Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings

TL;DR

This paper establishes a novel connection between high-dimensional multipartite quantum states and graph theory, showing how experimental setups correspond to graphs and superpositions relate to perfect matchings, with implications for quantum state creation and simulation.

Contribution

It introduces a framework linking quantum experiments to graph theory, enabling analysis of entangled states and properties of graphs through quantum setups.

Findings

Quantum states correspond to perfect matchings in graphs

Calculating the quantum state is #P-complete

Graph theoretical methods can analyze quantum experiment capabilities

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).

Tables

Table 1. Table 1: The analogies between Quantum Experiments involving multiple crystals and Graph Theory.

Quantum Experiment	Graph Theory
Optical Setup with Crystals	undirected Graph $G(V,E)$
Crystals	Edges $E$
Optical Paths	Vertices $V$
n-fold coincidence	perfect matching
#(terms in quantum state)	#(perfect matchings)
maximal dimension of photon	degree of vertex
n-photon d-dimensional GHZ state	n-vertex graph with d disjoint perfect matchings

Equations

$$\displaystyle|\psi\rangle=\frac{1}{2}\big{(}$$

$$\displaystyle+|2,2,2,2,2,2\rangle+|1,2,1,2,0,0\rangle\big{)}.$$

$$|{\psi }_{a,b}⟩$$

$$\displaystyle+\frac{p^{2}}{2}\left(\hat{a}_{a,\ell}^{\dagger}\hat{a}_{b,\ell}^{\dagger}-\hat{a}_{a,\ell}\hat{a}_{b,\ell}\right)^{2}+...\Big{)}|0\rangle.$$

$$|{\psi }_{\text{network}}⟩=\underset{e(i,j)\in E}{⨂}|{\psi }_{i,j}⟩$$

$$\displaystyle|\psi\rangle=\frac{1}{\sqrt{3}}\big{(}$$

Full text

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 wang1991induced ; zou1991induced . This phenomenon has found a manifold of applications such as in spectroscopy kalashnikov2016infrared , in quantum imaging lemos2014quantum , for the investigation of complementarity heuer2015induced , in superconducting cavities lahteenmaki2016coherence and for investigating quantum correlations hochrainer2017quantifying . By exploiting these ideas, the creation of a large number of high-dimensional multipartite entangled states has been proposed recently krenn2017entanglement (inspired by computer-designed quantum experiments krenn2016automated ).

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 valiant1979complexity . Futhermore, the link can be used as a natural implementation for the experimental investigation of quantum random networks perseguers2009quantum .

Experiments and Graph – The optical setup for creating a 3-dimensional generalization of a 4-photon Greenberger-Horne-Zeilinger state greenberger1989going ; lawrence2014rotational is shown in Fig. 1A krenn2017entanglement . The experiment consists of three layers of two down-conversion crystals each. Each crystal can create a pair of photons in the state $|0,0\rangle$, where the mode number could correspond to the orbital angular momentum (OAM) of photons allen1992orbital ; yao2011orbital ; krenn2017orbital or some other (high-dimensional) degree-of-freedom. A laser pumps all of the six crystals coherently, such that two pairs of photons are created in parallel. Four-fold coincidence (i.e. four photons are detected simultaneously in detector $a$, $b$, $c$ and $d$) can only happen if the two photon pairs are created in crystals I and II, or in crystals III and IV or in crystals V and VI. In every other case, there is at least one path without a photon, which is neglected by post-selection. Between each layer, the modes are shifted by $+1$. This example leads to the final state $|\psi\rangle=1/\sqrt{3}\left(|0,0,0,0\rangle+|1,1,1,1\rangle+|2,2,2,2\rangle\right)$.

The corresponding graph is shown in Fig. 1B. Every optical path $a$, $b$, $c$, $d$ in the experiment corresponds to a vertex in the graph, every crystal forms an edge between the vertices. A four-fold coincidence count happens if a subset of the edges contain each of the four vertices exactly once. Such a subset is called perfect matching of the graph. In the above example, there are three perfect matchings (two green edges, two blue edges and two red edges), thus there are three terms in the quantum state. We can therefore think of our quantum state as a coherent superposition of the perfect matchings in the corresponding graph. The correspondence between quantum optical setups and graph theoretical concepts are listed in Table 1.

