Noncooperative Quantum Networks

TL;DR

It is shown that for noncooperative LOCC protocols, the resulting fidelity may decrease as more entanglement is added to a network with non-pure states, a potential obstacle to the optimal use of resources in large quantum networks.

Abstract

Existing protocols for quantum communication networks usually assume an initial allocation of quantum entanglement resources, which are then manipulated through local operations and classical communication (LOCC) to establish high-fidelity entanglement between distant parties. It is generally held that the resulting fidelity would increase monotonically with the entanglement budget. Here, we show that for noncooperative LOCC protocols, the resulting fidelity may decrease as more entanglement is added to a network with non-pure states. This effect results from a quantum analog of selfish routing and constitutes a potential obstacle to the optimal use of resources in large quantum networks.

Figures (5)

• Figure 1: Schematic of a quantum communication network. Entanglement resources enable multiple users at the Alice node to establish independent end-to-end entangled states, along possibly different paths, with corresponding users at the Bob node. Orange links represent the initial entangled pairs, while the black links indicate the resulting Alice–Bob entangled states.
• Figure 2: Noncooperative behavior in an example quantum network. (a) Network with Bell-state (black) and Werner-state (gray) edges in equal proportion, in which 3 and 6 serve as end nodes. (b–d) User pairs’ paths for establishing end-to-end entanglement at the NE (b), at the global optimum (c), and at the NE after removing edge (5,7) (d). The color-labeled tables list the number of user pairs and/or fidelity associated with each path, where the paths are the same for (b) and (c). The parameters are $N=8$, $\nu_{\text{AB}}=3$, $M=1000$, and $F_0=0.95$.
• Figure 3: Equilibria and optima for all Alice-Bob pairs in the network of Fig. \ref{['8-node example']}(a). (a) Largest fidelity improvement from single-edge removals for both NE and WE results. (b) Average end-to-end fidelity at the NE before and after edge removal, global optimum, btN optimum, and fair optimum. (c) Histogram of the fidelity over all user pairs at the global and btN optima, scaled as $(F-F_{\text{NE}})/(F_{\text{Opt}}-F_{\text{NE}})$ for each Alice-Bob pair, where the indices indicate the NE value and global optimum average. The parameters are the same as in Fig. \ref{['8-node example']}.
• Figure 4: End-to-end fidelity improvement in ER networks from optimal one-edge removals, averaged over all Alice-Bob pairs. (a) Effect as a function of the initial Werner-state fidelity for different network sizes. (b) Cumulative fidelity improvement over all Alice-Bob pairs ordered by effect size for the 16-, 24-, and 32-node networks in (a) with $F_0=0.675$. (c) Effect as a function of the percentage of edges initially prepared in Bell states. (d) Effect in networks with only Werner states, where the initial fidelity $F_0$ of each edge is sampled from the discrete set $\{0.55,0.575, \ldots, 0.975\}$ with probabilities proportional to $e^{-[(F_0-\bar{F_0})/\sigma]^2/2}$ (a discrete truncated Gaussian). In (c-d), the network size is $N=16$. Each data point represents an average over 100 network realizations, where the bars indicate standard errors.
• Figure 5: End-to-end fidelity improvement under optimal edge removals, for varying numbers of removals and Alice-Bob pairs. (a) Improvement from optimal 2- or 3-edge removals compared with 1-edge removals [as in Fig. \ref{['fig:summary']}(a)]. (b) Improvement from single-edge removals for networks with multiple Alice-Bob end nodes.