Quantum Experiments and Graphs III: High-Dimensional and Multi-Particle Entanglement

TL;DR

This paper leverages graph theory to design experimental setups for creating complex high-dimensional and multi-particle entangled states, advancing quantum information science.

Contribution

It introduces a method connecting quantum experiments with graph theory to systematically generate various entangled states, including GHZ, W, and Dicke states.

Findings

Experimental setups for diverse entangled states identified

Enhanced understanding of entanglement producibility in photonic systems

Framework applicable to high-dimensional multipartite entanglement

Abstract

Quantum entanglement plays an important role in quantum information processes, such as quantum computation and quantum communication. Experiments in laboratories are unquestionably crucial to increase our understanding of quantum systems and inspire new insights into future applications. However, there are no general recipes for the creation of arbitrary quantum states with many particles entangled in high dimensions. Here, we exploit a recent connection between quantum experiments and graph theory and answer this question for a plethora of classes of entangled states. We find experimental setups for Greenberger-Horne-Zeilinger states, W states, general Dicke states, and asymmetrically high-dimensional multipartite entangled states. This result sheds light on the producibility of arbitrary quantum states using photonic technology with probabilistic pair sources and allows us to understand the underlying technological and fundamental properties of entanglement.

Tables

Table 1. Table 1: The analogies between graph theory and quantum experiments.

Graph Theory	Quantum Experiments
undirected Graph	optical setup with nonlinear crystals
Vertex	optical output path
Edge	nonlinear crystal
colors of the edge	mode numbers
perfect matching	$n$-fold coincidence
#(perfect matchings)	#(terms in quantum state)

Equations

$$|GH{Z}_n^d⟩=\frac{1}{\sqrt{d}}\sum _{i=0}^{d-1}|i{⟩}^{⨂n}$$

$$|\psi {⟩}_{abcd}=\frac{1}{\sqrt{2}}(|0000⟩+|1111⟩)$$

$$|D_n^k⟩=\frac{1}{\sqrt{\left(\genfrac{}{}{0ex}{}{n}{k}\right)}}\hat{S}(|0{⟩}^{⨂(n-k)}|1{⟩}^{⨂k})$$

$$|\psi {⟩}_{abcd}=\frac{1}{2}(|0001⟩+|0010⟩+|0100⟩+|1000⟩).$$

$$A=rank(T{r}_a(|\psi ⟩⟨\psi |))$$

$$B=rank(T{r}_b(|\psi ⟩⟨\psi |))$$

$$C=rank(T{r}_c(|\psi ⟩⟨\psi |))$$

$$|\psi {⟩}_{abc}=\frac{1}{2}(|000⟩+|101⟩+|210⟩+|311⟩).$$

$$1+min(1+(A-B),C)+min(1+(A-C),B-1)\ge A$$

$$|AME(3,d)⟩=\frac{1}{d}\sum _{\begin{array}{c}i=0\\ j=0\end{array}}^{d-1}|i,j,i+j⟩$$

$$|\psi {⟩}_{abc}=\frac{1}{2}(|000⟩+|011⟩+|101⟩+|110⟩).$$

$$|\psi {⟩}_{abct}=\frac{1}{2}$$

$$=\frac{1}{2}$$

$$|\psi {⟩}_{abc}=\frac{1}{3}$$

$$+$$

$$|\psi {⟩}_{abct}=\frac{1}{3}$$

$$+$$

Full text

Entanglement, which exhibits correlations without a classically analog einstein1935can ; bell1964einstein , is a very peculiar property of quantum states. It is of particular importance in understanding the foundations of quantum mechanics, especially for local realism. Nowadays it has been viewed as a prominently useful resource for quantum information applications, such as quantum computation and quantum communication.

The smallest entangled system consists of two particles, which share one bit of information (such as the polarization state of a photon) in a non-local-realistic way. Such a system is a cornerstone of research in quantum entanglement theory.

More particles or high-dimensional degrees of freedom can lead to more complex types of entanglement. A prominent example of multipartite entanglement is the so-called Greenberger-Horne-Zeilinger (GHZ) state greenberger1989going ; greenberger1990bell , which offers a new understanding in the study of our local and realistic worldview. Another famous class of entangled states is the Dicke state dicke1954coherence , with an important special case – the W state.

