Abstract

High-dimensional entanglement has demonstrated its potential for increasing channel capacity and resistance to noise in quantum information processing. However, distributing it is a challenging task, imposing severe restrictions on its application. Here we report the first distribution of three-dimensional orbital angular momentum (OAM) entanglement via a 1-km-long few-mode optical fiber. Using an actively stabilizing phase precompensation technique, we successfully transport one photon of a three-dimensional OAM entangled photon pair through the fiber. The distributed OAM entangled state still shows a fidelity up to 71% with respect to the three-dimensional maximally entangled state (MES). In addition, we certify that the high-dimensional quantum entanglement survives the transportation by violating a generalized Bell inequality, obtaining a violation of $ \sim 3 $ standard deviations from the classical limit with $ {I_3} = 2.12 \pm 0.04 $. The method we developed can be extended to a higher OAM dimension and larger distances in principle. Our results make a significant step towards future OAM-based high-dimensional long-distance quantum communication.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Increasing the channel capacity and tolerance to noise in quantum communications is a strong practical motivation for encoding quantum information in multilevel systems, qudits as opposed to qubits [110]. From a foundational perspective, entanglement in higher dimensions exhibits more complex structures [1115] and stronger nonclassical correlations [1618]. Despite these benefits, the distribution of high-dimensional entanglement is relatively new and remains challenging. We improve on previous works [19,20] by (1) distributing three-dimensional—rather than two-dimensional—entanglement, and (2) increasing the length of the fiber by 3 orders of magnitude—achieving the first OAM entanglement distribution via fiber in the kilometer regime.

Inside the laboratory, three-level qutrit quantum teleportation has been achieved [21,22]. To bring the benefit of qudits into practice, long-distance shared entanglement is required. To this end, a two-dimensional entangled state has been transmitted over a 24-km-long fiber using frequency modes [23]; a four-dimensional entangled state has been transmitted over 100-km-long fiber using time bins [24]. Polarization-time bin hyperentanglement has also been distributed across a 1.2 km free-space intracity link [25]. Distributing entanglement using spatial degrees of freedom (DOF), like path and transverse spatial modes, has also attracted much interest: four-dimensional path entanglement over a 30-cm-long multicore fiber [26] and two-dimensional transverse spatial mode entanglement over a 30 cm hollow photonic crystal fiber [19] and a 40 cm step-index fiber [20]. In very recent times, related works independent from our efforts have been demonstrated. In Ref. [27], hybrid vortex-polarization entanglement has been distributed through a 5 m air-core fiber. The distribution length of orbital angular momentum (OAM) is still limited, except in Ref. [28], which converts the two-level OAM state into orthogonal polarization before sending it through a 250 m single-mode fiber (SMF). However, the method in this work cannot be extended beyond two dimensions. Beyond two dimensions, six-dimensional entanglement in the transverse spatial mode has been transmitted through a 2-m-long commercial multimode fiber in Ref. [29]. Our work improves on previous results by showing qutrit entanglement transport through fiber in the kilometer regime.

Spontaneous parametric downconversion (SPDC) naturally conserves both energy and momentum. As a consequence, entanglement in temporal and spatial DOF comes naturally when the pump beam is coherent in the respective DOF. These two DOFs are both suitable to be used as qudits. We focus on the transverse spatial modes associated with photonic OAM. An OAM state is denoted by $ |\ell \rangle $, where $ \ell $ is an integer that describes the azimuthal phase dependence corresponding to $ \ell \hbar $ of orbital angular momentum [30]. High-dimensional OAM entanglement up to 50 dimensions can be generated by SPDC by tuning phase-matching [12], and even higher OAM states—up to $ \ell = 10,010 $—can be entangled with polarization [31]. As such, working with OAM or transverse spatial modes is a viable way of scaling up the dimensionality of entanglement. Our improving capabilities in preparing [32], measuring [33], and processing [34] these entangled states, together with extensions to multiple parties [15,35], make a strong case for transverse spatial mode as a platform for quantum computation and communication. However, these efforts will be in vain if we are not able to distribute entanglement over long distances.

The sensitivity of OAM to atmospheric turbulence makes it challenging—though not impossible with hybrid entanglement [36]—to distribute OAM entanglement over free-space. A free-space quantum channel is also subject to weather, line-of-sight, or time of day. Instead, we set out to use an optical fiber to distribute entanglement. Previous works were limited to short lengths of fibers and only restricted to two dimensions. [19,20,27].

