performance of other entanglement buffering systems.An interesting extension of this work would be to compare the performance of the 1G1B system to a bipartite entanglement buffering setup with  quantum memories per node.In such a system, one could employ more advanced purification protocols that consume more than two entangled states.We also expect that the mathematical framework developed in this work can be used to initiate the performance analysis of more complex systems.We leave this as future work.

A FORMULAE FOR PERFORMANCE METRICS
In this Appendix, we prove Proposition 4.2 and Theorem 4.4, which provide the formulae for our two performance metrics (availability and average consumed fidelity).First, in A.1, we describe the stochastic process in the 1G1B setup in a simplified form and we provide some intermediate results that are necessary for the main proofs.Then, in A.2, we employ the results from A.1 to prove Proposition 4.2 and Theorem 4.4.

A.1 Simplified 1G1B
We now only view the 1G1B system as taking one of two states: ∅ (no entangled link in memory G), or ¬∅ (link in memory G).The system then alternates between these two states.For an illustration, see Figure 8.
C Fig. 8.The simplified 1G1B process.The system alternates between the states ¬∅ (link in memory G) and ∅ (no link in memory G).The system starts in ¬∅.The times spent in ¬∅ and ∅ are denoted by   and   , respectively.
More formally, the simplified 1G1B process is the following.
Definition A.1 (Simplified 1G1B).Let  () ∈ {∅, ¬∅} denote the state of simplified 1G1B at time .Suppose that  (0) = ¬∅, i.e. the system starts when there is a link in memory F. Let  1 be the time until this first link is removed, and let  1 be the time for which the system is empty until a fresh link is produced again.Let {  }  ≥1 be the times spent in ¬∅ until the link was removed from memory G (due to consumption or failed purification), and {  }  ≥1 be the times which the system spent in ∅ until a link was produced.Then, according to our model of 1G1B, the   are i.i.d. and exponentially distributed with rate   + (1 − ), and the   are i.i.d and exponentially distributed with rate .
Recall that  is the rate of generation of new entangled links,  is the rate of consumption of links in memory G,  is the probability of immediately using new links for pumping, and  is the probability of successful pumping.
We will write the distribution functions as The process     +   defines a renewal process, which we introduce with the following definition.
The value   is referred to as the th arrival time of the process, and the values   are known as the interarrival times.From now on, we also use  0 = 0,   =  1 + .... +   to denote the th time at which a fresh link is produced, causing the system to move from ∅ into ¬∅.
The renewal function is central to renewal theory, which we define below.Throughout, we use the convention d() ≡  ′ ()d for differentiable functions .
We will derive formulae for the availability and average consumed fidelity using this mathematical framework.An important result that we will use in order to do this is the renewal theorem, which we state below.This result assumes that the   are not arithmetic.If  1 is arithmetic, this essentially means that  1 only takes values in a set { :  = 0, ±1, ...}, with  > 0. For more details of arithmetic random variables, see Chapter 10 of [19].
Theorem A.4 (Renewal Theorem/ Theorem 10.1.11from [19])).Consider a renewal process as given in Definition A.2. Let   be the distribution function of the random variable  1 , where  1 is not arithmetic.Let  () be a bounded function.Consider solutions  to the renewal-type equation Then, a solution is If  is bounded on finite intervals then  is bounded on finite intervals, and ( 32) is the unique solution of ( 31) with this property.
The renewal-type equation often arises when studying renewal processes, as we will see further on.The following result may be derived using Theorem A.4, and is useful when taking the infinite limit.
whenever  1 is not arithmetic.
We are now partially equipped to show the formulae for the availability and average fidelity.Next, we show a set of intermediate results that we will need for the main proofs.
