Quantum Capacities for Entanglement Networks

Abstract: We discuss quantum capacities for two types of entanglement networks: $\mathcal{Q}$ for the quantum repeater network with free classical communication, and $\mathcal{R}$ for the tensor network as the rank of the linear operation represented by the tensor network. We find that $\mathcal{Q}$ always equals $\mathcal{R}$ in the regularized case for the samenetwork graph. However, the relationships between the corresponding one-shot capacities $\mathcal{Q}_1$ and $\mathcal{R}_1$ are more complicated, and the min-cut upper bound is in general not achievable. We show that the tensor network can be viewed as a stochastic protocol with the quantum repeater network, such that $\mathcal{R}_1$ is a natural upper bound of $\mathcal{Q}_1$. We analyze the possible gap between $\mathcal{R}_1$ and $\mathcal{Q}_1$ for certain networks, and compare them with the one-shot classical capacity of the corresponding classical network.

• The paper establishes that, in regularized cases, quantum capacities and tensor network capacities are equivalent for identical network graphs.
• It reveals discrepancies in one-shot scenarios where the min-cut upper bound is seldom achieved due to complex entanglement distribution challenges.
• The authors introduce methods such as vertex splitting and teleportation to transform cyclic networks into acyclic ones, advancing quantum network coding.

Quantum Capacities for Entanglement Networks

Introduction

This paper examines the quantum capacities within two specific forms of entanglement networks: the quantum repeater network with free classical communication and the tensor network, wherein the linear function represented by the network's rank signifies its capacity. The authors establish that, in the regularized scenario, quantum capacities $Q$ and tensor network capacities $R$ are equivalent for identical network graphs. However, complications arise in one-shot scenarios, rendering the min-cut upper bound generally unattainable. In essence, the tensor network emerges as a stochastic protocol within the quantum repeater network, offering $R$ as a natural upper bound for $Q$.

Quantum Repeater Network and Tensor Network Capacities

The quantum repeater network, framed as $(G, d, S, T)$—where $G$ is an undirected graph, $d$ a dimensional function, and $S$, $T$ are source and sink nodes, respectively—operates on maximally entangled states distributed across network edges. The one-shot capacity $Q(N)$ is the maximum dimension of an entangled state generated between $S$ and $T$ using only LOCC operations. Its regularized capacity is the asymptotic lim-sup of $Q(N^n)^{1/n}$. For the tensor network, capacities are determined by the maximal rank of linear maps attained through all possible tensor assignments, with an analogous definition for regularized capacity $R(N)$.

Min-Cut Bound

A prominent result is that quantum and tensor network capacities, $Q(N)$ and $R(N)$, are generally restricted by the graph's min-cut bound $MC(N)$. Although this min-cut provides an upper bound in classical networks, achieving it in quantum networks remains complex. For directed acyclic graphs, classical capacities often achieve the min-cut bound, proving fundamental in network coding paradigms.

In the quantum field, while certain directed networks can simulate classical protocols, achieving min-cut equivalence universally with $Q(N)$ and $R(N)$ is more intricate. The paper affirms that regularized capacities $R(N)$ achieve the min-cut bound under certain configurational stipulations such as powers of integer dimensions. However, straightforward one-shot scenarios often show disparities between $R(N)$ and min-cuts.

One-shot Capacities and Gaps

The authors illuminate examples where one-shot capacities $Q(N)$ and $R(N)$ fall short of the min-cut bound, e.g., specific tensor network configurations where $MC(N)=15$ but $R(N)=14$. This exemplifies difficulties in translating theoretical bounds into practical quantum capacities, usually because of complexities in aligning $S$-$T$ maximally entangled states.

Directed Graphs and Cycles

The study advances the understanding of quantum network capacities by exploring transformations in networks with directed cycles. By employing concepts like splitting vertices and teleportation strategies, acyclic graph configurations are achieved. This has implications for practical application, especially in eliminating directed cycle issues, a task crucial for effective quantum network coding.

Achieving Conjectured Capacities

The concluding conjecture asserts that multiplying the network's dimensions by an integer factor allows min-cut achievement. The authors provided proof for specific networks and suggest this as a possible path for future exploration, bridging differences between one-shot capacities and their regularization, thus offering potential methodology for mesh and optimize large-dimensional networks.

Conclusion

This investigation comprehensively frames the complex interrelations between quantum and tensor networks' capacities, min-cut bounds, and classical coding analogs. With potential directions for overcoming observed gaps in one-shot capabilities and achieving conjectured efficiencies, this research underpins future applications in quantum communication networks and tensor algebraic operations. The study prompts deeper inquiry into operational transformations and strategic applications, advancing both theoretical understanding and technological quantum communication infrastructures.