Increasing the number of involved degrees of freedom in the entanglement significantly increases the number of different possible states and the complexity of studying them. For example, the question about all-versus-nothing violations of high-dimensional GHZ states has only been understood in 2014 ryu2014multisetting ; lawrence2014rotational , and these states have only been experimentally implemented in the very recent past erhard2018experimental . High-dimensional and multipartite entanglement can lead to new, asymmetric types of quantum correlations which are not seen in any qubit system huber2013structure ; goyeneche2016multipartite . Such a type of entanglement was first been investigated in the laboratory in 2016 malik2016multi and allows potentially different types of quantum communication scenarios pivoluska2018layered .

In the spirit of Richard Feynman, who once famously said ”What I cannot create, I do not understand,” here we ask, Which quantum entangled states can be created in the laboratories with current photonic technologies?

Using a recently uncovered bridge between quantum experiments with probabilistic photon pair sources and graph theory krenn2017quantum , we answer this question for many classes of entangled states. The correspondence is listed in Table 1. Our strategy is to translate the question about the construction of a quantum state into a question about the existence of a graph with certain properties. All of our affirmative answers are constructive, meaning that in these cases we show the graph and its corresponding quantum experimental setup.

In this paper, we briefly summarize the main results from krenn2017quantum 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. Afterwards, we extend the applications and find a construction for W state with arbitrary particles, and its generalization – the Dicke states. Furthermore, we present a general solution to producing high-dimensional 3-particle entangled states, which answers a question that has been raised more than 3 years ago.

Our investigation significantly enlarges the understanding of currently existing experimental technology and finds systematic solutions to a question that has previously investigated only with advanced automated search methods krenn2016automated ; melnikov2018active .

Generation of Greenberger-Horne-Zeilinger states

GHZ states form a very important class of entangled states and are denoted as

[TABLE]

where $n$ is the number of particles and $d$ is the dimension for every particle.

In Fig. 1A, we show an experimental setup to produce a 2-dimensional 4-particle GHZ state $|GHZ_{4}^{2}\rangle$ using Entanglement by Path Identity krenn2017entanglement . Photon pairs can be created by probabilistic photon pair sources (such as nonlinear crystals, depicted as gray squares) via the spontaneous parametric down-conversion (SPDC) process. The crystals are set up in such a way that crystals I and II produce photons with states $|00\rangle$, while crystals III and IV produce photons with states $|11\rangle$. Here the mode numbers [math] and $1$ correspond to the polarization of photons111A photon’s mode numbers can be changed by inserting variable mode-shifters in the photon’s paths. For convenience, we neglect the mode-shifters and label the mode numbers in the nonlinear crystal., the orbital angular momentum (OAM) allen1992orbital ; yao2011orbital ; krenn2017orbital or some other degree-of-freedom such as time-bin franson1989bell ; versteegh2015single or frequency olislager2010frequency .

The four crystals are pumped coherently and the pump power is set in such a way that two photon pairs are produced with reasonable probabilities222A higher number of photon pairs can be created in the down-conversion process. However, one can adjust the laser power such that these cases have a sufficiently low probability, which can be neglected.. In the experiment, the final quantum state is obtained by post-selection on 4-fold coincidences, which means that all $4$ detectors click simultaneously. This happens when two photon pairs origin either from crystals I and II or from crystals III and IV. No other event could contribute to the 4-photon coincidences. For example, if only the photon pairs are produced from crystals II and III, there will be two photons in path $c$ and no photon in path $b$.

One can translate such an optical setup into a graph krenn2017quantum , which is described in Fig. 1B. There the vertices depict the photon’s paths and the edges represent the nonlinear crystals. The graph contains two subsets of edges $(E_{ab},E_{cd})$ and $(E_{ac},E_{bd})$. Each subset contains all four vertices only once, which is called as a perfect matching of the graph. Therefore, the four-fold coincidences in the experiment are given by the coherent superposition of perfect matchings of the graph. The quantum state after conditioning on four-fold coincidences can be written as

[TABLE]

where values [math] and $1$ stand for photon’s mode numbers (such as the OAM modes of the photon), and the subscript $a$, $b$, $c$ and $d$ represent the photon’s paths.

Now we generalize this technique to 2-dimensional n-particle GHZ states $|GHZ_{n}^{2}\rangle$. One can arbitrarily increase the number of vertices of the graph in Fig. 1B, which means that the number333A probabilistic photon pair source (such as a nonlinear crystal) produces photon pairs, thus the number of particles $n$ is an even number. However, some of the photons can be seen as triggers such that the number $n$ can be an odd number. of particles can be arbitrarily large. We show the general graphs and experiments for creating 2-dimensional $n$-particle GHZ states $|GHZ_{n}^{2}\rangle$ in Fig. 2. These graphs can describe, for instance, the largest polarization GHZ state consisting of $n=12$ photons zhong201812 . 444Interestingly, the largest GHZ state ever produced in any platform is an 18-qubit state encoded in three degrees of freedom with six photons wang201818 . It would be interesting to extend the current graph language to cover such hyper-entangled multiphotonic quantum states.