Proposition A.6.Let  () =  ( () = ¬∅) be the probability that a link is available at time  in the simplified 1G1B process.Then, Proof.We proceed by conditioning on the value of  1 .Now, Notice that the event { () = ¬∅ ∩  1 >  } occurs if and only if  1 > .Further, if  < , then since the process starts afresh at time .Then, (34) becomes where d  () ≡  ′  ()d.We now see that this is of the form (31) with  () = 1 −   (), and so by Theorem A.4, Taking the infinite limit, lim It can be seen that  () = 1 −   () satisfies the conditions (a)-(c) required by Theorem A.5, so we may apply this Theorem to take the limit: Finally, using (33).□ Recall that the average fidelity of the system at a given time  is dependent on the time spent in each purification level leading up to this point.Therefore, in order to understand the average fidelity we first of all look at the current lifetime in this simplified setting.
Definition A.7 (Current lifetime).Consider the simplified 11 system.Let  () be the time spent so far in a state at time .More formally, 1G1B 0 time . Current lifetime of the simplified 1G1B process.The random variable  () denotes the time spent so far in the current state at time .This is most interesting when  () = ¬∅, because it tells us the age of a link in memory.
The first case ( () = ¬∅) is of most interest here, because it corresponds to when a link is in memory and is subject to decoherence.See Figure 9 for an illustration of this concept.In the following Lemma, we characterise the distribution of  (), conditional on being in the state ¬∅.
Lemma A.8. Consider the simplified 1G1B system.The limiting distribution of  () conditional on there being a link is given by which is an exponential distribution with parameter  when  1 ∼ Exp(). Proof.Writing we see that the bottom of the fraction has already been dealt with in Proposition A.6.We therefore focus on  (, )  ( () >  ∩  () = ¬∅).
(44) Conditioning on  1 , we see that Now, the event { () Moreover, if  <  then the process starts afresh from time , and Then, noting that  (, ) = 0 for  < , (45) becomes which is in the form of (31) with  () = 1 { ≥ } (1 −   ()).Then, by Theorem A.4,  (, ) is given by which has limit lim letting  =  − .Then, noting that () = 1 −   ( + ) satisfies conditions (a)-(c) of Theorem A.5, we may apply this to find lim From Proposition A.6, we observe that . We can use this to rewrite (52) as follows: which we notice is only dependent on the distribution of  1 .In the case  1 ∼ Exp(), as considered in the 1G1B system, lim and so conditional on there being a link, the current lifetime approaches an exponential distribution.□ We have now characterised the availability (Proposition A.6) and current lifetime (Lemma A.8) for the simplified 1G1B system.However, note that both Proposition A.6 and Lemma A.8 assumed that the system starts in the state  (0) = ¬∅.This was necessary in order to satisfy all of the conditions (a)-(c) of Theorem A.5.The result below states that Theorem A.5 still holds, even if the renewal process is delayed, which means that the first arrival has a different distribution to the others.For more details of delayed renewal processes, see [19] or [30].
Definition A.10.Let  d be a delayed renewal process.Then,  d () E[ d ()] is the delayed renewal function.
A consequence of Theorem A.11 is that even for delayed renewal processes, the limiting distribution is the same as for the non-delayed case.Therefore, the results of Proposition A.6 and Lemma A.8 hold even when the distribution of  1 is not the same as {  }  ≥2 .In particular, they still hold when the process starts in ∅.This is summarised with the following corollary.
Corollary A.12. Consider the simplified 1G1B process, now altered to start in  (0) = ∅.Let  0 be the time for which the system is empty until the first fresh link is produced.Let  1 be the time in which this link is present in memory until it is removed again, and so on.Let the probability of finding a link at time  be  () =  ( () = ¬∅).Then, and the distribution of the current lifetime of a link satisfies Recalling that  =  + (1 − ), we see that the formula for the availability in Proposition 4.2 is already shown by (56).

A.2 Availability and average consumed fidelity in 1G1B
Here, we compute the availability and the rest of the steady-state distribution of the 1G1B system (Proposition 4.2), as well as the average consumed fidelity (Theorem 4.4).
