Abstract
Quantum entanglement is one of the most important resources in quantum information. In recent years, the research of quantum entanglement mainly focused on the increase in the number of entangled qubits or the highdimensional entanglement of two particles. Compared with qubit states, multipartite highdimensional entangled states have beneficial properties and are powerful for constructing quantum networks. However, there are few studies on multipartite highdimensional quantum entanglement due to the difficulty of creating such states. In this paper, we experimentally prepared a multipartite highdimensional state \(\left{\Psi }_{442}\right\rangle =\frac{1}{2}(\left000\right\rangle +\left110\right\rangle +\left221\right\rangle +\left331\right\rangle )\) by using the path mode of photons. We obtain the fidelity F = 0.854 ± 0.007 of the quantum state, which proves a real multipartite highdimensional entangled state. Finally, we use this quantum state to demonstrate a layered quantum network in principle. Our work highlights another route toward complex quantum networks.
Introduction
Quantum entanglement^{1,2}, as one of the most important phenomena in quantum information, has been proven to play a central role in many applications: faulttolerant quantum computation^{3}, deviceindependent quantum communication^{4} and quantum precision measurements^{5}. In recent years, the research on quantum entanglement mainly focused on multipartite qubit systems^{6,7}, or twopartite highdimensional systems^{8,9}. For example, in optical systems, the preparation of quantum entanglement mainly develops in two directions: one is to increase the number of qubits of entanglement, such as 12photon entanglement^{6}, 18 qubit entanglement^{7}, the other is to increase the dimensionality of two photons, such as entanglement of 100 × 100 orbital angular momentum (OAM) degrees of freedom^{10}. Since the higherdimensional entanglement is naturally present in downconversion processes, it would be desirable to harness this highdimensionality for multiphoton experiments. Unfortunately, there are few experimental studies on multipartite highdimensional entangled quantum states. The main reason is that the preparation of such entangled states requires very delicate manipulation for highdimensional quantum systems.
In the field of quantum information, the most commonly used highdimensional degrees of freedom (DoFs) in photonic systems are as follows: OAM^{11}, time bin^{12}, path^{13,14,15,16,17}. The highdimensional DoFs of these photons have their advantages and disadvantages in the application of quantum information. For example, using OAM is easier to expand the dimension^{18}, but the fidelity of the preparation and operation is lower^{19} and the longdistance distribution is more difficult^{20}. The advantage of time bin DoF is that it is more suitable for longdistance distribution^{21}, however, it is difficult to implement arbitrary unitary operations on time bins. The path DoF has a very high fidelity and is easy to manipulate^{22,23}, and its dimension scalability^{16} and longdistance distribution were also demonstrated^{24}. Until now, multipartite highdimensional quantum entangled states have been successfully prepared only on OAM DoF. If classified according to Schmidt number vector^{25}, (3, 3, 2) state (\(1/\sqrt{3}(\left000\right\rangle +\left111\right\rangle +\left221\right\rangle )\))^{26}, (3, 3, 3) state (\(1/\sqrt{3}(\left000\right\rangle +\left111\right\rangle +\left222\right\rangle )\))^{27} and highdimensional Dicke states^{28} have been successfully prepared on OAM (See Fig. 1). Due to the difficulties of state preparation, the largest dimension encoded in each photon is 3, and the observed fidelities are a bit low compared to other multiphoton experiments.
Results
(4, 4, 2) Multiphoton highdimensional layered quantum states
For potential application in quantum key distribution, ref. ^{25,29} proposed to use a multipartite highdimensional quantum state, that has so far not been created in any experiment:
Notice that the first two photons, A and B, live in a fourdimensional space, whereas the third photon, C, lives in a twodimensional space. The state’s dimensionality is given by a vector of three numbers (4, 4, 2), which are the ordered ranks of the single particle reductions of the state density operator:
where \({\rho }_{i}={{\rm{Tr}}}_{\overline{i}}{\left\Psi \right\rangle }_{442}\left\langle {\left.\Psi \right}_{442}\right.\). This quantum state is obviously different from the general GHZ state (\(\left{\Psi }_{333}\right\rangle\)), and it contains quantum state \(\left{\Psi }_{332}\right\rangle\), see Fig. 1.
