1. Introduction

High-dimensional entangled states are very important for fundamental questions in quantum physics and have many curious phenomena due to their distinguishing properties compared with qubit states. Especially, in quantum information tasks, high-dimensional entanglement has many advantages over qubit-qubit entanglement, such as higher channel capacity, enhanced robustness against eavesdropping and quantum cloning [1], and larger violation of local-realistic theories [2, 3].

In recent years, growing interest has been devoted to generating high-dimensional entangled photon states. These include encoding based on energy-time [4–6], time-bins [7–10], orbital angular momentum [11–13], frequency modes [14–17], and path modes [18–20]. A natural space for exploring large dimension is photon’s spatial mode. The orbital angular momentum (OAM) of a photon is a spatial property that provides a discrete and unbounded state space. However, how to generate high quality of high-dimensional entanglement and operate it effectively remains challenges for the utilization of OAM. Another effective method is to use the photon’s energy of time. But limited by the scale of Franson interferometer, there exist the same problems as the OAM mode. Differing from other format degree of freedom of photon (such as OAM or time bin), the freedoms of path and polarization are easy to prepare and control in linear optical systems.

Here we report on the experimental preparation of a high quality path-polarization hybrid high-dimensional entanglement in photonic system. Moreover, we demonstrate the ability to generate arbitrary high-dimensional pure state on it. First, we use β-barium borate (BBO) crystal, half wave plates (HWPs) and beam displacers (BDs) based Mach-Zehnder interferometer to prepare high quality three and four-dimensional maximally entangled states, with fidelities exceeding 0.99. Then, we vary the relative proportions of Schmidt coefficients of these states by simply changing the pump photon state. Combining this with local unitary operation, we can then generate arbitrary high-dimensional pure state [21, 22]. Finally, we show the scalability of our method to generate even higher-dimensional states. Our high quality path-polarization entanglement source has many advantages over other sources: (1) extremely high fidelity high-dimensional entanglement can be achieved; (2) arbitrary high-dimensional pure state can be obtained with presented technology; (3) efficient single qudit measurement can be constructed with linear optics. These properties make our source reliable for fundamental research of quantum mechanics and realization of high-dimensional quantum information tasks.

2. Scheme

Generally, the simplest quantum system is composed of two components: the generation and measurement of a quantum state. A source creates two particles, each containing d modes (so-called a qudit) and forming the entangled state. Here, as an example, we show how to generate qutrit-qutrit entanglement.

As shown in Fig. 1(a), the pump photon is divided into two paths, the upper path (u) and the lower path (l). By carefully adjusting the half-wave plates (HWPs), the pump photon state reads

 

Fig. 1 Experimental setup. (a) Entanglement source. A continuous wave laser at 404 nm serves as the pump source. To generate qutrit-qutrit entanglement, the pump laser is separated into two paths by three HWPs and a BD, where HWP1 is set at 17.63°, HWP2 is set at 22.5° and HWP3 is set at 0°. Then the pump laser is directed into two 0.5 mm thick type-I cut β-borate (BBO) crystal. After the BD and three HWPs, the pump state is prepared on $({|V〉}_u+{|H〉}_u+{|V〉}_l)/\sqrt{3}$. The pump light is then focused on two spots of the crystals to generate spatial and polarizing hybrid entangled state. Thus the state $(|00〉+|11〉+|22〉)/\sqrt{3}$ is prepared if we encode the upper path |H〉 as |0〉, |V〉 as |1〉 and the lower path |H〉 as |2〉. (b) A typical single-observable measurement device. PBSs, HWPs, and BDs are used to construct the observable. The angles of HWPs are chosen to project the state to the eigenstates of the corresponding observable. (c) Three-dimensional gate. By applying different voltages, the liquid crystal phase plates (LCs) will introduce different phases between the fast axis and the slow axis. In our scheme, the optical axes of LCs are set at 0°. The setup realizes an I gate when we set HWP7-12 at 45°, 0°, 45°, 45°, 0°, 45°. The setup realizes an X gate when we set HWP7-12 at 0°, 0°, 45°, 0°, 0°, 45°. The setup realize an X² gate when we set HWP1-6 at 45°, 0°, 45°, 0°, 45°, 45°. Combining the gates and phase controller, we can operate locally on one photon of the maximally entangled state $(|00〉+|11〉+|22〉)/\sqrt{3}$, producing all nine Bell-states. Notation for optical elements: half-wave plate (HWP), beam displacer (BD), polarizing beamsplitter (PBS).

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Then the pump photon is directed into nonlinear crystals and split into two photons (idler and signal). If we use type I spontaneous parametric down-conversion (SPDC) process [23], we obtain a superposition state

If encoding the |H〉_u → |0〉, |V〉_u → |1〉 and |H〉_l → |2〉, we finally get the state

When changing the component of the pump light, we can obtain high-dimensional entangled state of this form a |00〉 + b |11〉) + c |22〉, where the coefficients satisfy |a|² + |b|² + |c|² = 1. Moreover, we can obtain arbitrary three-dimensional pure state through local unitary operations [21, 22] as shown in Fig. 1(c).