In order to calculate the average fidelity, we not only need the time spent in ¬∅, but also the times spent in each pumping level leading up to the current one.
From 1G1B (Definition 3.1), one may define a simplified 1G1B system as 10.Number of pumping rounds.We are interested in the number of pumping rounds that have been carried out while a link is in memory.Here,  is the (limiting) distribution of the current lifetime in memory (see Figure 9), and   is the time between the ( − 1)th and th pumping round.
For the characterisation of the fidelity of the link in memory at time ,  (), we are interested in the successful pumping attempts that occur in the the time interval [  ( ) ,   ( ) +  ()), where  () is the current lifetime (Definition A.7).In 1G1B, the successful pumping attempts are a Poisson process with rate  .Since the rate is constant for all , the number of successful pumping attempts within the interval [  ( ) ,   ( ) +  ()) has the identical distribution as the number of successful pumping attempts in the time interval [0,  ()).From Corollary A.12, we see that  () converges in distribution to  ∼ Exp().In the following Lemma, we characterise the number of successful pumping attempts that occur within the time , and the time spent between each pair of consecutive pumping rounds.See Figure 10 for an illustration.An observation that we use below is that within the time interval [0, ), the times at which pumping occurs form a separate renewal process, which is convenient for notation.Lemma A.13. Consider a renewal process Ñ () with arrival times  0 = 0,   =  =1   , with  1 ∼ Exp().Let  ∼ Exp() be independent of the   .Let  =  () be the number of arrivals that have occurred by time .Let C  −   be the current lifetime at time .Then, (1) The distribution of  is given by or equivalently (2) Conditional on  = , the random variables ( 1 , ...,   , C) are mutually independent and identically distributed as Exp( + ). Proof.
(1) We proceed by induction.Letting    ( ≤ ) We have In step (), we have used the inductive assumption.In step () we have made use of the memoryless property of the exponential distribution: Since  +1 and   are positive and independent random variables, this has as a consequence (2) We firstly note that for any events  1 ,  2 ,  3 , it holds that where ¬ the complement of the event .Now, consider the events where in () we have used the definition of C, and in () we have used the definition of .Then, by (61), we see that By the independence of the   , this is equivalent to and we now use the memoryless property of the exponential distribution (see the argument leading to (60)) to rewrite as which, using that  (  ≥     >   ) (  >   ) =  (  ≥   >   ), becomes where we have again made use of (61) to rewrite the factor on the left.Now, and by symmetry We therefore see that It therefore follows that which suffices to show the second result.□ Recalling that  () converges in distribution to , we now adapt Lemma A.13 to apply to  ().In order to do this, we use the following result (for a proof, see Chapter 7 of [19]).Since this is true for all such sequences {  }, the result follows.□ In the following corollary, we let the current lifetime be dependent on the parameter  to avoid confusion with the time of the renewal process (which is denoted as ).
Corollary A. 16.Consider a renewal process  () with arrival times  0 = 0,   =  =1   , with  1 ∼ Exp().Suppose that  () converges in distribution to  ∼ Exp() as  → ∞.Let  () =  ( ()) be the number of arrivals that have occurred by time  ().Let C ()  () −  ( ) be the current lifetime at time  ().Then, the results of Lemma A.13 still hold in the limit  → ∞.In particular, (1) The limiting distribution of  () is that of , (2) Conditional on  () = , the random variables ( 1 , ...,   , C) converge in distribution to mutually independent and identically distributed Exp( + ) as  → ∞, i.e. (2) One may use exactly the same arguments as were used to obtain (63), only replacing  with  () and  with  (), to show that By Corollary A.15, in the limit  → ∞ this satisfies lim It then follows that − (+ )  , by Lemma A. 13. □ For the case when  () is the current lifetime of simplified 1G1B, the random variable ( 1 , . . .,   , C ()) by definition has the same distribution as ì  ().Recall that ì  () contains the times spent in each purification level leading up to the current one at time  in 1G1B (Definition 3.2).This leads to the following results.
