Entanglement distillation and entanglement swapping have been extensively studied assuming perfect quantum memories. However, near-term quantum networks will be fundamentally limited by quantum memories with a finite coherence time, resulting in complex choices for the timing and ordering of these operations. In this work, we study entanglement distillation and entanglement swapping at the level of the elementary building blocks of noisy quantum repeater networks, with the goal of elucidating the fundamental tradeoffs induced by memory decoherence. First, we focus on a minimal one-hop setting, where we analytically compare ``distill-as-soon-as-possible'' and ``distill-as-late-as-possible'' strategies against a baseline strategy that simply discards the older entangled state. We find that in the low memory coherence time regime, discarding the older entangled state achieves higher expected output fidelity, while in the high coherence time regime, delaying distillation until the end achieves the highest expected output fidelity and, over most of the deadline range, the highest weighted coherent information, at the expense of a lower success probability. We then extend our analysis to two-hop repeater chains using Monte Carlo simulation. In this setting, we find that the highest weighted coherent information is achieved by strategies that defer distillation to the end of the time window, with \textsf{Distill-ALAP-then-Swap-ALAP} and \textsf{Swap-ASAP-then-Distill-ALAP} leading at different operating points, while \textsf{Discard-Oldest-then-Swap} never reaches positive weighted coherent information in any regime tested. Together, these results clarify how decoherence reshapes the optimal operation timing in quantum networks and provide instructive insights into the link-level principles that govern larger-scale architectures.
Comments
11 pages, 9 figures, under review for publication at QCE26 as of arXiv submission
Entanglement distillation and entanglement swapping have been extensively studied assuming perfect quantum memories. However, near-term quantum networks will be fundamentally limited by quantum memories with a finite coherence time, resulting in complex choices for the timing and ordering of these operations. In this work, we study entanglement distillation and entanglement swapping at the level of the elementary building blocks of noisy quantum repeater networks, with the goal of elucidating the fundamental tradeoffs induced by memory decoherence. First, we focus on a minimal one-hop setting, where we analytically compare distill-as-soon-as-possible'' and distill-as-late-as-possible'' strategies against a baseline strategy that simply discards the older entangled state. We find that in the low memory coherence time regime, discarding the older entangled state achieves higher expected output fidelity, while in the high coherence time regime, delaying distillation until the end achieves the highest expected output fidelity and, over most of the deadline range, the highest weighted coherent information, at the expense of a lower success probability. We then extend our analysis to two-hop repeater chains using Monte Carlo simulation. In this setting, we find that the highest weighted coherent information is achieved by strategies that defer distillation to the end of the time window, with \textsf{Distill-ALAP-then-Swap-ALAP} and \textsf{Swap-ASAP-then-Distill-ALAP} leading at different operating points, while \textsf{Discard-Oldest-then-Swap} never reaches positive weighted coherent information in any regime tested. Together, these results clarify how decoherence reshapes the optimal operation timing in quantum networks and provide instructive insights into the link-level principles that govern larger-scale architectures.
Long-distance quantum communication relies on quantum repeaters to overcome exponential photon attenuation in optical fiber links. Connecting adjacent elementary links requires entanglement swapping, while preserving end-to-end fidelity above cryptographic thresholds requires entanglement distillation. In physical hardware, however, quantum memories suffer from decoherence during the classical round-trip signaling intervals mandated by distillation protocols. This work investigates the architectural trade-off between two core operational strategies in a dual-hop quantum repeater: Strategy A (Purify-then-Swap), wherein elementary links are distilled prior to swapping, and Strategy B (Swap-then-Purify), wherein raw links are swapped immediately and distillation is executed end-to-end. We develop an analytical and numerical simulation pipeline tracking Werner state transformations under bilateral CNOT purification and non-Markovian memory dephasing. Our results reveal an exact crossover tipping point at tau_RTT / T_2 ≈ 0.018: whenever classical signaling delay exceeds 1.8% of the memory coherence time, Strategy B strictly outperforms Strategy A. Furthermore, we show that local purification becomes actively destructive beyond tau_RTT / T_2 ≈ 0.070, delivering fidelities inferior to unpurified swapping. Finally, we establish the minimum hardware boundary, showing that raw elementary links must achieve F_init >= 0.880 for single-round purification to cross the threshold for secret key distillation (F > 0.810).
The quantum switch is a vital component for the development of the quantum internet. In an entanglement distribution network, the function of a quantum switch is to generate elementary entanglement with its clients followed by entanglement swapping to distribute end-to-end entanglement of sufficiently high fidelity between clients. The threshold on entanglement fidelity is any quality of service requirement specified by the clients as dictated by the application they run on the network. We consider a discrete time model for a quantum switch that attempts generation of fresh elementary entanglement with clients in each time step in the form of maximally entangled qubit pairs, or Bell pairs, which succeed probabilistically; the successfully generated Bell pairs are stored in noisy quantum memories until they can be swapped. We focus on establishing the value of entanglement distillation of the stored Bell pairs prior to entanglement swapping in presence of their inevitable aging, i.e., decoherence: For a simple instance of a switch with two clients, an exponential decay of entanglement fidelity, and a wellknown probabilistic but heralded two-to-one distillation protocol, given a threshold end-to-end entanglement fidelity, we use the Markov Decision Processes framework to identify the optimal action policy - to wait, to distill, or to swap that maximizes throughput. We compare the switch's performance under the optimal distillation-enabled policy with that excluding distillation. Simulations of the two policies demonstrate the improvements that are possible in principle via optimal use of distillation with respect to average throughput, average fidelity and jitter of end-to-end entanglement, as functions of fidelity threshold. Our model thus helps capture the role of entanglement distillation in mitigating the effects of decoherence in a quantum switch in an entanglement distribution network, adding to the growing literature on quantum switches.