As we have familiarized ourselves with the connection between graphs and quantum experiments krenn2017quantum , we can use it to create higher-dimensional GHZ states, such as a 3-dimensional 4-particle GHZ state $|GHZ_{4}^{3}\rangle$. The corresponding graph is described in Fig. 3A.

It has been shown in bogdanov267013 ; krenn2017quantum that such a graph is the only graph which can be constructed where all perfect matchings are independent555Independent perfect matchings (which are called disjoint perfect matchings in graph theory), means that every edge appears exactly once in a perfect matching. If the perfect matchings contain common edges, we call them nonindependent perfect matchings.. That means the quantum state $|GHZ_{4}^{3}\rangle$ is the only high-dimensional GHZ state which can be experimentally implemented in this way, while one can produce arbitrary 2-dimensional n-particle GHZ states $|GHZ_{n}^{2}\rangle$.

Generation of Dicke states

One very large important class of states has been introduced by Robert H. Dicke, – Dicke states $|D^{k}_{n}\rangle$ dicke1954coherence . The states are defined as

[TABLE]

where $n$ and $k$ stand for the number of particles and excitations, respectively. $\hat{S}$ is the symmetrical operator that gives summation over all distinct permutations of the $n$ particles.

W states $|W_{n}\rangle$ – The special case with only one excitation is the well-known $n$-particle W state (denoted as $|D^{1}_{n}\rangle$ or $|W_{n}\rangle$) zeilinger1992higher ; bourennane2004experimental , which is highly persistent against photon loss. It is interesting that W states cannot be transformed into GHZ states with local operation and classical communication (LOCC) dur2000three , meaning that they reside in different classes of entangled states.

Firstly we start with a 4-particle W state $|W_{4}\rangle$, which is

[TABLE]

There are four terms in the quantum state, which correspond to four perfect matchings in the graph. For a complete graph666If every pair of vertices is connected with edges exactly once in a graph, we call such a graph as a complete graph. A complete graph with $n$ vertices is denoted as $K_{n}$. $K_{4}$, the number of perfect matchings is three. However, we can use multiple edges to increase the number of perfect matchings. These graphs are denoted as multigraphs.

We show such a multigraph for the W state $|W_{4}\rangle$ in Fig. 4A. There, every edge can contain two colors (black and red [dark gray]). For example, a red [dark gray] edge stands for that the corresponding crystal produces photon pairs in a state $|11\rangle$. Thus the edges with colors black-black, black-red [black-dark gray], red-black [dark gray-black] and red-red [dark gray-dark gray] represent the states $|00\rangle$, $|01\rangle$, $|10\rangle$ and $|11\rangle$, respectively.

We find that every perfect matching contains only one half-red [half dark gray] (black-red [black-dark gray] or red-black [dark gray-black]) edge and no more other red [dark gray] edges can be involved. That means every term in the quantum state contains exactly one excitation and their coherent superposition describes a W state. The corresponding optical setup is described below the graph. Therefore, one can experimentally produce 4-particle W state $|W_{4}\rangle$ krenn2017quantum .

Now we generalize the graph for arbitrary $n$-particle W state $|W_{n}\rangle$. We connect all the half-red [half dark gray] edges to vertex $a$ and describe the graphs in Fig. 4. Thereby, every perfect matching contains exactly one half-red [half dark gray] edge because of the fact that vertex $a$ can be used only once in a perfect matching. This gives exactly one excitation in every term of the quantum state. Thus one can construct such graphs for producing arbitrary W states. A 3D printed graph for a 26-particle W state $|W_{26}\rangle$ is shown in Fig. 4C.

Interestingly, the structure of the graph for creating $n$-particle W state $|W_{n}\rangle$ can be seen as a strong product of graphs sabidussi1959graph ; weissteingraphproduct . The general graph for state $|W_{n}\rangle$ is a special book graph weissteinbookgraph , which consists of $n/2-1$ complete graphs $K_{4}$ with common edges $E_{ab}$ (for details see the Appendix A). The multiple common edge $E_{ab}$ is the so-called base of the book graph and the $n/2-1$ complete graphs form the pages of our book graph. Hence such a graph can also be called a (n/2-1)-page 2-base $K_{4}$-book graph drorgraphname294174 . For simplicity, we denote such a graph as an Olivern graph. Thus, the graph for W state $|W_{8}\rangle$, which is shown in Fig. 4C, is a book graph with three pages.