We now continue with the formulae for the performance metrics.The availability in the 1G1B system was given in Proposition 4.2 and the average consumed fidelity was given in Theorem 4.4.Next, we prove both of them.
Proof of Proposition 4.2.From Corollary A.12, we see that Letting  () denote the current lifetime of simplified 1G1B at time , and  () denote the number of purifications that have occurred within this time, by Corollary A.16 it follows that Recalling the distribution of  as found in Lemma A.13, we obtain lim We note that this result can also be derived with the global balance equations of a CTMC.Here, we chose to use the derivation with renewal theory since it offers a more general formula for the availability (see (56)) and ties in more neatly with the derivation of the formula for the average consumed fidelity, as we will see below. □ The following proposition will be helpful in the proof of Theorem 4.4 (formula for average consumed fidelity).Proposition A.18.Let {  ()}  ≥0 and {  ()}  ≥0 be such that for all , lim  →∞   () =   and lim  →∞   () =   .Suppose also that for all , 0 ≤ Proof of proposition A.18.To show (68), it suffices to show that for any  > 0, there exists a  such that for all  >  , We firstly bound the sum using the triangle inequality, We then show that () → 0 and () → 0 as  → ∞.We firstly show that lim . This suffices to show (71).Combined with the fact that the   are bounded, it follows that () → 0. We now show that () → 0, i.e.This is possible since the LHS is a finite sum.Then, which shows (73).□ Combining these results (() → 0 and () → 0) in ( 70)) suffices to show Proposition A.18.We are now ready to prove Theorem 4.4 (formula for average consumed fidelity).
Proof of Theorem 4.4.We firstly expand out the average consumed fidelity (Definition 4.3) as a sum by conditioning on the value of  (), Recall that we are interested in the limit  → ∞ of the above.Note that from Proposition 4.2, we know the limiting values of  ( () ≠ ∅) and  ( () = ).We now claim that lim where  0 , where   = E  ( ) ( 0 ,  1 , ...,   ) and   = lim  →∞  ( () = ).□ where the constant terms are no longer in the sum since by the normalisation of the steady state distribution.Recalling the distribution of  from Proposition 4.2, we may evaluate the sum as a geometric series, We may now substitute this into (80) to obtain a closed-form solution for the average fidelity, (82) Since the sign of the derivative does not depend on , we conclude that  linear is monotonic in .
To show (), we proceed similarly: linear  =  4( −  new ) (Γ +  + ) +  Γ + 4 new ( + ) Since the sign of this derivative does not depend on , we conclude that  linear is monotonic in .□ Note that  linear can be monotonically increasing or decreasing in  and in  depending on the values of the other parameters.For a pumping protocol with a good enough jump function,  linear becomes increasing in .A sufficient condition is for the jump function to satisfy  ≥  new (1 − ), as we show next.The partial derivative with respect to  can be written as follows: where  = 2(Γ + +(1−)),  = 4 new (1−) + (4 +) −1, and  =  − new (1−).Using the fact that  ≥ (1 − )/4, we find that  ≥ 0. A sufficient condition for the partial derivative to be positive is that  ≥ 0, i.e., if  ≥  new (1 − ), then the average consumed fidelity is monotonically increasing in .Moreover, we can conclude that, if the noise is above certain threshold (Γ > −4/), the derivative is positive and the pumping is always beneficial, even if it succeeds with a very small probability. min =  3 and  max =  1 .In order to show this bound, we make use of the following collection of results.It is important to note that when showing all of the following results,  BD is fixed.