Universal single-observable three-dimension measurement can be constructed in a simple way. For any single-observable measurement, there are three orthogonal bases. We now show that the projection onto one of the eigenstates can be done in two steps as shown in Fig. 1(b). To perform the projection onto a |0〉 + b |1〉 + c |2〉, for example, we firstly accomplish the projection in the subspace composed of |0〉 and |1〉. And then, we finish the process by accomplishing the projection in the space composed of the result state above and |2〉. As shown in [3], projection onto the other two eigenstates can be accomplished simultaneously, for its measurement setup has three orthogonal exports.

3. Experiment

In principle, a photonic qudit system can be encoded using any degree of freedom (like OAM, time bin, or path). Here, path and polarization hybrid encoding will be utilized. The system consists of BBO, HWPs, BDs and can be divided into two parts: entanglement source (Fig. 1(c)) and measurement (Fig. 1(b)).

The maximum qutrit-qutrit entanglement experimental setup is shown in Fig. 1. A continuous wave laser (the wavelength is 404 nm and the power is 100 mW and the beam waist is about 1mm) is separated to two paths to pump two 0.5mm-thick type-I cut β-barium borate (BBO) crystals and generate photon pairs at 808nm. To prepare the two-photon state $(|00〉+|11〉+|22〉)/\sqrt{3}$, the pump photon is prepared on $(|0〉+|1〉+|2〉)/\sqrt{3}$ by half wave plates (HWPs) and beam displacers (BDs), where we encode the upper path |H〉 as |0〉, |V〉 as |1〉 and the lower path |H〉 as |2〉. Here, we use BDs to construct the phase-stable interferometers. BDs operating at 404 nm (808 nm) are approximate 36.41 mm (39.70 mm) long, introducing 4.21 mm separation between the horizontally and vertically polarized photons at 404 nm (808 nm). If we need to generate other format of entanglement state, such as $|00〉/\sqrt{2}+|11〉/2+|22〉/2$ or $|00〉/2+|11〉/\sqrt{2}+|22〉/2$, HWP1-3 should be set at corresponding angles, and the pump light is prepared on ${|V〉}_u/\sqrt{2}+{|H〉}_u/2+{|V〉}_l/2$ or ${|V〉}_u/2+{|H〉}_u/\sqrt{2}+{|V〉}_l/2$.

Four-dimensional entanglement can also be realized with the above device in Fig. 1. The only thing to do is encoded the state |V〉_l as |3〉. For example, the four-dimensional maximally entangled state

The standard quantum state tomography process is used to reconstruct the density matrix of maximum three and four-dimensional entanglement as shown in Fig. 2 and Fig. 3. For three-dimensional entangled state $(|00〉+|11〉+|22〉)/\sqrt{3}$, the fidelity observed in our experiment is F = 0.994 ± 0.001. As for the four-dimensional maximally entangled states, we also obtain a very high fidelity with F = 0.990 ± 0.001. The fidelities of 3 and 4 dimensional entanglement obtained in our experiment exceed the highest value which read approximately 0.98 that has ever been reported [18, 20, 24]. Detailed descriptions of tomographic measurements are presented in [25, 26]. Here the integration time for each data is 60 seconds to reduce the statistic error, and the error bars are estimated by Monte Carlo simulation.

 

Fig. 2 Graphical representation of the reconstructed density matrix of the three-dimensional maximally entangled state. The density matrix of the two qutrits is reconstructed from a set of 81 measurements represented by the operators u_i ⊗ u_j (with i, j=1, 2,…,9) and u_k = |Ψ_k 〉(Ψ_k |. The kets |Ψ_k 〉 for both the idler and the signal photons are selected from the following set: {|0〉, |1〉, |2〉, $(|0〉+|1〉)/\sqrt{2}$, $(|0〉+i|1〉)/\sqrt{2}$, $(|1〉+|2〉)/\sqrt{2}$, $(|1〉+i|2〉)/\sqrt{2}$, $(|0〉+|2〉)/\sqrt{2}$, $(|0〉+i|2〉)/\sqrt{2}$ }. Detailed descriptions of the tomographic measurements are presented in Ref. [25, 26].

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Fig. 3 Graphical representation of the reconstructed density matrix of the four-dimensional maximally entangled state. The method of constructing four-dimensional density matrices is similar to that of the three-dimensional ones.

