## Abstract

The Bell basis is a distinctive set of maximally entangled two-particle quantum states that forms the foundation for many quantum protocols such as teleportation, dense coding, and entanglement swapping. While the generation, manipulation, and measurement of two-level quantum states are well understood, the same is not true in higher dimensions. Here we present the experimental generation of a complete set of Bell states in a four-dimensional Hilbert space, comprising 16 orthogonal entangled Bell-like states encoded in the orbital angular momentum of photons. The states are created by the application of generalized high-dimensional Pauli gates on an initial entangled state. Our results pave the way for the application of high-dimensional quantum states in complex quantum protocols such as quantum dense coding.

© 2017 Optical Society of America

## 1. INTRODUCTION

Quantum entanglement is not only a curious phenomenon that radically deviates from our everyday experiences; it is also essential for many quantum information applications. High-dimensional entanglement [1–7] offers specific advantages over qubit entanglement, which is conventionally used in quantum information applications. The use of high-dimensional entanglement can enhance quantum communication schemes by increasing their channel capacity [8,9] and offering improved robustness against sophisticated eavesdropping attacks [10,11]. High-dimensionally entangled states can potentially be used for the teleportation of quantum states of ever-increasing complexity [12]. In addition, such states can enhance the capacity of quantum dense-coding schemes that allow for the sharing of more information than is classically possible [13,14].

Quantum communication schemes such as these usually require control over a basis of maximally entangled quantum states—the so-called *Bell basis*. For example, both quantum teleportation [15] and entanglement swapping [16] require one to unambiguously distinguish between at least one of the Bell states. In quantum dense coding, the generation, manipulation, and discrimination of Bell states is essential, as they form the basis in which information is encoded [14]. Polarization-entangled states have proved to be the most versatile implementation of two-dimensional Bell states thus far, and have found use in several quantum communication protocols. In addition to the polarization degree of freedom, two-dimensional Bell states have also been demonstrated with time-bin entangled states [17] and recently with photons entangled in their orbital angular momentum [18,19].

Here we demonstrate the experimental generation of a complete basis of four-dimensional Bell-like entangled states. Our experiment constitutes the first demonstration to our knowledge of a complete Bell basis beyond qubits. In the case of two-dimensional polarization entanglement, it is well known that one can rotate between all four Bell states with a half-wave plate (which performs a Pauli-X transformation) and a quarter-wave plate (performing a Pauli-Z transformation). We generate all 16 Bell states by generalizing this method to a higher-dimensional Hilbert space.

Our states consist of two photons entangled in their orbital angular momentum (OAM) [20,21]. The 16 orthogonal states in this basis are created by applying high-dimensional generalizations of Pauli gates on an initial OAM-entangled state that is produced via spontaneous parametric down-conversion. A four-dimensional X-gate is applied to one photon of the entangled pair, which corresponds to a cyclic transformation between the four basis states [22–24]. The second photon is sent through a four-dimensional Z-gate, which imparts a mode-dependent phase shift on the photon. We quantify the quality of our generated states by measuring their overlap with ideal states from a four-dimensional Bell basis and verify the presence of four-dimensional entanglement by measuring an appropriate entanglement witness.

## 2. TECHNIQUE

The $D$-dimensional Bell basis of a bipartite system AB, as generalized in the original teleportation paper by Bennett *et al.* [15], can be written in the form

In the first set of states ${\psi}_{0n}$ in Fig. 1(a), photons A and B share the same state, while the relative phase between the probability amplitudes varies according to $n$. The other three sets of states are obtained by performing specific transformations on the first group. To obtain the second group ${\psi}_{1n}$, photon B is transformed using clockwise cyclic mode transformation $X$ $(-2\to -1\to 0\to 1\to -2)$. For the groups ${\psi}_{2n}$ and ${\psi}_{3n}$, the state of photon B is transformed by an ${X}^{2}$ transformation $(-2\leftrightarrows 0,-1\leftrightarrows 1)$ and an anticlockwise cyclic transformation ${X}^{\u2020}$ $(-2\to 1\to 0\to -1\to -2)$, respectively. To transform among states within each group, photon A undergoes a mode-dependent phase transformation. In this manner, all 16 states in the four-dimensional Bell basis are obtained (Fig. 1). Since this is a bipartite maximally entangled system, it does not matter on which photon or in which order the phase and cyclic transformations are applied.