Now, what will happen when we add more crystals in each layer? As an example, in Fig. 1D, three crystals in each layer produce 6 photons, there are three layers which make the photons 3-dimensionally entangled. Surprisingly however, in contrast to the natural generalisation of the 4-photon case in Fig. 1A-C (and in contrast to what some of us wrote in krenn2017entanglement ), the resulting state is not a high-dimensional GHZ state. In contrast to the previous case, there are four perfect matchings, thus the resulting quantum state has four terms (Fig. 1F). One perfect matching comes from each of the layers (which are the terms expected for the GHZ state), and one additional arises due to a combination of one crystal from each layer (which we call Maverick-term). If the mode shifter between the layers is $+1$ as before, the Maverick term has $|1_{a},1_{c}\rangle$ from the blue layer, $|2_{b},2_{d}\rangle$ from the green layer and $|0_{e},0_{f}\rangle$ from the red layer. This leads to the final state

[TABLE]

A GHZ state can only appear when all perfect matchings are disjoint, meaning that every edge appears only in one perfect matching. Otherwise, additional terms are present in the quantum state.

When the number of layers of crystals is increased to four (with 3 cystals per layer) and modes are shifted by +1 as before (and no phase-shifters are used), there are 8 terms in the resulting quantum state: 4 GHZ-like terms and 4 additional Maverick terms. For five layers, the resulting 6-photon quantum state consists of 15 terms (5 GHZ-like terms and 10 additional Maverick terms), entangled in 5 dimensions (see Appendix). In general, $n$ crystals in one layer produce 2$n$ photons. One can design setups with $d=(2n-1)$ layers, which correspond to a complete graph $K_{2n}$ (in a complete graph, every vertex is connected with every other one exactly once). It produces a state with $\frac{(2n)!}{n!2^{n}}$ terms, $(2n-1)$ of them are GHZ-like (see Appendix). By changing the mode shifters and phase shifters between the layers, a vast amount of different quantum states can be created.

Now one could ask what types of GHZ states are possible in general using the experimental scheme above. We show a proof based on Graph Theory which answers that question. For that, we first translate the quantum physics question Which $d$-dimensional GHZ states can be created? into the graph theory question Which undirected graphs exist with $d$ perfect matchings which all are disjoint?. The proof strategy is to construct a graph with a maximum number of disjoint perfect matchings, starting from $n$ vertices 267013 . The concept and the proof are described in Fig. 2. We find that one can create arbitrarily large 2-dimensional GHZ states, and a 3-dimensional 4-photon GHZ state. In an analogous way, different questions in such quantum experiments can be translated and answered with Graph Theory.

In order to build 3-dimensional GHZ-type experiments with 6 photons (without extra terms), one can use two copies of the 3-dimensional 4-photon GHZ state (presented in Fig. 1A), and combine them with a 3-dimensional Bell-state measurement bennett1993teleporting ; sych2009complete . In the graph this is represented by two graphs that are merged (see Appendix). Many other classes of entangled states, such as two-dimensional W-state zeilinger1992higher ; bourennane2004experimental or asymmetrically entangled Schmidt-Rank Vector (SRV) huber2013structure ; malik2016multi can be created by exploiting multigraphs (graphs with more then one edge between two vertices), as shown in the Appendix.

An important result is that calculating the final quantum state cannot be done efficiently: Counting the number of perfect matchings in a graph (i.e. calculating the number of terms in the resulting quantum state) is in the complexity class #P-complete. In a bipartite graph, it is equivalent to computing the permanent of the graph’s biadjacency matrix valiant1979complexity (see Appendix for such an experimental setup). Furthermore, for general graphs, counting the number of perfect matchings corresponds to calculating the Hafnian (a generalisation of the permanent) of the graph’s adjacency matrix. Even for approximating the Hafnian there is no known deterministic algorithm which runs in polynomial time bjorklund2012counting ; barvinok2017approximating . An example is given in Fig. 3A for a random graph, its corresponding perfect matching and Hafnian in Fig. 3B-C, and the corresponding quantum setup in Fig. 3D.

