Figure 2
Application of the bridge between quantum experiments and graph theory: As a concrete example, we ask which -dimensional -photon GHZ states can be created experimentally with this method. The idea of the proof is to construct a graph starting with vertices without edges. We try to maximize the number of disjoint perfect matchings (PMs) by adding appropriate edges to the graph [24]. In disjoint PMs, every edge appears in only one perfect matching. The example in the figure is for , but the proof works for any arbitrary even . Step I: In , we add the first PM to a set of eight vertices (green). Step II: In , we add more edges to construct a second PM (red). Whenever the new PM, together with the first (green) PM, creates more than one cycle (here: edges 1-6,6-7,7-8,8-1; and 2-3,3-4,4-5,5-2), we immediately find an additional Maverick PM (indicated with white boundary, edges 1-6,2-3,4-5,7-8). Thus the graph cannot represent a GHZ state (as a GHZ state has only disjoint perfect matchings). The only choice for the second PM is to create together with the first PM one cycle that visits every vertex—a Hamilton cycle, shown in . Hamilton cycles consist of 2 PMs, and therefore correspond to 2-dimensional GHZ states. It can be arbitrarily large, and thus there can be arbitrarily large -photon 2-dimensional GHZ states. Step III: Starting with the Hamilton cycle, we try to add a third PM with blue edges. In , we observe that if the new edge splits the graph into an even number of vertices (upper part: vertices 7,8; lower part: vertices 2,3,4,5), we always find a new Maverick PM. It consists of the new edge (here: 1–6) and edges from the Hamilton cycle (here edges 2-3,4-5,7-8). We learn—as we require only disjoint perfect matchings—no edge of a new PM should split the graph into even numbers of vertices (otherwise Maverick PMs appear). Finally, in we try to add edges that split the graph into an odd number of vertices. We observe that in every additional PM there are at least two neighboring edges that intersect (neighboring edges start from consecutive vertices; here—shown in blue—they start at vertex 1 and vertex 2). This pair always forms a new Maverick PM with additional edges from the Hamilton cycle (here: 1-5,2-6,3-4,7-8). There is one exception for the case of : There can be a 3rd disjoint PM, because a Maverick PM needs at least 3 edges (2 blue ones and one from the Hamilton cycle). Therefore, a 4-photon 3-dimensional GHZ state can be created, while for , GHZ states can only be created with .
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