## 3. EXPERIMENT

As shown in Fig. 2, photons entangled in their OAM are generated via a frequency-degenerate type-II spontaneous parametric down-conversion (SPDC) process in a periodically poled potassium titanyl phosphate (ppKTP) crystal. The photons are produced in the state

By orienting the DP at different angles $(\alpha =0,\pi /4,\pi /2,3\pi /4)$, we obtain all four states in one group.

A fourfold clockwise cyclic transformation of OAM modes was recently developed through the use of the computer algorithm Melvin [22] and implemented with coherent light as well as single photons [23,24]. The principle idea of cyclic transformations is to split even and odd OAM modes into two different paths and manipulate them independently. Finally, the two paths are recombined coherently. In our experiment, we implement three such cyclic transformations ($X$, ${X}^{2}$, ${X}^{\u2020}$) at the single-photon level.

As shown in Fig. 2, we use a double-path Sagnac interferometer containing two DPs [25] to split even and odd OAM modes into two different paths (green frame). In the path for even OAM modes, different OAM manipulations are performed that are necessary for the three cyclic transformations. The two paths are probabilistically recombined with a beam splitter (BS), forming a Mach–Zehnder (MZ) interferometer. In principle, the two paths can be recombined with another parity sorter in a deterministic way. To perform the $X$ transformation $(-2\to -1\to 0\to 1\to -2)$, a spiral phase plate (SPP) adds an OAM quantum of $+1$ before the OAM sorter. After the sorter, one of the paths of the MZ interferometer undergoes an additional reflection. A final additional reflection completes the $X$ transformation. For the ${X}^{2}$ transformation $(-2\leftrightarrows 0,-1\leftrightarrows 1)$, an SPP is inserted within the MZ interferometer, adding an OAM quanta of $+2$ for even OAM modes only. A single reflection at the end completes the ${X}^{2}$ transformation. In the case of the ${X}^{\u2020}$ transformation $(-2\to 1\to 0\to -1\to -2)$, an additional reflection is introduced within the MZ interferometer, after which an SPP subtracts one OAM quantum from the recombined modes. Finally, an additional reflection completes the ${X}^{\u2020}$ transformation. Together, the three cyclic transformations on photon B and the three phase transformations on photon A allow us to obtain all four groups of states in the four-dimensional Bell basis.

The detection system consists of a spatial light modulator (SLM), a single-mode fiber (SMF), and a single-photon detector. The SLM is used to flatten the phase of an incoming photon, transforming it into an $\ell =0$ mode that efficiently couples to the SMF [26]. In this manner, the OAM content of single photons can be measured for specific modes or mode superpositions.

## 4. RESULTS

The 16 experimentally generated Bell states are analyzed using two different quantitative measures—their overlap with the theoretically expected Bell states and a witness of four-dimensional entanglement. The overlap allows us to estimate how close we are in our experiment to the ideal Bell basis, and the witness allows us to verify the presence of genuine four-dimensional entanglement in our generated states. Figure 3 shows the overlap of states within each of the four groups ${\psi}_{0n}\text{-}{\psi}_{3n}$. The overlap is measured by calculating the fidelity ${F}_{\mathrm{exp}}=\mathrm{Tr}({\rho}_{\mathrm{exp}}|{\psi}_{mn}\u27e9\u27e8{\psi}_{mn}|)$, where ${\rho}_{\mathrm{exp}}$ denotes the experimentally created state and $|{\psi}_{mn}\u27e9$ the ideal Bell states. Taking the non-flat spiral bandwidth [27] of the SPDC state into account, the maximum expected fidelity is limited to 93%. The average fidelity to the ideal state for the first group (without any cyclic transformation) is $82.1\%\pm 1.1\%$. The decrease of the measured fidelity of about 11% is mainly due to intermodal crosstalk. The other three groups combined show an average fidelity to the ideal state of $76.6\%\pm 2.2\%$. This shows that the cyclic transformation lowers the average fidelity by approximately 5.5%, which can be attributed to additional misalignments within the interferometers that comprise the $X$-gates.

Next, we certify the entanglement dimensionality of our generated states by using a bipartite entanglement witness for $d$-dimensional systems [28,29]. We search for the maximal overlap of an arbitrary three-dimensional quantum state with a maximally entangled four-dimensional state (which is not necessarily a Bell state but might have different phases). The theoretical maximum overlap is ${\overline{F}}_{\mathrm{max}}=75\%$. If we exceed this bound in the experiment, the state is certified to be (at least) four-dimensionally entangled.

