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

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.

Figure 1

Experiment for producing a two-dimensional four-particle GHZ state $|{\text{GHZ}}_4^2\rangle$ based on Ref. [16] and the corresponding graph [13]. (a) Four nonlinear crystals (gray squares) are pumped coherently, and the pump laser can be set such that two photon pairs are created with reasonable probabilities. The final quantum state is created conditionally on simultaneous clicks in all four detectors. (b) In the graph, every vertex stands for a photon's path, and every edge represents a nonlinear crystal. The color depicts the mode number of a photon. Here black and red (dark gray) edges correspond to a state with $|00\rangle$ and $|11\rangle$, respectively. A fourfold coincidence in the experiment can be seen as a subset of edges that contains every vertex only once, which is called a perfect matching in the graph. Thus, the coherent superposition of two perfect matchings leads to fourfold coincidences, which describes the quantum state ${|\psi \rangle }_{abcd}=\frac{1}{\sqrt{2}}\left(\right|0000\rangle +|1111\rangle )$.

Figure 2

General graphs and experimental implementations for creating two-dimensional $n$-particle GHZ states $|{\text{GHZ}}_n^2\rangle$. In Fig. 1 we have shown a graph for a two-dimensional four-particle GHZ state $|{\text{GHZ}}_4^2\rangle$. One can arbitrarily extend that graph, which means one can arbitrarily increase the number of particles in the quantum state. Panels (a)–(c) show the general graphs and experiments for producing two-dimensional six-, eight-, and $n$-particle GHZ states. On the right side, we also show a 3D printed graph, which corresponds to a two-dimensional 26-particle GHZ state $|{\text{GHZ}}_{26}^2\rangle$. There the mode numbers 0 and 1 are represented with white and red (dark gray), and the vertices are depicted in black.

Figure 3

Graph and optical setup for producing a 3D four-particle GHZ state $|{\text{GHZ}}_4^3\rangle$. (a) We add one perfect matching of the graph in Fig. 1. The black, red (dark gray), and green (light gray) edges stand for the corresponding crystals producing the photon pairs with states $|00\rangle , |11\rangle$, and $|22\rangle$, respectively. (b) The corresponding experimental setup of the graph. All crystals are pumped coherently and the laser power can be set such that two photon pairs are produced. The coherent superposition of three perfect matchings leads to the quantum state, which is ${|\psi \rangle }_{abcd}=\frac{1}{\sqrt{3}}\left(\right|0000\rangle +|1111\rangle +|2222\rangle )$.

Figure 4

General graphs and experiments for producing $n$-particle $W$ states $|W_n\rangle$. (a) A colored multigraph with four perfect matchings. Every perfect matching contains only one half-red (half-dark gray) [black-red (black-dark gray) or red-black (dark gray-black)] edge, meaning that every term in the quantum state has exactly one excitation. The coherent superposition of all perfect matchings leads to a four-particle $W$ state $|W_4\rangle$. The corresponding experimental setup is described below the graph. (b, c) In an analogous way, we show the graphs and experiments for generating six- and eight-particle $W$ states. On the right side, we show a 3D printed graph for a 26-particle $W$ state $|W_{26}\rangle$. There the mode numbers 0 and 1 are represented with white and red (dark gray), and the vertices are depicted in black. We call the graph for producing an $n$-photon $W$ state an $Olive{r}_n$ graph.

Figure 5

General experimental scheme and graph for producing a symmetric Dicke state $|D_n^{n/2}\rangle$. (a) One can pump a nonlinear crystal to create $n/2$ photon pairs. The photons of a photon pair have orthogonal mode numbers (such as horizontal and vertical polarization), which are denoted as mode numbers 0 and 1. These photon pairs are probabilistically separated by beam splitters. The Dicke state $|D_n^{n/2}\rangle$ is created conditionally on a click in every detector. (b) In the general graph $K_n$, every blue (light gray) edge stands for a double edge with coloring black-red (black-dark gray) and red-black (dark gray-black), which corresponds to the state $|01\rangle +|10\rangle$. There every perfect matching contains exactly $n/2$ half-red (half dark gray) [black-red (black-dark gray) or red-black (dark gray-black)] edges, which means each term in the quantum state includes $n/2$ excitations. Thus the superposition of all perfect matchings describes the Dicke state $|D_n^{n/2}\rangle$.

Figure 6

General graph for general Dicke states $|D_n^m\rangle$ ($0<m<n$). For better visualization, we show such a graph in a 3D viewpoint. The graph consists of two complete graphs, $K_{n-m}$ and $K_m$. The first graph $K_{n-m}$ contains black edges, while the second $K_m$ (upper) involves red (dark gray) edges. Each vertex of $K_m$ is connected to every vertex of $K_{n-m}$ with a blue (light gray) edge, which stands for a double edge consisting of a black-red (black-dark gray) edge and a red-black (dark gray-black) edge. We call a black-red (black-dark gray) edge or red-black (dark gray-black) edge as a half-red (half dark gray) edge. Thus a red (dark gray) edge [or red-red (dark gray-dark gray) edge] represents for two half-red (half dark gray) edges. Because of this construction, every perfect matching contains exactly $m$ half-red (half dark gray) edges, meaning that each term in the quantum state has $m$ excitations.

