Improved Routing of Multiparty Entanglement over Quantum Networks

TL;DR

This paper proposes two graph state-based routing protocols for sharing GHZ states, achieving larger sizes than the existing works, for given network topologies and shows that if such a tree is balanced, it achieves a larger size than unbalanced trees.

Abstract

Effective routing of entanglements over a quantum network is a fundamental problem in quantum communication. Due to the fragility of quantum states, it is difficult to route entanglements at long distances. Graph states can be utilized for this purpose, reducing the need for long-distance entanglement routing by leveraging local operations. In this paper, we propose two graph state-based routing protocols for sharing GHZ states, achieving larger sizes than the existing works, for given network topologies. For this improvement, we consider tree structures connecting the users participating in the final GHZ states, as opposed to the linear configurations used in the earlier ones. For arbitrary network topologies, we show that if such a tree is balanced, it achieves a larger size than unbalanced trees. In particular, for grid networks, we show special constructions of the above-mentioned tree that achieve optimal results. Moreover, if the user nodes among whom the entanglement is to be routed are pre-specified, we propose a strategy to accomplish the required routing.

Figures (9)

• Figure 1: A graphical representation of a network. In general, a network can be thought of as a combination of different layers. The top layer consists of at least one main server connecting the whole network. The next layer has a few regional servers $R_i$ providing service to a big region that may be covering one or multiple states. The following layer may have some local servers $L_j$ providing service within a city. Finally, the last layer consists of users. A server may be connected with another nearby server. This graphical representation has a hidden tree structure with the main server as root. Therefore, this tree structure can be used to route entanglement.
• Figure 2: Reconstructing GHZ state using generalized $X$ protocol proposed by Mannalath and Pathak PhysRevA.108.062614. Only the last two columns from a $11\times 11$ grid are shown here. The blocks $B_i^{(3)}$ and $B_j^{(0)}$ contains the remaining $9$ columns constructed as in Ref. PhysRevA.108.062614. Dotted lines have been used to denote the edges between vertices from two different blocks. There are a total of $45$ vertices from the final GHZ state inside the blocks. The last two columns cannot have more than $9$ vertices in the final GHZ state as shown in this figure. This creates a $54$-party GHZ state. However, the conjecture in Ref. PhysRevA.108.062614 claims it as $55$-party GHZ.
• Figure 3: Application of $X$ measurement on vertex $u$. (a) A part of graph $G$, that will change due to $X$ measurement. The remaining part (omitted with the dashed line) of the graph will remain unchanged. (b) Local complement on $v_2$. Red edges (dashed lines) have been deleted and green edges have been added. (c) Local complement on $u$. (d) $Z$ measurement on $u$. Since $v_2$ has only one neighbor $v_1$, this is the final state. (e) Effect of $X$ measurement on vertex $u$ (degree $3$) of an arbitrary graph. $G_i(i=1,2,3)$ in squares denotes the rest of the graph. The neighbors of $v_3$ in $G_2$ are the same as the neighbors of $v_1$ in $G_2$.
• Figure 4: All memories from the multi-memory leaf cannot be part of the final GHZ state. Here we consider a simple tree with one $h$-type node and one single root-to-leaf path. Note that, After $X$ measurement on $h_1$, only one memory, $c$, from the leaf vertex becomes part of the GHZ state (star graph). After applying X measurement on $c$, the remaining memories are in the final GHZ state.
• Figure 5: Size of GHZ states that can be extracted from a network starting from different initial trees connecting the participants in the GHZ state. $m$ is the number of children of the $g$-type nodes. The linear graph corresponds to the result given in Ref. PhysRevA.108.062614.

Theorems & Definitions (16)

• Definition 1: Vertex Deletion
• Definition 2: Local Complementation
• Definition 3: Vertex-minor
• Definition 4: Graph state
• Theorem 1: $X$ measurement
• proof
• Corollary 1
• proof
• Corollary 2
• Definition 5: Repeater Tree