(1) Proposition C.1, Corollory C.2, Proposition C.3 -the formula for  * is derived (Equation 22).This is the maximum achievable fidelity achievable in the 1G1B system, with fixed Bell-diagonal input state  BD .Therefore, at any given time , the fidelity  () of the stored link in the 1G1B system (see Definition 3. In particular, the third result means that any straight line taken between two points on   must lie below the curve itself.The linear lower bound is the linear function connecting the points ( * ,  1 ( * ,  BD )) , 1 4 ,  3 where  min = min{ 1 ,  2 ,  3 }.(92) To check the sign of these functions, we must therefore check the sign of  − .Recalling that   may be rewritten as Proof of Lemma C.6.We will analyse the entanglement of a Bell-diagonal state using the Peres-Horodecki criterion, which states that a bipartite quantum state  is entangled if and only if the partial transpose of  has at least one negative eigenvalue [21,33].A Bell-diagonal state can be written in the Bell basis as  =  0  +  + +  1  +  + +  2 | − ⟩⟨ − | +  3 | − ⟩⟨ − | .
Fig.1.Illustration of the entanglement buffering system with two quantum memories (1G1B system).Each of the nodes has two memories (G and B).Memory G is used to store the buffered link.An entangled link is generated at a rate  in memory B. If memory G is empty when the new link is generated in B, the link is immediately transferred to G. If memory G is occupied, the new link generated in B is immediately used to purify the buffered link with probability  (otherwise, the new link is discarded).The pumping protocol consumes the link in B to increase the quality of the buffered link in G, and it succeeds with probability  (otherwise, it destroys the link in G).The buffered link is consumed at a fixed rate .The quality of the entanglement stored in G decays exponentially with rate Γ.Formal definitions of the problem parameters can be found in Section 3.

Proposition 5 . 3 .
The quantity  linear has the following properties: (a)  linear is a monotonic function of ; (b)  linear is a monotonic function of ;

Fig. 7 .
Fig. 7. Noise in the memories decreases the average consumed fidelity but does not affect the availability.Bounds on the performance of a 1G1B system with a bilocal Clifford protocol, and with (a) noiseless memories (Γ = 0) or (b) noisy memories (Γ = 5 • 10 −2 a.u.).For a given target availability, the average consumed fidelity is within the blue/yellow region (see Corollary 5.5).Availability is maximized for  = 0 ( is the probability of purification after successful entanglement generation), and it decreases for increasing .White regions cannot be achieved by bilocal Clifford protocols.Striped regions cannot be achieved by any pumping protocol.Red star: performance of the replacement protocol (buffered link is replaced by new links).Dotted line: fidelity of newly generated entangled links.Parameters used in this example (times and rates in the same arbitrary units):  = 1,  = 0.1,  new = 0.8,  new =  new  +  + + (1 −  new )  +  + + | − ⟩⟨ − | /2.

Table 1 .
Parameters of the 1G1B system.See main text for detailed explanations. new ) Jump function: fidelity of the output state following successful pumping ( is the fidelity of the Werner state stored in the good memory) 1 , . ..,   are i.i.d.random variables with  0 ∼ Exp( +), and  ( ) is given in Definition 3.5.We use the following result: Theorem A.19 (Theorem 7.2.19 of [19]).Let   be a sequence of random variables.Then,   →  in distribution if and only if E[(  )] → E[( )] for all bounded continuous functions .Recall that conditional on  () = , we have  () =  ( ) ì  () (from Definition 3.6).As mentioned in Section 3,  ( ) is a continuous and bounded function.Therefore, (75) follows by combining Theorem A.19 and Corollary A.17.From Proposition A.18, we therefore see that lim  →∞ From Proposition C.1 and Corollary C.2, we know which of  1 ,  2 and  3 provide the best fidelity for  ∈ [1/2, 1].With the following proposition, we see that for some lower fidelities, it is better to replace with the bad link rather than choose to pump.Proposition C.3.The largest fidelity obtainable by pumping a low-fidelity Werner state with  BD and bilocal Clifford protocols is  * = 2 BD − 1 + √︁ (2 BD − 1) 2 + 2 min (2 BD − 1 + 2 min ) 2(2 BD − 1 + 2 min )