Entanglement buffering with two quantum memories

TL;DR

The paper tackles buffering entanglement in two-node quantum networks subject to time-dependent decoherence. It develops a CTMC framework for a 1G1B memory architecture and derives closed-form expressions for availability $A$ and average consumed fidelity $\overline F$, revealing a trade-off between keeping links available and maintaining high fidelity. It then analyzes a linear jump-function purification regime to obtain explicit $\overline F_{linear}$ and shows how pumping can improve fidelity under realistic memory noise, including a noise threshold. Finally, it provides bounds for bilocal Clifford purification protocols, supported by numerical simulations that confirm the analytic predictions and demonstrate the practical utility of the framework for evaluating and guiding entanglement buffering strategies in near-term quantum networks.

Abstract

Quantum networks crucially rely on the availability of high-quality entangled 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.

Figures (12)

• Figure 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 $\lambda$ 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 $q$ (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 $p$ (otherwise, it destroys the link in G). The buffered link is consumed at a fixed rate $\mu$. The quality of the entanglement stored in G decays exponentially with rate $\Gamma$. Formal definitions of the problem parameters can be found in Section \ref{['sec:1G1B_system']}.
• Figure 2: The transitions of the 1G1B system.
• Figure 3: 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 $s(t)=i$, the random variables $\{ T_j(t): j=0,1,...,i-1 \}$ are the times spent in each state of the CTMC immediately leading up to the current state, and $X_i(t)$ is the time so far spent in state $i$.
• Figure 4: An example timeline of the 1G1B process. Black dashes are link generation and removal. Shorter purple dashes are pumping rounds. If there is a link present at time $t$, the random variable $C(t)$ is the total time spent so far in $\neg \emptyset$ (link present). Pumping rounds occur within the time $C(t)$ as a Poisson process with rate $\lambda q p$. This may be used to characterise the distribution of $\vec{T}(t)$ in the limit $t\rightarrow \infty$, which is needed to prove Theorem \ref{['thm:average_fidelity_formula']}.
• Figure 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 $r_{ BA}+r_{ BC}$.

Theorems & Definitions (61)

• Definition 3.1: 1G1B system
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Definition 3.5
• Definition 3.6
• Definition 4.1: Availability
• Proposition 1
• Definition 4.2: Average consumed fidelity
• Theorem 1