Dicke states $|D^{n/2}_{n}\rangle$ – Another special case of Dicke states, which has been experimentally investigated, are the states $|D^{n/2}_{n}\rangle$. By splitting probabilistically photons, experimental implementations for Dicke states $|D^{2}_{4}\rangle$ and $|D^{3}_{6}\rangle$ have been successfully realized in laboratories kiesel2007experimental ; prevedel2009experimental ; wieczorek2009experimental ; hiesmayr2016observation . The general experimental scheme for symmetric Dicke states $|D^{n/2}_{n}\rangle$ is described in Fig. 5A.

The corresponding graph for such experimental setup is a complete graph $K_{n}$, which is described in Fig. 5B. There every pair of vertices is connected with a blue [light gray] edge, which stands for multiple edges colored with black-red [black-dark gray] and red-black [dark gray-black]. Therefore, every perfect matching contains $n/2$ half-red [half dark gray] edges, meaning that each term in the quantum state involves $n/2$ excitations. The coherent superposition of all perfect matchings describes the symmetric Dicke state $|D^{n/2}_{n}\rangle$.

General Dicke states $|D^{m}_{n}\rangle$ – A natural question is whether we can experimentally create arbitrary Dicke states $|D^{m}_{n}\rangle$ ($0<m<n$). We answer the question affirmatively, and show the construction of a graph in Fig. 6. In general, we use two complete graphs $G_{1}=K_{n-m}$ and $G_{2}=K_{m}$, where all edges of $G_{1}$ are black while all edges of $G_{2}$ are red [dark gray]. Each vertex of $G_{1}$ is connected to every vertex of $G_{2}$ with a blue [light gray] edge, which is a double edge with red-black [dark gray-black] and black-red [black-dark gray].

While in all constructions before, all terms of the resulting quantum state had the same amplitude (which we call maximally entangled), that is not the case here anymore. In quantum experiments, one can adjust the pump power to make nonmaximally entangled states into maximally entangled states, which means adjusting all amplitudes to be the same values. For such quantum state, the total number of terms in the quantum state is given by the number of perfect matchings of the corresponding graphs, which holds for the rest of the paper. These introduce weights in the graphs, which have been investigated in gu2019quantum . We show some examples of maximally entangled Dicke states in the Appendix B.

generation of high-dimensional multipartite entangled states

The generalization of high-dimensional entangled states allows very rich types of nonclassical correlations. One method to characterize these states is the so-called Schmidt-Rank Vector (SRV) huber2013structure ; huber2013entropy ; cadney2014inequalities . These states give rise to asymmetrically entangled states that exist only if both the number of particles and the dimensions are larger than two. We study one important special case of 3-particle entangled states with an additional particle as a trigger. These states recently have been investigated experimentally malik2016multi ; erhard2018experimental , and studied extensively in the form of computer-designed experiments krenn2016automated ; melnikov2018active .

The SRV represents the rank of the reduced density matrices of each particle. In the quantum state of three parties $a$, $b$ and $c$, the rank of the reduced density matrices

[TABLE]

together form the SRV $d_{\psi}=(A,B,C)$, where $A\geq B\geq C$. The values $A$, $B$ and $C$ stand for the dimensionality of entanglement particle $a$, $b$ and $c$ with the other two parties.

The classification with different SRVs provides an interesting insight that one can transform quantum states from higher classes to lower classes with LOCC, and not vice versa777The dimensionality $i$ ($i=A,B,C$) cannot be increased with LOCC..

As an example, we show a maximally entangled state with SRV=(4, 2, 2), which is

[TABLE]

There the first particle $a$ is 4-dimensionally entangled with the other two particles $bc$, whereas particle $b$ and $c$ are both only 2-dimensionally entangled with the rest.

We are interested in maximally entangled states (as before, all amplitudes are the same). Furthermore, we want that the quantum state with $SRV(A,B,C)$ has $A$ terms. Thereby, the structure of the SRV is clearly visible in the computation basis, which is convenient experimentally. We call such an entangled state an $SRV(A,B,C)$ state.

Searching experimental implementations for producing $SRV(A,B,C)$ states has been investigated with the computer algorithm MELVIN krenn2016automated . In Fig. 7, for the strong green cells, MELVIN has found experimental setups after several months of runtime. All other cases have remained open.