While the information about the number of terms is encoded in every $n$-photon quantum state emerging from the setup, the question is how one can obtain this information (or approximate it) efficiently. Measurements in the computation basis are not sufficient, otherwise it could be calculated classically as well. One direction would be to investigate frustrated generation of multiple qubits herzog1994frustrated (for instance, by using phase shifters instead of mode shifters between each crystal), or by analysing multi-photon high-dimensional entanglement detections huber2010detection . A detailed investigation of the link between the outcome of such experiments and complexity classes would be valuable, but is outside the scope of this article.

As it is possible to generate experimental setups for arbitrary undirected graphs, the presented scheme is also a natural and inexpensive implementation of quantum (random) networks (see Fig. 3). This could be used to experimentally investigate entanglement percolation acin2006entanglement ; cuquet2009entanglement ; perseguers2010multipartite and critical exponents which lead to phase transitions in quantum random networks perseguers2009quantum . As an example, it has been shown that for large quantum networks with $N$ nodes, every quantum subgraph can be extracted with local operations and classical communication (LOCC) if the edges are connected with a probability $p\geq N^{-2}$ perseguers2009quantum . In close analogy to the experimental schemes here, $N$ is the number of output paths of photons, and $p$ corresponds to the probability for a down-conversion event in a single crystal. The quantum state for the edge between vertices $a$ and $b$, with mode number $\ell$ can be written as

[TABLE]

where $p$ is the SPDC probability. The complete quantum (random) network is a combination of all crystals being pumped coherently, which is a tensor product over all existing edges in the form of

[TABLE]

where $i$ and $j$ are the vertices which are connected by the edge $e\in E$.

Finally, to strengthen the link between quantum experiments and graph theory, we show that theorems from Graph theory can be translated and reinterpreted in the realm of quantum experiments. In Fig. 4A and B, we show Hall’s marriage theorem, which gives a necessary and sufficient condition in a bipartite graph for the existence of at least one perfect matching hall1935representatives . A generalisation to general graphs, Tutte’s theorem tutte1947factorization ; akiyama2011factors , is shown in the Appendix. Both Graph theory theorems can be understood in the language of quantum experiments.

To conclude, we have shown a strong link between quantum experiments and Graph Theory. It allows to systematically analyse the emerging quantum states with methods from graph theory. The new link immediatly opens up many new directions for future research. For example, the analysation of the number of maximal matchings and matchings in a graph (called Hosoya index and often used in chemistry hosoya2002topological ; jerrum1987two ) in the contect of quantum experiment.

A detailed investigation of links between these experiments and computation complexity classes, in particular the relation to computation complexity with linear optics would be interesting aaronson2011computational ; aaronson2005quantum ; hamilton2016gaussian .

Furthermore it would be interesting how the merging of graphs can be generalized with non-destructive measurements wang2015quantum , whether it leads to larger classes of accessible states and how that can be described in the Graph theoretical framework.

The generalisation to other graph theoretical methods would be interesting, such as weighted graphs (which could correspond to variable down-conversion rates via modulating the laser power), hypergraphs (which would correspond to creation of tuples of photons, for instance via cascaded down-conversion hubel2010direct ; hamel2014direct ) or 2-Factoriations (or general $n$-Factorizations, which would lead to $n$ photons in one single arm).

Experimental implementations could not only create a vast array of well-defined quantum states, but could also investigate striking properties of quantum random networks in the laboratory.

Finally, we suggest that recent developments of integrated optics implementations of quantum experiments, where the photons are generated on a photonic chip silverstone2014chip ; jin2014chip ; krapick2016chip , could be particularly useful to realize setups of the type proposed here.

Acknowledgements

The authors thank Manuel Erhard, Armin Hochrainer and Johannes Handsteiner for useful discussions and valuable comments on the manuscript. X.G. thanks Lijun Chen for support. This work was supported by the Austrian Academy of Sciences (ÖAW), by the European Research Council (SIQS Grant No. 600645 EU-FP7-ICT) and the Austrian Science Fund (FWF) with SFB F40 (FOQUS). XG acknowledges support from the Major Program of National Natural Science Foundation of China (No. 11690030, 11690032), the National Natural Science Foundation of China (No.61272418).