Figure 7

A list of experimentally possible $SRV(A,B,C)$ states. Strong green (light gray) cells show that these states have been found with the computer algorithm melvin [14]. For all remaining cases, using graph theory we find the corresponding experimental setups [light green (light gray in the solid box)] or that states cannot be created [red (dark gray)] with probabilistic sources (without additional ancillary photons). States represented by black cells cannot exist even theoretically, because of combinatorial reasons [9].

Figure 8

Graph operations for constructing the graph corresponding to an $n$-particle $W$ state, the $Olive{r}_n$ graph. The structure of graph $G$ for the $W$ state $|W_8\rangle$ can be seen as a union of graphs $G_1$ and $G_2$, which is $G=G_1\bigcup G_2$. Graph $G_1$ is a strong product of a star graph $S_4$ and a path graph $P_2$, which is $G_1=S_4⊞P_2$. Therefore, the graph for the $n$-particle $W$ state $|W_n\rangle$ can be depicted as $G=(S_i⊞P_2)\bigcup G_2$, where $i=n/2$.

Figure 9

Weighted graphs for $n$-particle $W$ states $|W_n\rangle$. (a) Graph for $W$ state $|W_4\rangle$. The weights for black-black, black-red (black-dark gray), red-black (dark gray-black), and red-red (dark gray-dark gray) edges are $\delta , \alpha , \beta$, and $\gamma$, respectively. For simplicity, we write the black-red (black-dark gray) and red-black (dark gray-black) edges as a blue (light gray) edge, which corresponds to a state $\alpha |01\rangle +\beta |10\rangle$. From Fig. 6, we know that such a graph can be redrawn as two complete graphs $K_1$ and $K_3$ connected with blue (light gray) edges. The coherent superposition of all the perfect matchings in such a graph leads to the final quantum state, which is ${|\psi \rangle }_{abcd}=\gamma \left(3\alpha \right|0001\rangle +\beta |0010\rangle +\beta |0100\rangle +\beta |1000\rangle )$ (without normalization). In order to obtain the maximally entangled $W$ state, the coefficients should be the same, meaning $3\alpha =\beta$. For example, we can set the weights ($\alpha =1,\beta =3,\gamma =1$). (b) Graph for $W$ state $|W_6\rangle$. In an analogous way, we need to calculate all the perfect matchings of the graph. First, we start with edge $E_{af}$. The graph can be decomposed into the edge $E_{af}$ and a graph $K_{n-2}$ with vertices $b, c, d$, and $e$. In this case, the superposition of the perfect matchings are calculated, which is $3{\gamma }^2\left(\alpha \right|000001\rangle +\beta |100000\rangle )$. We calculate all the perfect matchings and require that $5\alpha =\beta$. Then we obtain the $W$ state. In general, we can obtain $n$-particle $W$ states with $(n-1)\alpha =\beta$.

Figure 10

Weighted graph for Dicke states $|D_n^2\rangle$. (a) Graph for Dicke state $|D_6^2\rangle$. This graph can be redrawn as two complete graphs $K_4$ and $K_2$ connected with blue (light gray) edges. (b) First, we consider the term $|000011\rangle$ in the Dicke state, and we find that there are two cases in the graph where the perfect matchings lead to that term. One is that the perfect matchings contain red (dark gray) edge $E_{ef}$. The other case is that the red-black (dark gray-black) edges connect to vertices $e$ and $f$. Thereby, we find the number of perfect matchings leading to the term of the quantum state. (c) Next we consider the terms where one of the vertices $e$ and $f$ and one of vertices ($a, b, c$, and $d$) are related to the red (dark gray) edge. Then we enumerated the cases where two of vertices ($a, b, c$, and $d$) are connecting to red (dark gray) edges. Finally, by solving these equation systems, one can create Dicke states $|D_n^2\rangle$.

Figure 11

Restriction on the generation of three-particle maximally entangled states with different Schmidt-rank vectors (SRVs) based on graph theory. (a) We ask which $SRV(A,B,C)$ states can be created. (b) Each term of an $SRV(A,B,C)$ state is given by a perfect matching of the corresponding graph. If a graph with four vertices involving more than $A$ perfect matchings can be constructed, a possible experimental setup for such a state exists. Therefore, when the parameters $A, B$, and $C$ fulfill the condition $1+\text{min}(1+(A-B),C)+\text{min}(1+(A-C),B-1)\ge A$, one could experimentally produce an $SRV(A,B,C)$ state. (c) Two examples with different SRVs. In the case of $SRV(6,3,3)$, we apply the restriction, and the parameters fulfill the condition derived in panel (b); thus such a state can be created. However, in the case of $SRV(6,3,2)$, the parameters do not fulfill the requirement, and one needs four disjoint perfect matchings of a graph with four vertices. Such a graph does not exist. Therefore, this quantum state cannot be produced with probabilistic photon pair sources in such a way.

Figure 12

A graph for a possible experiment producing the $AME(3,2)$ state with an additional trigger. There are four perfect matchings in the graph, which are described in the solid box. The coherent superposition of all perfect matchings leads to the quantum state ${|\psi \rangle }_{abct}=\frac{1}{2}\left(\right|000\rangle +|011\rangle +|101\rangle +|110\rangle )|0\rangle$.