Now one could ask which $SRV(A,B,C)$ states are experimentally possible to create with probabilistic photon pair sources? We apply our connection between graphs and quantum experiments to answer the question. In krenn2017quantum , the authors have shown that graphs with four vertices can contain maximally three independent perfect matchings. We extend that technique and find whether one can experimentally create an $SRV(A,B,C)$ state without additional particles with probabilistic pair sources (details see the Appendix C)888All of the experimental setups are based on Ref. krenn2017entanglement . It is an open question how to create these setups with nonlinear crystals producing photon pairs and linear optics.. This finally answers a question that has been open for 3 years.

Our technique can be applied to find experimental implementations for another type of high-dimensional multipartite quantum states such as absolutely maximally entangled state scott2004multipartite ; goyeneche2014genuinely ; goyeneche2015absolutely ; fhuberAME35 . We show more interesting examples in the Appendix D. Many related questions remain open, and are summarized elsewhere krenn2019questions .

Conclusion

We have presented a method to experimentally create large classes of entangled quantum states that are theoretically well studied but unexplored in laboratories, by extending recent ideas in Ref. krenn2017entanglement and the bridge between quantum experiments and graphs krenn2017quantum .

An exciting extension of our work would be a full classification of which quantum states are achievable with current photonic technology involving probabilistic pair sources.

One particular important class of photonic entangled states are so-called graph states, which are resources for measurement-based quantum computation raussendorf2001one ; raussendorf2003measurement . Despite the similarity of names, graph states are not related to the techniques explained here. It would be very interesting to investigate which type of graph states can be experimentally generated with probabilistic pair sources. A starting point will be the introduction of complex weights, which has been discussed in Ref. gu2019quantum .

Motivated by our results, another purely physical question raises: What does it mean physically that some entangled quantum states cannot be created? Is the producibility or lack thereof connected to a property of entanglement, such as entanglement of formation wootters1998entanglement ? While the graph theoretical representation covers the mathematical results in an excellent way, a physical interpretation of these results is still missing. It would be an exciting research project to shed more light on that question.

Acknowledgements

This work was supported by the Austrian Academy of Sciences (ÖAW), by the Austrian Science Fund (FWF) with SFB F40 (FOQUS), the National Natural Science Foundation of China (No.61771236) and its Major Program (No. 11690030, 11690032), the National Key Research and Development Program of China (2017YFA0303700), and from a Scholarship from the China Scholarship Council (CSC).

Appendix A Strong Product of Graphs

Here we explain the structure of graphs for $n$-particle W states $|W_{n}\rangle$ and show a graph $G$ for the W state $|W_{8}\rangle$ in Fig. 8.

The graph $G$ can be seen as the result of a union operation of graphs $G_{1}$ and $G_{2}$. There the graph $G_{1}$ is a strong product999The strong product $G\boxplus H$ of graphs $G$ and $H$ is the graph with vertex set $V(G)\times V(H)$ and u=(u1,v1) is adjacent with v=(u2,v2) whenever (v1=v2 and u1 is adjacent with u2) or (u1=u2 and v1 is adjacent with v2) or (u1 is adjacent with u2 and v1 is adjacent with v2). sabidussi1959graph ; weissteingraphproduct ($S_{4}\boxplus P_{2}$) of a star graph101010A star graph $S_{i}$ is a graph with $i$ vertices, where ($i-1$) vertices are only connected, with one edge, to a single central vertex. $S_{i}$ and a path graph111111A path graph $P_{j}$ is a graph with $j$ vertices, where $j$ vertices and $(j-1)$ edges lie on a single line. $P_{i}$. The graph $G_{1}$ can be seen as a special case of book graph weissteinbookgraph . The graph $P_{2}$ is the base of the book graph and the number of edges of the graph $S_{i}$ gives the number of pages in the book graph. Therefore, the graph for the quantum state $|W_{8}\rangle$ is a book graph with three pages.

Appendix B Graphs for General Dicke States

We have shown a general graph for arbitrary nonmaximally Dicke states in Fig. 6. Each term in the quantum state corresponds to a number of perfect matchings. The number of perfect matchings is not necessarily the same number of the terms in the quantum state. Experimentally this leads to different coefficients for each term of the state and thereby to nonmaximally entanglement.

In the laboratories, one can adjust the pump power to change the amplitudes in order to obtain the maximally entangled states. This will introduce weights in the corresponding graph gu2019quantum . We show how to make the nonmaximally Dicke states $|D^{1}_{n}\rangle$ and $|D^{2}_{n}\rangle$ to the maximally entangled Dicke states in Figs. 9 and 10.