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To prepare arbitrary three-dimensional pure state of the form $a|00〉+b{e}^{i{\phi }_1}|11〉+c{e}^{i{\phi }_2}|22〉$, it is vital to control the real coefficients {a, b, c} and the phases {φ₁,φ₂,φ₁ − φ₂} accurately. The proportions of a, b, and c can be adjusted by controlling the pumping light through HWP1-3 in Fig. 1(a). We have prepared three states where $a:b:c=1:1:\sqrt{2},1:\sqrt{2}:1$, and $\sqrt{2}:1:1$. The classical terms of its density matrix 〈i, j|ρ|i, j〉 were recorded as shown in Fig. 4 which shows the ability of accurate adjustment of the proportions of a, b and c. As for the relative phases {φ₁, φ₂, φ₁ − φ₂}, all of them are scanned within 0 to 2π using liquid crystal phase plate (which can change the relative phase of H and V from 0 to 2π). The current qutrit system involves three two-qubit subspaces. For the state when a : b : c = 1 : 1 : 1, each of them shows a visibility exceeding 0.990 as presented in Figs. 5(a)–5(c). Figs. 5(d)–5(f) shows the fringes when $a:b:c=\sqrt{2}:1:1$. In this way, we experimentally proved that we could prepare state $a|00〉+b{e}^{i{\phi }_1}|11〉+c{e}^{i{\phi }_2}|22〉$.

 

Fig. 4 The coincidence results of varying proportion of Schmidt coefficients of state a|00〉+b|11〉+c|22〉 when both photons are measured with computational basis {|0〉, |1〉, |2〉}. (a–c) We have prepared three states with $a:b:c=1:1:\sqrt{2},1:\sqrt{2}:1,\sqrt{2}:1:1$ respectively.

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Fig. 5 Coincidence fringes for state a|00〉 + b|11〉 + c|22〉. (a–c) show the phase variation of the three two-qubit subspaces of the state with a : b : c = 1 : 1 : 1. (a) For example, taking the subspace composed of |0〉 and |1〉, the two photons are subjected to the measurement base (|0〉 + e^iφ |1〉) ⊗ (|0〉 + |1〉) with varied φ. (d–f) present the fringe results of the state with $a:b:c=\sqrt{2}:1:1$.

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4. Arbitrary high-dimensional pure state

Starting from the generated state $a|00〉+b{e}^{i{\phi }_1}|11〉+c{e}^{i{\phi }_2}|22〉$, we can prepare arbitrary three-dimensional pure state [21] by local operations. As an example, we presented the high-dimensional Pauli-X operation, which transforms each mode to the nearest mode in a cyclic way, [27] and its integer powers as shown in Fig. 1(c). These operations are efficiently to transform a high-dimensional Bell-state to the others [28]. Such gate operations based on BDs and HWPs have very high fidelity up to 0.99 [29].

Our scheme is also scalable to higher-dimensional states. Here we give a protocol of preparing eight-dimensional entanglement as shown in Fig. 6. The core idea of generating eight dimensional entanglement is to divide the pump light into four beams, making the pump photon takes the form ${(|H〉}_1+e^i{ }^{{\phi }_1}|V{〉}_1+e^i{ }^{{\phi }_2}|H{〉}_2+e^i{ }^{{\phi }_3}|V{〉}_2+e^i{ }^{{\phi }_4}|H{〉}_3+e^i{ }^{{\phi }_5}|V{〉}_3+e^i{ }^{{\phi }_6}|H{〉}_4+e^i{ }^{{\phi }_7}|V{〉}_4)/\sqrt{8}$. When all the phases φ_i, i = 1, 2, … 7 equal 0, the state $(|00〉+|11〉+|22〉+|33〉+|44〉+|55〉+|66〉+|77〉)/\sqrt{8}$ is prepared. The measurement setup can also be designed along the idea of Fig. 1(b).

 

Fig. 6 Design of experimental device for eight dimensional entanglement. After transmitting through the BDs, the state of pump light is $({|H〉}_1+e^{i{\phi }_1}{|V〉}_1+e^{i{\phi }_2}{|H〉}_2+e^{i{\phi }_3}{|V〉}_2+e^{i{\phi }_4}{|H〉}_3+e^{i{\phi }_5}{|V〉}_3+e^{i{\phi }_6}{|H〉}_4+e^{i{\phi }_7}{|V〉}_4)/\sqrt{8}$. When all the phases φ_i(i = 1, 2, …, 7〉 equal 0, the eight-dimensional maximally entangled two-photon state $(|00〉+|11〉+|22〉+|33〉+|44〉+|55〉+|66〉+|77〉)/\sqrt{8}$ is prepared. Specially, the length of BD2 is half of BD1, thus introducing half displacement compared with BD1.

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5. Outlook and conclusion

High-dimensional entanglement has many advantages compares with two-dimensional entan­glement and attracts increasing attentions in recent years. However, generating high-quality high-dimensional entanglement, operating on high-dimensional qudit, and efficient measuring high-dimensional qudit in one system are still challenges. Here, we use photon’s path-polarization degrees of freedom, demonstrate the generation of high-quality high-dimensional entanglements and the ability of generating arbitrary high-dimensional pure states. We also give promising way for high-dimensional operations and measurements in our systems. These move the obstacles on the applications of high-dimensional entanglement in fundamental quantum physics [3, 30] or in quantum information field [31–33]. Our method is also scalable to higher-dimensional systems and is possible to connect with integrated optical circuits [20] in the future.