Entanglement Distribution in the Quantum Internet: Knowing when to Stop!

TL;DR

This work addresses the challenge of determining when to stop entanglement distribution in a noisy Quantum Internet. It formulates the process as a Markov decision process with two actions, continue ($C$) and stop ($Q$), over a finite time horizon $N$, and derives optimal decision rules leveraging reward structures and one-step look ahead. The authors establish the MD P formulation, provide properties of the optimal policy via reward majorants/minorants, and show that a simple OLA rule is optimal for several reward forms while remaining near-optimal in general. The framework offers a flexible, design-oriented tool for engineers to trade off average cluster size and distribution time by tailoring reward functions to specific network goals, thereby enabling efficient, robust entanglement distribution under realistic decoherence constraints.

Abstract

Entanglement distribution is a key functionality of the Quantum Internet. However, quantum entanglement is very fragile, easily degraded by decoherence, which strictly constraints the time horizon within the distribution has to be completed. This, coupled with the quantum noise irremediably impinging on the channels utilized for entanglement distribution, may imply the need to attempt the distribution process multiple times before the targeted network nodes successfully share the desired entangled state. And there is no guarantee that this is accomplished within the time horizon dictated by the coherence times. As a consequence, in noisy scenarios requiring multiple distribution attempts, it may be convenient to stop the distribution process early. In this paper, we take steps in the direction of knowing when to stop the entanglement distribution by developing a theoretical framework, able to capture the quantum noise effects. Specifically, we first prove that the entanglement distribution process can be modeled as a Markov decision process. Then, we prove that the optimal decision policy exhibits attractive features, which we exploit to reduce the computational complexity. The developed framework provides quantum network designers with flexible tools to optimally engineer the design parameters of the entanglement distribution process.

Figures (11)

• Figure 1: Pictorial representation of the considered quantum network architecture. The quantum network comprises several clusters of nodes. Within each cluster, client nodes are connected to a super-node. The super-nodes are specialized nodes equipped with dedicated hardware able to generate the entanglement resources. Client nodes obtain access to the multipartite entangled states, generated at the super-nodes, through the entanglement distribution process.
• Figure 2: Pictorial representation of the system model. The legend of the figure available in Fig. \ref{['fig:1']}. Subfigure (a) represents a zoomed-view of the cluster of nodes in the lower-right part of Fig. \ref{['fig:1']}. We consider a scenario where a super-node is connected through quantum channels to a set of quantum nodes, referred to as clients. The super-node is in charge of generating and distributing EPR pairs to the clients. The aim of the process is to distribute the multipartite entangled state through teleportation. For this, the super-node performs multiple attempts of ebit distribution, as represented in Fig. \ref{['fig:01b']}. Fig. \ref{['fig:01b']} constitutes the ideal scenario with the successful distribution of all the EPR pairs between the super-node and the clients, but clearly multiple distribution attempts might be required depending on the noise level affecting the quantum channels. Finally, the super-node can exploit the distributed EPR pairs for teleporting the multipartite entangled state. As represented in Fig. \ref{['fig:01c']}, after the teleportation, the client nodes share a multipartite entangled state.
• Figure 3: Pictorial representation of the model for the entanglement distribution process. The overall goal is to decide when to stop the entanglement distribution.
• Figure 4: Representation of the two functioning regimes for a network with $S=3$ clients: (a): regime of the action $C$. (b): regime of the action $Q$.
• Figure 5: Expected total reward $v_{\pi}$ as a function of the ebit propagation probability $p$ for $S = 100$, $N=100$ and $g(s_n) = \frac{s}{n}$. Logarithmic scale for axis $y$.

Theorems & Definitions (27)

• Remark
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• Definition 1: Time horizon
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• Definition 2: Action Set
• Definition 3: State Space
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• Definition 4: Allowed Action Set