There is a trade-off between intermodal mixing—the coupling between degenerate modes—and intermodal dispersion—the difference in group velocities for modes of different orders. The latter is a significant factor contributing to decoherence of superposition states even for very short fibers. Ideally, one could minimize dispersion by using modes from the same order, but doing so increases the intermodal mixing. In Ref. [37], single photons from an attenuated laser, encoded with OAM and OAM mode superpositions, were transmitted through custom-made fiber designed to have low cross talk between higher-order OAM modes. We extend their results by using an entangled photon source. We choose to work with a step-index fiber as in Ref. [38] because of their ubiquity in optical communications—we show that a step-index fiber can be used to distribute OAM entanglement over long distances.

We first needed a source of high-dimensional maximally entangled states (MES) for photons at the telecom wavelength. SPDC is a natural source of photons entangled in their OAM, albeit the entanglement is often nonmaximal [10,39]. The quality of the MES increases with the spiral bandwidth—the number of entangled OAM modes generated. The spiral bandwidth is wider for thin crystals [40], but the lower photon counts (compared to longer crystals) and wider spectral bandwidth present a challenge. Instead of a thin bulk crystal for SPDC, we used a 10-mm-long periodically poled potassium titanyl phosphate (PPKTP) crystal. We experimentally determined the intermodal dispersion and employed an actively stabilizing precompensation module to eliminate it. With these measures, we successfully transmitted one of the photon pairs through 1 km of fiber and demonstrated the entanglement via a generalized Bell inequality [16] violation of three standard deviations.

2. EXPERIMENT

We now give more details of the experiment: The entangled photons are produced by degenerate Type-II collinear SPDC. Figure 1 shows the schematic of the experiment. A continuous-wave (CW) 775 nm laser beam is coupled to a SMF to obtain a pure fundamental Gaussian mode pump beam. At the output of the SMF, the pump beam is demagnified by a pair of lenses (not shown in Fig. 1) and is incident on a 10-mm-long PPKTP crystal. The power of the pump beam is 36 mW, and the beam waist on the crystal is $ {\omega _0} \sim 175\; {\unicode{x00B5}{\rm m}} $. The center wavelengths of the generated signal and idler photons are $ {\lambda _A} = {\lambda _B} = 1550\;{\rm nm} $, where subscripts A and B refer to Alice and Bob, respectively. The pump beam is then blocked by a dichroic mirror.

 figure: Fig. 1.

Fig. 1. Schematic of the experimental setup. The entangled photons are generated by Type-II SPDC in a PPKTP crystal. The idler photon is directly measured (Alice) while the signal photon is fed to a precompensation module and coupled into a 1-km-long few-mode fiber and finally analyzed (Bob). The single photon detectors we used are InGaAs detectors. Here the dotted box is an actively locked unbalanced MZ interferometer (more details are shown in the inset). Both the 1550 nm single photon and classical 775 nm reference beam go through the Mach–Zehnder interferometer.

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The OAM state of the photon pairs can be described in terms of Laguerre–Gaussian (LG) modes, following

$$\left| {{\Psi _{\rm SPDC}}} \right\rangle = \sum\limits_{\ell , {p_s}, {p_i}} C_{\ell , - \ell }^{{p_s},{p_i}}\left| {\ell ,{p_s}} \right\rangle \left| { - \ell , {p_i}} \right\rangle ,$$
where $ C_{\ell , - \ell }^{{p_s},{p_i}} $ denotes the complex weightings of the states $ | {\ell , {p_s}} \rangle $ and $ | { - \ell ,{p_i}}\rangle $, the quantum state of signal and idler, respectively [41]; the indices $ \ell $ and $ p $ denote the azimuthal and radial index of the LG mode. We limit measurements only to $ {p_s} = {p_i} = 0 $; hence, we drop these indices and focus only on $ \ell $: we analyze in the three-dimensional Hilbert space spanned by $ \ell = 0, \pm 1 $. The postselected state can be written as
$$\left| {{\Psi _{{\rm SPDC}}}} \right\rangle = {C_{0,0}}\left| 0 \right\rangle \left| 0 \right\rangle + {C_{1, - 1}}\left| 1 \right\rangle \left| { - 1} \right\rangle + {C_{ - 1,1}}\left| { - 1} \right\rangle \left| 1 \right\rangle .$$
Here, $ {C_{0,0}}$, $ {C_{1, - 1}} $ and $ {C_{ - 1,1}} $ depend on the $ z$ ratio between pump waist and downconverted beam waist [40,42]. The quantum state right after the source was reconstructed via standard quantum state tomography using maximum likelihood estimation [43], and we show the density matrix in Figs. 2(a) and 2(b). In order to characterize the quality of the entangled source, we calculate the fidelity with respect to a 3 * 3 MES. Several factors such as pump beam profile [44], propagation through the nonlinear crystal, and the relative rotation between the two SLMs may introduce an additional phase to the entangled state. Thus, we performed a global search to find an MES closest to the reconstructed quantum state. The MES is described by
$$\left| {{\Psi _{{\rm MES}}}\left( {\theta ,\varphi } \right)} \right\rangle = \left( {{e^{i\theta }}\left| { - 1} \right\rangle \left| 1 \right\rangle + \left| 0 \right\rangle \left| 0 \right\rangle + {e^{i\varphi }}\left| 1 \right\rangle \left| { - 1} \right\rangle } \right)/\sqrt 3 ,$$
with $ \theta = 1.02\pi $, $ \varphi = 0.98\pi $, the highest fidelity between the constructed density matrix $ \rho $ and $ {\Psi _{{\rm MES}}} $ being $ \langle {{\Psi _{{\rm MES}}}} |\rho | {{\Psi _{{\rm MES}}}} \rangle = 0.888 \pm 0.007 $, and the purity being $ {P_{{\rm source}}} = {\rm Tr}( {{\rho ^2}} ) = 0.83 \pm 0.01 $. We note that MESs are more easily achieved with thin crystals—these result in a wider spiral spectrum for the same pump characteristics [40], thus leading to states with higher fidelity to an MES. However, thin crystals also result in lesser photon counts that are detrimental to the signal-to-noise ratio, and wider bandwidth not ideal for distribution. We thus chose to generate photons in a PPKTP crystal as in Ref. [45] to balance the fidelity and photon counts, and also to give a narrower bandwidth. Details of the experimental parameters are shown in Supplement 1 (Section 1).
 figure: Fig. 2.

