We investigate how end-to-end fidelity in quantum networks depends on the relative entropy of coherence under amplitude damping and depolarizing noise, as well as how it degrades with an increasing number of swapping operations. Analytical models reveal that while purification improves fidelity, it offers diminishing returns as the number of purifications increases. Our results show that depolarizing noise yields higher fidelity than amplitude damping for the same relative entropy of coherence. This framework enables coherence-aware routing using only fidelity measurements, providing a foundation for future routing protocols in quantum networks.
We investigate how end-to-end fidelity in quantum networks depends on the relative entropy of coherence under amplitude damping and depolarizing noise, as well as how it degrades with an increasing number of swapping operations. Analytical models reveal that while purification improves fidelity, it offers diminishing returns as the number of purifications increases. Our results show that depolarizing noise yields higher fidelity than amplitude damping for the same relative entropy of coherence. This framework enables coherence-aware routing using only fidelity measurements, providing a foundation for future routing protocols in quantum networks.
Ensuring high end-to-end fidelity is crucial for quantum networks, yet the impact of relative entropy of coherence on end to end fidelity remains unexplored. This study investigates how amplitude damping, and depolarizing noise affect coherence and fidelity as distance increases, with and without entanglement purification. Using Kraus operators, we model noise and compute relative entropy of coherence. By applying curve fitting, we extract coherence decay time and integrate it into the end-to-end fidelity equation, deriving a new analytical expression. Results show that noise models significantly impact fidelity, with purification improving performance but with varying effectiveness. These findings provide new insights into coherence-fidelity relationships, emphasizing the need for coherence-aware strategies in quantum routing.
In quantum networks, routing mechanisms designed to deliver successfully multiple simultaneous requests to the destination under limited channel capacity should simultaneously consider resource usage optimization, entanglement-generation performance, and the quality of the end-to-end delivered entangled states. The routing approaches developed for quantum networks in the literature mostly consider the change in fidelity during routing. However, it should be noted that quantum states with the same fidelity value may have different levels of coherence. In this study, for the first time, a novel routing and purification approach for quantum networks is presented, using the end-to-end (E2E) relative entropy of coherence (REC) together with E2E fidelity to determine the purification level and the feasibility of candidate paths. In our study, the coherence- and fidelity-aware routing algorithm (CAFARA) is proposed. In CAFARA, the BBPSSW purification levels are determined for each candidate path and ordered from the lowest to the highest. This information is stored in a lookup table. Then, the lowest BBPSSW purification level that simultaneously satisfies the requested fidelity, REC, latency, and capacity constraints is selected. The E2E fidelity, REC, and required number of raw Bell pairs mentioned here are all stored together in the previously mentioned lookup table. This lookup table was constructed using imperfect initial Werner states, one-sided amplitude damping, Werner-state twirling, BBPSSW purification, and density-matrix-based entanglement swapping, and all calculations were completed before the routing process. As the final step, CAFARA selects the best path with the highest entanglement generation rate (EGR) from the selected feasible paths. In our study, the FARA-PostREC and FARA-NoREC algorithms were developed for comparison with CAFARA. While the FARA-PostREC algorithm uses the REC constraint during the final validation stage of the request, FARA-NoREC does not use any REC constraint; it uses only fidelity as the quality parameter of the paths and as the parameter for determining the purification level. In our simulations under varying link distance, channel capacity, network size, and request load, CAFARA achieved a better average request success rate than FARA-PostREC by reducing late-stage request drops related to coherence because it also uses REC during the purification-level determination stage while maintaining the required E2E fidelity and REC values. Although FARA-NoREC often accepts more requests and assumes that they are successfully delivered, the average final fidelity (AFF) and average final coherence (AFC) values of the requests considered successful are lower than those of the other algorithms; the AFC does not even satisfy the REC threshold. Overall, CAFARA provides a balanced trade-off between the quality-guaranteed request success rate, latency, purification overhead, and resource consumption.
Entanglement purification with two entangled resource pairs is widely employed in the literature on quantum repeater networks to counteract fidelity degradation introduced by noisy quantum memories and entanglement swapping across multiple hops. Standard purification protocols assume both resource pairs carry identical fidelity. In practice, entanglement generation is stochastic, the two resource pairs are heralded at different times, and so the first pair decoheres in memory while the second is being generated. Thus a fidelity asymmetry is a structural feature of any network operating under realistic memory conditions, leading to the question: when is it beneficial to perform purification? We derive a closed-form fidelity asymmetry tolerance delta(F) that governs whether a purification attempt is beneficial. We determine a universal upper bound delta_max of approximately 0.076 beyond which purification is always counterproductive. Our simulations show that with exponential memory decoherence, purification yields benefits in only approximately 14% of purification attempts on two resource pairs in a two-hop repeater chain. We define three network objectives: fidelity only, time only, and a combination of time and fidelity, to deliver end-to-end entanglement. We show that when the application fidelity requirement is achievable through swapping alone, no-purification is the superior policy, with its advantage increasing with the number of hops. When the fidelity requirement cannot be met with swapping alone and purification is necessary, to be effective, it must be conditioned on delta(F) between resource pairs. We introduce DeltaPurify, a policy that conditions purification decisions on local fidelity information, and show it reduces time-to-serve relative to both naive purification and no-purification across several fidelity thresholds and hops of a repeater chain.
Quantum networks rely on the efficient distribution of entanglement to enable long-distance quantum communication and information processing. A key challenge in these networks is the design of routing protocols capable of maintaining high quality entanglement in the presence of noise, decoherence, and imperfect operations, which progressively degrade the fidelity of entangled states through entanglement swapping. Entanglement purification provides an effective mechanism to mitigate this degradation at the cost of additional resources. In this work, we study purification-aware quantum routing and formulate the problem of selecting optimal purification strategies as an optimization task. By employing dynamic programming techniques, we identify strategies that optimally balance resource consumption and end-to-end fidelity, demonstrating the effectiveness of our approach across different scenarios.
Abstract Quantum state purification is crucial in quantum communication and computation, aiming to recover a purified state from multiple copies of an unknown noisy state. This work introduces a general state purification framework designed to achieve the highest fidelity with a specified probability and characterize the associated trade-offs. For i.i.d. quantum states under depolarizing noise, our framework can replicate the purification protocol proposed by Barenco et al (1997 SIAM J. Comput. 26 1541–57) and further provide exact formulas for the purification fidelity and probability with explicit trade-offs. We prove the protocols’ optimality for two copies of noisy states with any dimension and confirm its optimality for higher numbers of copies and dimensions through numerical analysis. Our methodological approach paves the way for proving the protocol’s optimality in more general scenarios and leads to optimal protocols for other noise models. Furthermore, we present a systematic implementation method via block encoding and parameterized quantum circuits, providing explicit circuits for purifying three-copy and four-copy states under depolarizing noise. Finally, we estimate the sample complexity and generalize the protocol to a recursive form, demonstrating its practicality for quantum computers with limited memory.
We investigate how noise impacts nonstabilizerness-a key resource for quantum advantage-in many-body qubit systems. While noise typically degrades quantum resources, we show that amplitude damping, a nonunital channel, can generate or enhance nonstabilizerness, whereas depolarizing noise provably cannot. In an encoding-decoding protocol, we find that, unlike in the coherent-noise case, a sharp decoding fidelity transition is not accompanied by a transition in nonstabilizerness. Although amplitude damping locally injects nonstabilizerness, this resource is washed out at the collective level after encoding, decoding, and postselection. Our results reveal that realistic incoherent noise can suppress many-body nonstabilizerness criticality even while generating it microscopically.