Probabilistic Cutoffs in Homogeneous Quantum Repeater Chains

TL;DR

This work investigates probabilistic cutoffs in homogeneous quantum repeater chains, aiming to reduce the need to track link ages while maintaining entanglement distribution performance. By modeling a discrete-time, barrier-free chain and comparing a new probabilistic cutoff policy with a traditional deterministic cutoff, the authors derive exact rate and fidelity expressions for three-node chains and use Markov-chain analysis for larger chains. They show that probabilistic cutoffs generally yield lower fidelity at fixed rate but can outperform deterministic cutoffs when a high end-to-end fidelity is required, and that secret-key rates under probabilistic cutoffs are within the same order of magnitude as deterministic cutoffs for short chains and/or high $p_g$, with Monte Carlo results extending to longer chains. The findings suggest probabilistic cutoffs as a practical alternative in certain regimes, particularly where state tracking or classical coordination is costly, and point to potential benefits in multiplexed repeater networks and future extensions to longer chains and different cutoff strategies.

Abstract

We study quantum repeater chains in which entangled links between neighbouring nodes are created through heralded entanglement generation and adjacent links are swapped as soon as possible. Since heralded entanglement generation attempts succeed only probabilistically, some links will have to be stored in quantum memories at the nodes of the chain while waiting for adjacent links to be generated. The fidelity of these stored links decreases with time due to decoherence, and if they are stored for too long then this can lead to low end-to-end fidelity. Previous work has shown that the end-to-end fidelity can be improved by deterministically discarding links when their ages exceed some cutoff value. Such deterministic cutoff policies provide strict control of the fidelity of all links, but they come at the expense of having to track link ages. In this work, we introduce a probabilistic cutoff policy that does not require tracking link ages, at the cost of abandoning strict control of the fidelity. We benchmark this new probabilistic cutoff policy against a deterministic cutoff policy. We compare the policies in terms of the end-to-end rate and fidelity, and the secret-key rate. We find that even though the probabilistic cutoff policy keeps track of less state, it can provide secret-key rates of the same order of magnitude as the deterministic cutoff policy in chains with few nodes or high elementary link generation probabilities. Moreover, we identify a scenario in which the probabilistic cutoff policy can deliver end-to-end links that are required to have some minimum threshold fidelity at a higher rate than the deterministic cutoff policy.

Figures (12)

• Figure 1: Schematic representation of a quantum repeater chain. A quantum repeater chain generates an entangled link between two end nodes by using heralded entanglement generation (HEG) and entanglement swaps. Dashed arrows indicate an HEG attempt and solid arrows indicate swaps. Qubits are depicted as blue circles. Qubits that share an entangled link are filled in and connected by an arc.
• Figure 2: An entanglement swap. The qubits $1_l$ and $1_r$ are measured in the Bell basis. The measurement outcome must be communicated to nodes $0$ and $2$. By local operations at nodes $0$ and $2$ the entangled link $\rho_{02}$ is created.
• Figure 3: Deterministic cutoffs provide higher fidelities than probabilistic cutoffs. Rates and fidelities of quantum repeater chain with probabilistic or deterministic cutoff policy. Results shown are for parameter values $n_\mathrm{node}=3$, $p_\mathrm{g}=0.1$, $\tau_\mathrm{coh}=20$, $w_0=1$ and $p_\mathrm{s}=1$. Horizontal dashed line indicates comparison of rates above minimum threshold fidelity $F_{\mathrm{min}}$.
• Figure 4: Probabilistic cutoffs provide secret-key rates of the same order of magnitude as deterministic cutoffs over a large range of link generation probabilities when $n_\mathrm{node}\leq 5$. Maximized secret-key rates $\max_{p_\mathrm{c}}\mathrm{SKR}$ and $\max_{t_\mathrm{c}}\mathrm{SKR}$ for probabilistic ($p_\mathrm{c}$) and deterministic ($t_\mathrm{c}$) cutoff policy, respectively, are shown as function of elementary link generation probability $p_\mathrm{g}\in [10^{-3},1]$ for $\tau_\mathrm{coh}=50$ and $p_\mathrm{s}=1$.
• Figure 5: Probabilistic cutoffs provide orders of magnitude higher secret-key rate than trivial cutoff policies, and are competitive with deterministic cutoffs when link generation probability is large. Maximized secret-key rates $\max_{p_\mathrm{c}}\mathrm{SKR}$ and $\max_{t_\mathrm{c}}\mathrm{SKR}$ for probabilistic and deterministic cutoff policy, respectively, are shown as function of $n_\mathrm{node}=3,4,\dots,10$ for chains with $p_\mathrm{g}=0.25$, $\tau_\mathrm{coh}=50$ and $p_\mathrm{s}=1$. The secret-key rates of the trivial policies to never discard a link ($p_\mathrm{c}=0$) or never store a link $(p_\mathrm{c}=1$) are also shown. No secret-key can be generated with $p_\mathrm{c}=0$ for $n_\mathrm{node}>6$. Errorbars show one standard deviation, but almost all are smaller than marker (see \ref{['appendix: SKR']} for details).

Theorems & Definitions (8)

• Proposition A.1
• Remark A.2
• proof : Proof of \ref{['prop: linear sytem for expected werner matrix']}
• Remark A.3
• Lemma A.4
• proof
• Lemma A.5
• proof