The measured fidelity witnesses ${F}_{\text{wit}}$ for all 16 states are plotted in Fig. 4. Each of the 16 Bell states individually exceeds the bound of 0.75 by at least three standard deviations, and is thus certified to be four-dimensionally entangled. The error in the fidelity is calculated by propagating the Poissonian error in the photon-counting rates via a Monte Carlo simulation.

## 5. HIGH-DIMENSIONAL QUANTUM DENSE CODING

By replacing the SPP in our experiment with computer-generated holograms implemented on SLMs, our technique can be extended for rapidly switching between all 16 states in the four-dimensional Bell basis in an on-demand manner. This would constitute the first step in a high-dimensional quantum dense coding protocol [14]: by implementing both phase and cyclic transformations on one photon, Bob can encode 4 bits of information using the two-photon four-dimensional Bell basis. In a real application, the encoded information is lower due to crosstalk between the measured states. The experimental results of Fig. 3 lead to a mutual information of 2.6 bits (assuming that the probabilities for the not measured values, which are zero in the ideal case, are equally distributed). That exceeds the theoretical upper limit for two-dimensionally entangled states of 2 bits. The subsequent step where Alice must distinguish between all 16 Bell states in order to decode this information provides a significant challenge. It has been shown that it is impossible to unambiguously discriminate a single high-dimensional Bell state from the others with just linear optics [30]. However, it is possible to sort 16 Bell states into seven classes of states that can be distinguished with a linear optical setup, as was recently demonstrated with hyperentangled time-polarization states [31].

With linear optics only, we could therefore extract $\frac{7\xb72.6}{16}=1.14$ bits of information. However, full Bell state measurements are possible, for instance, using nonlinear optics [32].

Our experimental technique solely involves the photonic spatial degree of freedom and can thus be readily combined with well-developed techniques for polarization [33] and time-bin encoding [31], allowing for a significant increase in Hilbert space dimensionality.

## 6. CONCLUSION

Here we have shown the application of recently developed high-dimensional quantum gates to photonic quantum entanglement. By doing so, we were able to create a complete set of four-dimensionally entangled quantum states for which no method of creation was previously known. The quantum states we created are a high-dimensional generalization of the Bell basis, arguably the most commonly used set of entangled quantum states in two dimensions. Access to the complete high-dimensional Bell basis allows for the exploration of strong nonclassical correlations and their application in quantum information protocols such as quantum dense coding. Furthermore, our technique can be used for the generation of complete sets of high-dimensional Greenberger–Horne–Zeilinger states [6,35]. Another possible application is to create multi-qubit cluster states, which can be generated by merging several different Bell states [36]. By analogy, different types of high-dimensional multi-photon states can be produced using the technique presented here.

## APPENDIX A: SUPPLEMENTARY MATERIAL

## 1. Deviations from the Ideal States

Deviations from the ideal states can be explained by three main effects: non-equal distribution of modes from the original state, crosstalk between modes, and loss of coherence in the interferometers. The spiral bandwidth of the OAM distribution is not flat, thus the state created in the down-conversion process is not maximally entangled. In our experiment, we measure an initial state of $|\psi \u27e9=\alpha |0,0\u27e9+\beta |1,1\u27e9+\beta |-1,-1\u27e9+\gamma |2,2\u27e9$ with $\alpha /\beta =0.69$ and $\alpha /\gamma =0.45$. Thus the maximum possible fidelity with a maximally entangled Bell state is limited by 93%. This inherent unbalancing of the created modes can be overcome with a Procrustean filtering technique [2,37]. Another issue that lowers the fidelity is the crosstalk between different modes, and we find that in the computation basis, $\frac{\text{cross}\text{-}\text{talkcounts}}{\text{allcounts}}=0.11(3)$. The crosstalk limits the fidelity to 91%. These impurities mainly stem from misalignments of the OAM sorter and the Mach–Zehnder interferometer, which can be reduced by active stabilization. The coherence of the off-diagonal elements in this experiment has been measured to be 0.97(6)% on average. Taking these three limiting factors into account, the expected fidelity witness values are given by ${F}_{\text{wit}}=0.81(5)$. Hence, the observed average fidelity witness of ${F}_{\text{wit}}=0.808\pm 0.016$ is mainly due to unbalancing and crosstalk in the diagonal elements.