This quantum state exhibits different properties from other multipartite entangled states. If we observe the twodimensional subspaces of this quantum state, we find that there is a perfect correlation between particle A, B, and C in \(\{\left0\right\rangle ,\left1\right\rangle \}\) space (\(\{\left000\right\rangle ,\left111\right\rangle \}\)), simultaneously there is also a perfect correlation between A, B, and C in \(\{\left2\right\rangle ,\left3\right\rangle \}\) space and C in \(\{\left0\right\rangle ,\left1\right\rangle \}\) space (\(\{\left220\right\rangle ,\left331\right\rangle \}\)). So there is always a perfect GHZ correlation between A, B, and C if we observe the quantum state in a specific subspace. On the other hand, if C is detected in mode \(\left0\right\rangle\), then A, B are perfectly correlated in modes \(\left0\right\rangle\) and \(\left2\right\rangle\); if C is detected in mode \(\left1\right\rangle\), then A, B are perfectly correlated in modes \(\left1\right\rangle\) and \(\left3\right\rangle\). This property is quite different from all the previous states^{26,27,28}, and enables a layered quantum network and exhibits the advantage of highdimensional systems. For this reason this quantum state has been called layered quantum state. In this letter, we use the path and polarization DoFs of photons to build fourdimensional systems, demonstrate the creation and verification of one such entangled state with a fidelity of 0.854, which is higher than previous experiments^{26,27,28}. Owing to the highest fidelity, we demonstrate its application in a highly efficient layered quantum communication protocol in principle. This technique can be applied to construct complex quantum networks in the future.
Because the polarization DoF of photons has only two levels, it is impossible to construct highdimensional quantum states only by using polarization DoF in a single photon^{30,31}. So we use a beam displacer (BD) to additionally use the path DoF and combine it with the polarization DoF to complete hybrid highdimensional coding. The experiment setup is shown in Fig. 2, experimental details are presented in the “Methods” section.
Experiment result
We have witnessed the fidelity F_{exp} of the ideal quantum state \(\left{\Psi }_{442}\right\rangle\) with the state ρ_{exp}. One can conclude that the multipartite entangled state is genuinely (4, 4, 2) entangled from the obtained fidelity F_{exp}. This method relies on proving that the measured (4, 4, 2) state cannot be decomposed into entangled states of a smaller dimensionality structure (4, 3, 2). We found the best achievable overlap of a \(\left{\Psi }_{432}\right\rangle\) state with an ideal \(\left{\Psi }_{442}\right\rangle\) state to be F_{max} = 0.750 (see Supplementary Note 2). If F_{exp} > F_{max}, our state is certified to be entangled in 4 × 4 × 2 dimensions. To calculate F_{exp}, it is sufficient to measure the 32 diagonal and 6 unique real parts of offdiagonal elements of ρ_{exp} as shown in Fig. 3 (see Supplementary Note 3 for details). The coincidence efficiencies of 2–8 nm and 3–2 nm entangled sources are 19% and 13%, the total success probability of the postselection is 25%. Our fourphoton counting rate is 0.66 per second, and the integration time of each measurement setting is 1800 s. From the experimental data, F_{exp} is calculated to be 0.854 ± 0.007, which is above the bound of F_{max} = 0.750 by 14 standard deviations. This certifies that the threephoton state is indeed entangled in 4 × 4 × 2 dimensions.
