Quantum Experiments and Graphs: Multiparty States as Coherent Superpositions of Perfect Matchings

Abstract

We show a surprising link between experimental setups to realize high-dimensional multipartite quantum states and Graph Theory. In these setups, the paths of photons are identified such that the photon-source information is never created. We find that each of these setups corresponds to an undirected graph, and every undirected graph corresponds to an experimental setup. Every term in the emerging quantum superposition corresponds to a perfect matching in the graph. Calculating the final quantum state is in the complexity class #P-complete, thus cannot be done efficiently. To strengthen the link further, theorems from Graph Theory -such as Hall's marriage problem -are rephrased in the language of pair creation in quantum experiments. We show explicitly how this link allows to answer questions about quantum experiments (such as which classes of entangled states can be created) with graph theoretical methods, and potentially simulate properties of Graphs and Networks with quantum experiments (such as critical exponents and phase transitions).When a pair of photons is created, and one cannot -even in principle -determine what its origin is, the resulting quantum state is a coherent superposition of all possibilities [1,2]. This phenomenon has found a manifold of applications such as in spectroscopy [3], in quantum imaging [4], for the investigation of complementarity [5], in superconducting cavities [6] and for investigating quantum correlations [7]. By exploiting these ideas, the creation of a large number of highdimensional multipartite entangled states has been proposed recently [8] (inspired by computer-designed quantum experiments [9]).Here we show that Graph Theory is a very good abstract descriptive tool for such quantum experimental configuration: Every experiment corresponds to an undirected Graph, and every undirected Graph is associated with an experiment. On the one hand, we explicitly show how to translate questions from quantum experiments and answer them with graph theoretical methods. On the other hand, we rephrase theorems in Graph Theory and explain them in terms of quantum experiments.An important example for this link is the number of terms in the resulting quantum state for a given quantum experiment. It is the number of perfect matchings that exists in the corresponding graph -a problem that lies in the complexity class #P-complete [10]. Futhermore, the link can be used as a natural implementation for the experimental investigation of quantum random networks [11].Experiments and Graph -The optical setup for creating a 3-dimensional generalization of a 4-photon Greenberger-Horne-Zeilinger state [12,13] is shown in Fig. 1A [8]. The experiment consists of three layers of two down-conversion crystals each. Each crystal can create a pair of photons in the state |0, 0 , where the mode number could correspond to the orbital angular momentum (OAM) of photons [14][15][16] or some other (high-dimensional) degree-of-freedom. A laser pumps all of the six crystals coherently, such that two pairs of photons...

Quantum experiments and graphs. III. High-dimensional and multiparticle entanglement

“…Using a recently uncovered bridge between quantum experiments with probabilistic photon pair sources and graph theory [13], we answer this question for many classes of entangled states. The correspondence is listed in Table I.…”

“…In this paper, we briefly summarize the main results from [13] and explain the connection between quantum experiments and graphs. Then we show graphs and experimental setups for creating 2-dimensional and 3-dimensional GHZ states as well as 4-particle W state.…”

Quantum experiments and hypergraphs: Multiphoton sources for quantum interference, quantum computation, and quantum entanglement

“…Interestingly, this indicates that the dimension of GHZ states grows when more crystals are added in the case of 3-photon sources. This is in stark contrast to the case of 2photon sources where the maximum dimension d = 3 [15,60,61]. If we restrict ourselves to two n-photon sources firing simultaneously, then the maximal possible dimension for a GHZ state grows as n = 2 − 7; d = 3, 10, 35, 126, 462, 1716, 6435, which is potentially connected to the integer sequence in OEIS A001700 [62] (number of ways to put n + 1 indistinguishable balls into n + 1 distinguishable boxes).…”

“…Those photon pairs are then interpreted as two vertices connected by an edge. This simple idea has been exploited in [15][16][17] to understand better the generalization of quantum states, quantum information protocols and for gaining new insights towards quantum computation.…”

Progress on Integrated Quantum Photonic Sources with Silicon

“…These authors used four paths to prepare two entangled photon‐pair sources and subsequently arbitrarily regulated these sources to construct various four‐photon graph states. By further reducing the loss and increasing the rates, we can display more complex graph state construction with more photon numbers directly on the silicon chip …”