## 2. Overlap Between States

Here we show the data from which Fig. 3 has been created. It shows the overlap between different states from the same class, with the same OAM values but different phases.

${\psi}_{0,0}$ | ${\psi}_{0,1}$ | ${\psi}_{0,2}$ | ${\psi}_{0,3}$ | |

${\psi}_{0,0}$ | 0,810 | 0,063 | 0,011 | 0,048 |

${\psi}_{0,1}$ | 0,024 | 0,823 | 0,041 | 0,002 |

${\psi}_{0,2}$ | 0,015 | 0,060 | 0,818 | 0,049 |

${\psi}_{0,3}$ | 0,027 | 0,006 | 0,032 | 0,835 |

${\psi}_{1,0}$ | ${\psi}_{1,1}$ | ${\psi}_{1,2}$ | ${\psi}_{1,3}$ | |

${\psi}_{1,0}$ | 0,762 | 0,004 | 0,058 | 0,046 |

${\psi}_{1,1}$ | 0,053 | 0,748 | 0,002 | 0,074 |

${\psi}_{1,2}$ | 0,030 | 0,053 | 0,780 | 0,009 |

${\psi}_{1,3}$ | 0,005 | 0,025 | 0,044 | 0,811 |

${\psi}_{2,0}$ | ${\psi}_{2,1}$ | ${\psi}_{2,2}$ | ${\psi}_{2,3}$ | |

${\psi}_{2,0}$ | 0,788 | 0,100 | 0,020 | 0,003 |

${\psi}_{2,1}$ | 0,010 | 0,784 | 0,008 | 0,017 |

${\psi}_{2,2}$ | 0,027 | 0,041 | 0,751 | 0,066 |

${\psi}_{2,3}$ | 0,098 | 0,020 | 0,001 | 0,764 |

${\psi}_{3,0}$ | ${\psi}_{3,1}$ | ${\psi}_{3,2}$ | ${\psi}_{3,3}$ | |

${\psi}_{3,0}$ | 0,773 | 0,042 | 0,076 | 0,003 |

${\psi}_{3,1}$ | 0,022 | 0,745 | 0,056 | 0,089 |

${\psi}_{3,2}$ | 0,055 | 0,076 | 0,740 | 0,031 |

${\psi}_{3,3}$ | 0,076 | 0,061 | 0,018 | 0,747 |

From there, the average expected fidelity can be calculated to be ${\overline{F}}_{\mathrm{exp}}=0.78\pm 0.03$.

## 3. Entanglement Witness

First, we calculate the overlap ${F}_{\text{wit}}$ between our state and a $d$-dimensional maximally entangled target state. Then, we compute a $d$-dimensional entanglement bound $\mathcal{B}(d)=\sum _{\ell =0}^{d-1}{\lambda}_{i}^{2}$, which is the sum of the squares of all but the smallest Schmidt coefficients of the target state. If the overlap, ${F}_{\text{wit}}$ exceeds the bound for a $d$-dimensional entangled state; then the measurement data can only be explained with a ($d+1$)-dimensionally entangled state.

## 4. Combining the Beams Probabilistically

In our experiments, we combine the two photon paths for photon B probabilistically. The beam splitter in Fig. 2 is implemented via a half-wave plate at 45° in the horizontal arm (after which the polarization is diagonal), and PBS. In order to erase the *which-path information*, we could use a polarizer at 45°. However, we use half-wave plate at 45°, which rotates horizontal to diagonal and vertical to anti-diagonal; afterwards we use the SLM as an effective polarizer, as the SLM only works with horizontally polarized light.

## Funding

Austrian Academy of Sciences (OeAW); H2020 European Research Council (ERC) (SIQS 600645 EU-FP7-ICT); Austrian Science Fund (FWF) (SFB F40, CoQuS W1210-N16, Y879-N27); National Natural Science Foundation of China (NSFC) (11534008); Joint Czech-Austrian project MultiQUEST (I 3053-N27).

## Acknowledgment

F. W. was supported by the National Natural Science Foundation of China. M. M. acknowledges support from the Austrian Science Fund (FWF) through the START project and the joint Czech–Austrian project MultiQUEST.

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