In order to prove the layered property of (4, 4, 2) state, we calculate the fidelity of the twodimensional subspace GHZ state. There are six maximally entangled states in twodimensional subspaces. Four of them are maximally entangled states of three photons (A, B, C), and two of them are maximally entangled states of two photons (A, B). We still use the fidelity witness to certify the correlation of those subspaces. As shown in Fig. 3b–g, we measure all diagonal and unique offdiagonal elements of the density matrix. Through them, we can calculate the fidelity of each state and the maximally entangled state. The fidelity of these entangled states are F = 0.910 ± 0.029, 0.906 ± 0.030, 0.914 ± 0.030, 0.922 ± 0.028, 0.933 ± 0.030, 0.941 ± 0.031 (F > 0.5 is the bound for genuinely multipartite entangled states). The results proved that the (4, 4, 2) entangled state we prepared has good correlations in different subspaces.
Our method can be easily extended to generate morepartite highdimensional layered entangled states. Compared with the OAM DoF in photons, the path DoF is easier to manipulate. According to the method in ref. ^{32}, arbitrary multipartite highdimensional GHZ quantum states can also be realized by our experimental scheme.
The layered quantum communication network
As shown in Fig. 4(a), if there is a banking system including central bank, branches, and ATMs, which have different communication security levels, a layered quantum communication network will be formed. There are many ways to complete this network, such as establishing QKD between two bodies many times or GHZ state^{29}. However, in order to maximize the average key bits per photon, multiphoton highdimensional quantum states can be used to implement layered quantum key distribution protocol^{29}. The state \(\left{\Psi }_{442}\right\rangle\) we prepared can complete the simplest layered quantum communication network (central bank, branch1, and ATM1). Due to the dimensionality of four, one can share secret keys among all the three parties, and share secret keys among A (central bank) and B (branch1) simultaneously independent of the measurement results of C (ATM1)^{29} as shown in Fig. 4(b). We take the simplest layered quantum communication network as an example.
As shown in Fig. 4b, consider the quantum states \(\left{\Psi }_{442}\right\rangle\) we prepared. Each of the four possible outcome combinations {000, 111, 220, 331} is distributed uniformly to A, B, and C. Moreover, the outcomes of A, B {00, 11, 22, 33} are perfectly correlated and partially independent of the outcomes of C. Between A, B, and C, the following measurements can be used to complete the tripartite quantum communication k_{ABC}.
At the same time, A and B can communicate with each other in the following way k_{AB}.
k_{ABC} is completely correlated with C’s measurement results; therefore, it constitutes a random string shared by the three users. On the other hand, string k_{AB} is completely independent of C’s data conditioned on either of C’s two measurement outcomes, the value of k_{AB} is 0 or 1, each with probability 1/2. So the simplest layered quantum communication can be achieved by the above method. Of course, we can use the highdimensional GHZ state or (3, 3, 2) state to complete the same quantum communication protocol, but it is proved that the communication efficiency is lower^{29}.
In order to assess usefulness of the produced (4, 4, 2) state for quantum key distribution, we calculate asymptotic key rates R for each layer. We use a method developed in ref. ^{33}, which considers security against adversaries using coherent attacks. More precisely we use their equation (23), which uses parameters QBER_{Z}, QBER_{X}, QBER_{Z}(AB), and QBER_{Z}(AC). These are in order: quantum bit error rate between all three parties in Z and X basis, and quantum bit error rate in Z basis between pairs of users A, B and A, C. All of these parameters can be obtained directly from experimental data. We present values of these experimental parameters for all four layers in Table 1. Plugging the highest values from these intervals into equation (23) of ref. ^{33} yields a lower bound for the asymptotic key rate (bits per round), which also listed in the table.
Discussion
We have created a (4, 4, 2) quantum state using photons’ path and polarization DoFs. This state exhibits different correlations from all the previously reported state^{26,27,28} because we have promoted the dimension from 3 to 4. The postselection scheme we employed to increase the dimension allows a heralded generation of the (4, 4, 2) state, with an overhead of 1/2 compared to the canonical (2, 2, 2) generation. We have also experimentally demonstrated, as a proofofprinciple, that this quantum state can complete an efficient layered quantum communication network. Compared with OAM DoF or time bin DoF, the path DoF has higher fidelity and more controllability, so many physical phenomena^{15,17,23} and quantum information tasks^{22,34,35} are realized in the path DoF. Our experimental method can be effectively extended to produce more kinds of multipartite highdimensional entanglement^{32,36}, and to construct more complex highdimensional quantum networks.