I Appendix

I.1 Appendix I. Examples of Path Identity

Twenty-five years ago, Wang, Zou and Mandel (originally suggested by Zhe-Yu Ou) have demonstrated a remarkable idea: They coherently overlapped one of the output modes from each crystal ($|b\rangle=|d\rangle$ in Fig. 5A), such that the which-crystal information for the photon in $d$ never exists in the first place. That leads to $|\psi\rangle=1/\sqrt{2}\left(|a\rangle+|c\rangle\right)|d\rangle$, where one photon is in $d$ and the second photon is in a coherent superposition of being in $a$ and in $c$. When both output modes from the two crystals are overlapped such that the paths of the photons are identical (Fig. 5B). By adding phases between the two crystals, one obtains $|\psi\rangle=(|a,b\rangle+\exp(i\phi)|a,b\rangle)=(1+\exp(i\phi))|a,b\rangle$, which means that by changing the phase $\phi$, one can enhance or suppress the creation of photons – a phenomenon denoted as frustrated generation of photon pairs [26]. If instead of phase shifters one would add mode shifters between the crystals (for instance, the crystal produces two horizontal polarized photons, and the mode-shifter changes horizontal to vertical), one creates an entangled two-photon state $|\psi\rangle=1/\sqrt{2}(|H_{a},H_{b}\rangle+|V_{a},V_{b}\rangle)$ [45].

I.2 Appendix II. Multi-Graphs for different structured entangled states

The examples in Fig. 1 and Fig. 2 in the main text have aimed at producing GHZ states. One can use multi-graphs (which have two edges between two vertices) to create a vast array of different entangled states, such as the W-state or high-dimensional asymmetrically entangled states – as shown in Fig. 6.

I.3 Appendix III. High-dimensional Multipartite Entanglement Swapping for State merging

The merging of two 4-photon 3-dimensional entangled states with a 3-dimensional Bell-state measurement (as shown in Fig. 7A and B) leads to a 6-photon 3-dimensional entangled state:

[TABLE]

which is a 6-photon, 3-dimensional GHZ state. This can be generalized to multi-photon 3-dimensional GHZ states with more copies chained together. The operation is a generalisation of entanglement swapping [46, 47] to multi-photonic systems [48] with more than two dimensions.

I.4 Appendix VI. Entanglement of 6 photons in 5 dimensions - complete graph $K_{6}$

An experiment with five layers and three crystals in each layer is shown in Fig. 8A. It corresponds to the complete graph $K_{6}$, which has one edge between each of its six vertices Fig. 8B. It has 15 perfect matchings, which are shown in Fig. 8C. For complete graphs $K_{2n}$ with $2n$ vertices, the number of perfect matchings is $\#(PM)=\frac{(2n)!}{n!2^{n}}$ [49].

A 1-Factorization of the graph G(V,E) is a partitioning of the graph’s edges into disjoined subgraphs (called 1-factors), such as the first line of Fig. 8C. Each of these 1-Factor of the 1-Factorization can be controlled independently in the quantum experiment, the additional Maverick terms arise then automatically. Interestingly, in contrast to factorization of natural numbers, 1-Factorizations of graphs are not unique [50]. This allows for a lot of extra variability in the generation of entangled states.

I.5 Appendix V. Bipartite Graphs

Counting the number of perfect matchings in a bipartite graph is in the complexity class #P-complete. In Fig. 9A, an experimental setup is shown which corresponds to the bipartite graph in Fig. 9B. The perfect matchings for this case can be found in Fig. 9C. They correspond to the number of terms in the resulting quantum state. The mode number of the different terms can be set for each crystal individually, thus one can simply see which states are possible.

I.6 Appendix VI. Perfect matchings in general Graphs: Tutte’s theorem

A different important result in Graph theory about perfect matchings is Tutte’s theorem. It gives a necessary and sufficient condition for general graphs, when one can find perfect matchings (but not talking about how many). It is a generalisation of Hall’s marriage theorem, which answers the same question for bipartite graphs. In Fig. 10A, the theorem is explained based on an example. That theorem can be understood with quantum experiments, as shown in Fig. 10B.

I.7 Appendix VII. From Graph to Experimental Setup

We show one simple example how to construct the experimental setup and the wiring of the paths for one specific graph. We use the random graph shown in Fig3 in the main text.