Bell-state measurement (BSM) on entangled states shared between quantum repeaters is the fundamental operation used to route entanglement in quantum networks. Performing BSMs on Werner states shared between repeaters leads to exponential decay in the fidelity of the end-to-end Werner state with the number of repeaters, necessitating entanglement distillation. In this work, we use quantum error correcting codes for deterministic entanglement distillation to route Werner states on a chain of repeaters. To maximize the end-to-end distillable entanglement, we utilize global link-state knowledge to determine the optimal policy for scheduling distillation and BSMs at the repeaters. We observe that low-rate codes produce high-fidelity end-to-end states owing to their excellent error-correcting capability, whereas high-rate codes yield a larger number of end-to-end states but of lower fidelity. The number of quantum memories used at repeaters increases with the code rate as well as the classical computation time of the decoder.
Quantum network applications impose a variety of requirements on entanglement resources in terms of rate, fidelity, latency, and more. The repeaters in the quantum network must combine good methods for entanglement generation, effective entanglement distillation, and smart routing protocols to satisfy these application requirements. In this work, we focus on entanglement distillation in a linear chain of quantum repeaters. While conventional approaches reuse the same distillation scheme over multiple hop lengths after entanglement swaps, we propose a novel adaptive quantum error correction (QEC) scheme that boosts end-to-end metrics. Specifically, depending on the network operating point, we adapt the code used in distillation over successive rounds to monotonically increase the rate while also improving fidelity. We demonstrate the effectiveness of this strategy using three codes, with parameters [[9,1,3]], [[9,2,3]], [[9,3,3]], and a new performance metric, efficiency, that incorporates both overall rate and fidelity. Since the minimum input fidelity for QEC-based distillation is high, we then extend our study to include non-QEC-based purification protocols, specifically DEJMPS since it outperforms others. We compare the performance of end-to-end DEJMPS against adapting from DEJMPS to QEC once DEJMPS improves the initial fidelity to the threshold for QEC. Through a refined efficiency metric, we illuminate the regime where QEC is beneficial. These results provide a detailed outlook for entanglement purification and distillation in first and second generation quantum repeaters.
Entanglement distillation is a fundamental operation in quantum information processing used to obtain higher-fidelity entangled pairs from a supply of less entangled quantum states using local operations aided by classical communication (LOCC). In a physically relevant setting, where states with an initial fidelity of f0, probabilistically generated over multiple, m, memory pairs distributed between two parties, Alice and Bob, are pairwise distilled, the optimal policy identifies the system-configuration dependent sequence of entanglement generation and distillation operations that need to be performed in order to minimize the expected time to reach some target fidelity fT>f0. Here, we formulate and systematically analyze this task as a Markov decision problem and using a value iteration algorithm, obtain optimal deterministic policies that minimize the expected waiting time required to reach a target fidelity. Our results show that the expected waiting time under the optimal policy decreases with increasing generation probability p and number of quantum memories m - as expected. In contrast, it exhibits non-monotonic behavior with respect to f0 for a fixed fidelity gap, (Δf=fT−f0). While the optimal policy consistently outperforms baseline policies such as the greedy, nested and entanglement pumping policies, its relative advantage is regime-dependent, being determined by the system parameters (p,f0,fT,m), and exhibits a nontrivial dependence on the fidelity gap Δf. Our results highlight the value of formulating entanglement distillation as a Markov decision problem, enabling the systematic design of policies that achieve target fidelity thresholds for quantum information tasks in realistic resource-constrained settings.
Losses are one of the main bottlenecks for the distribution of entanglement in quantum networks, which can be overcome by the implementation of quantum repeaters. The most basic form of a quantum repeater chain is the swap ASAP repeater chain. In such a repeater chain, elementary links are probabilistically generated and deterministically swapped as soon as two adjacent links have been generated. As each entangled state is waiting to be swapped, decoherence is experienced, turning the fidelity of the entangled state between the end nodes of the chain into a random variable. Fully characterizing the (average) fidelity as the repeater chain grows is still an open problem. Here, we analytically investigate the case of equally-spaced repeaters, where we find exact analytic formulae for all moments of the fidelity up to 25 segments. We obtain these formulae by providing a general solution in terms of a generating function ; a function whose n'th term in its Maclaurin series yields the moments of the fidelity for n segments. We generalize this approach as well to a global cut-off policy – a method for increasing fidelity at the cost of longer entanglement delivery times – allowing for fast optimization of the cut-off parameter by eliminating the need for Monte Carlo simulation. We furthermore find simple approximations of the average fidelity that are exponentially tight, and, for up to 10 segments, the full distribution of the delivered fidelity. We use this to analytically calculate the secret-key rate, both with and without binning methods.