The biggest remaining challenge is the efficiency of promoting bipartite sources of downconversion into multipartite states. Correlated singlephoton sources would provide an obvious route toward more efficient production and would also be compatible with our postselection scheme.
Methods
Experiment setup
First, we prepare a standard threephoton qubit GHZ state, as shown in Fig. 2. The ultraviolet pulse laser (390 nm) from the doubler sequentially pumps two entanglement source (780 nm). Then, the two entanglement sources, both are prepared in the state \((\leftHH\right\rangle +\leftVV\right\rangle )/\sqrt{2}\). Afterward, the output photon 2 and 3 are directed to a polarizing beam splitter (PBS1). Here, all the PBSs are set to transmit horizontal polarization (H) and reflect vertical polarization (V). If there is one and only one photon in each output of the fourphoton source part, the state \((\leftHHH\right\rangle +\leftVVV\right\rangle )/\sqrt{2}\) is generated (one of the photons acts as a trigger)^{37}.
Then, one photon is directly measured in the polarization basis (particle 3), and the other two photons incident to a postselection setup consisting of BD1, 2, PBS2 and halfwave plates (HWPs) set at 22.5^{∘}. At this time, we define the polarization DoF of particle 1 and particle 2 in the upper path as \(\leftH\right\rangle \longrightarrow \left0\right\rangle\), \(\leftV\right\rangle \longrightarrow \left2\right\rangle\), and lower path \(\leftH\right\rangle \longrightarrow \left1\right\rangle\) and \(\leftV\right\rangle \longrightarrow \left3\right\rangle\). The function of this device is to postselect the twodimensional entangled state \((\leftHH\right\rangle +\leftVV\right\rangle )/\sqrt{2}\) into the fourdimensional entangled state \((\left00\right\rangle +\left11\right\rangle +\left22\right\rangle +\left33\right\rangle )/2\)^{13,38,39} (see Supplementary Note 1).
After this postselection, the quantum state becomes:
Since the witness of quantum states and layered quantum communication protocols only needs twodimensional subspace projection measurements, we only perform the measurement in twodimensional subspaces. The measurement device consists of HWPs, QWPs, BDs, and PBSs. These setup can also be used to construct measurement setups of any dimension^{40}.
Data availability
All data not included in the paper are available upon reasonable request from the corresponding authors.
Code availability
All code not included in the paper are available upon reasonable request from the corresponding authors.
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Acknowledgements
This work was supported by the National Key Research and Development Program of China (Nos. 2017YFA0304100, 2016YFA0301300, and 2016YFA0301700), NSFC (Nos. 11774335, 11734015, 11874345, 11821404, 11904357, and 61490711), the Key Research Program of Frontier Sciences, CAS (No. QYZDYSSWSLH003), Science Foundation of the CAS (ZDRWXH20191), the Fundamental Research Funds for the Central Universities, Science and Technological Fund of Anhui Province for Outstanding Youth (2008085J02), and Anhui Initiative in Quantum Information Technologies (Nos. AHY020100, AHY060300). MH acknowledges funding from the Austrian Science Fund (FWF) through the START project Y879N27. M.H. and M.P. acknowledge the joint CzechAustrian project MultiQUEST (I 3053N27 and GF1733780L).
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X.M.H performed the experiment with assistance from W.B.X., C.Z. B.H.L., and Y.F.H.; M.P. and M.H. supported the theoretical part; all of the authors analyzed the data and discussed the contents; X.M.H, B.H.L., M.P. and M.H wrote the paper with input from all authors; C.F.L. and G.C.G. supervised the project.
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Hu, XM., Xing, WB., Zhang, C. et al. Experimental creation of multiphoton highdimensional layered quantum states. npj Quantum Inf 6, 88 (2020). https://doi.org/10.1038/s41534020003186
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