We do this by computing all perfect matchings that correspond to individual terms and then require that the corresponding weights lead to a constant value. This leads to an algebraic equation system. For the example mentioned above, that system can be solved.

Appendix C Restriction on the Generation of $\mathbf{SRV(A,B,C)}$ States

Here we apply the connection between graphs and experiments to answer which maximally entangled $SRV(A,B,C)$ states can be created. As we have described in the main text, for an $SRV(A,B,C)$ state with an additional trigger $t$ ($t$ stays the same mode number), the dimensionality of particles $a$, $b$ and $c$ are given by the values $A$, $B$ and $C$. That means particles $a$, $b$ and $c$ must contain $A$, $B$ and $C$ different mode numbers ($A\geq B\geq C$).

In the graph description, every perfect matching of the graph corresponds to a term in the quantum state. Thus we need to construct a graph with exactly $A$ perfect matchings, as this is part of our definition of maximally entanglement. We now use the three disjoint perfect matchings that exist in the complete graph $K_{4}$ to find a possible experimental implementation for different $SRV(A,B,C)$ states.

The main idea is, when there is more than one term with the same mode number for a particle $a$, $b$ or $c$, we could combine the trigger $t$ together with the particle of the repeated mode number to form a multiedge. That will allow us to create more than three terms in the quantum state (Note: we can always create three arbitrary terms, as we have full control of edges in the three perfect matchings.). In total we need to create $A$ terms.

First we consider the edge $E_{t,a}$. The mode number in each term of particle $a$ needs to be different, thus we can only use $E_{t,a}$ to create one term.

Now we consider the edge $E_{t,b}$. Photon $b$ has $B$ different mode numbers, therefore in $A-B$ terms, the mode numbers can be the same. So in addition to the one term that we always create, we have the possibility to create $A-B$ additional terms, leading to $1+(A-B)$ terms producible using $E_{t,b}$. However, in the cases when we use the same mode number for particle $b$, the mode number for particle $c$ needs to be different (otherwise it would reduce the dimensionality of the state, for example: $|0\rangle_{a}|0,0\rangle_{b,c}$ + $|1\rangle_{a}|0,0\rangle_{b,c}$ = $\left(|0\rangle_{a}+|1\rangle_{a}\right)|0,0\rangle_{b,c}$. That means there is a tradeoff between the number of repetitions in particle $b$ that we can use, and the number of different modes particle $c$ has (which is $C$). So in total, using edge $E_{t,b}$, we can create $\min(1+(A-B),C)$ terms.

Finally, we apply the same argument to the terms that we can create using $E_{t,c}$. We use the $(A-C)$ repetitions to create $1+(A-C)$ terms, again conditioned that there are enough usable mode numbers of photon $b$. That usable numbers of different modes in $b$ is now $(B-1)$, because one mode number was already used in the perfect matchings using $E_{t,b}$. Therefore we find that, using the edge $E_{t,c}$, we can create $\min(1+(A-C),B-1)$ terms.

Overall we find the following condition explaining whether the $SRV(A,B,C)$ can be created:

[TABLE]

We illustrate that conditions in Fig. 11 and describe two concrete examples in Fig. 11C.

Appendix D Generation of Absolutely Maximally Entangled States

The absolutely maximally entangled (AME) states are another type of multipartite states, which give the maximally mixed states by tracing out half or more of the parties. Such state is defined as $AME(n,d)$ with $n$ particles of local dimension $d$.

Here we only consider the experimentally most significant cases with $n=3$, which is written as goyeneche2018entanglement

[TABLE]

where sums inside kets are computed to be modulo $d$.

Firstly, we consider the 2-dimensional 3-particle AME state, which is

[TABLE]

Here, we apply the technique from the restriction for creating $SRV(A,B,C)$ states in Fig. 11. Thus we can rewrite such a state in Eq. 8 as

[TABLE]

We show such a graph in Fig. 12, which means that the quantum state can be experimentally produced.

Now we consider a 3-dimensional 3-particle AME state, which is

[TABLE]

In an analogous way, such a state can be rewritten as

[TABLE]

There we would need more than three independent perfect matchings for a graph with 4-vertices. However such a graph does not exist. Thus one cannot experimentally produce the state $|AME(3,3)\rangle$ in such a way. Similarly, the state $|AME(4,d)\rangle$ with $d\geq 2$ cannot be created in this way.