Fig. 2. Reconstructed density matrices (a), (b) before and (c), (d) after distribution. The real parts of the reconstructed density matrix are shown in (a) and (c), and the imaginary parts are shown in (b) and (d). For comparison, the closest maximally entangled state (MES) and the difference of the density matrix elements between the experimental result and theoretical MES are shown together in the orange-outlined insets.

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The signal and idler photons are separated by a polarizing beam splitter (PBS). The idler photons are directly analyzed by a spatial light modulator (SLM) via phase-flattening measurements [46] (Alice in Fig. 1, green region). The signal photon, before going through the 1 km fiber, is fed to the precompensation module (red region in Fig. 1). This part is critical in the experiment because the intermodal dispersion among different OAM modes acts as a dephasing channel, inevitably leading to decoherence. Therefore, high-dimensional superpositions and entanglement would be destroyed in a few centimeters [19,20]. This is in contrast to deterministic transmission of classical OAM information, where one can simply tune the electrical delay to compensate for the delays of the different modes.

We experimentally determine the intermodal dispersion and devise a setup to reverse it, to precompensate before entering the 1 km fiber (blue region in Fig. 1). The precompensation module consists of two cascaded interferometers and a locking system. The first interferometer is an OAM sorter, which serves as a parity check to convert the different OAM to polarization according to their topological charge $ \ell $. We redesigned the OAM sorter from the Mach–Zehnder (MZ) configuration [47] into a Sagnac interferometer for more robust phase stability. A half-wave plate (HWP) is used to rotate the polarization of the signal photon into $ ( {| H \rangle + | V\rangle } )/\sqrt 2 $ before entering the OAM sorter. This first interferometer is designed such that OAM modes with odd topological charge ($ \ell = \pm 1 $) are converted to horizontal polarization, and the even some $ ( {\ell = 0} ) $ are converted to vertical. The second interferometer is an unbalanced MZ interferometer designed to separate the different OAM modes into unequal path lengths: the odd OAM modes enter the short arm and even some enter the long arm. A dove prism in the long arm is mounted on a translation stage for arm-length tuning to compensate for the intermodal dispersion. To stabilize the path difference between the long and short arms, a phase locking system is used (depicted in the Fig. 1 inset). This is composed of a 775 nm laser beam, two photodetectors, and a piezoelectric transducer (PZT) mounted dove prism. Here the phase between the two arms is locked by a 775 nm classical light separated from the pump laser beam. The power of reference light fed into the U-shape interferometer is monitored by a tunable beam splitter consisting of a HWP, a PBS at 775 nm, and a detector. Both the single photons and the reference 775 nm beam go through the unbalanced MZ inteferometer, and the PBSs at the input and output work for both 775 nm and 1550 nm. A HWP (22.5°) and PBS combined with the detector at the output of the laser beam acquire the feedback power signal. Any fluctuation in path difference perturbs the relative phase between $ | 0 \rangle $ and $ | { \pm 1} \rangle $, which could be tracked through the readout of the detector. The feedback power signal drives the PZT to stabilize the phase relation with the help of the proportional-integral-derivative (PID) control module. After the cascaded interferometer, the single photon is sent to a HWP and a PBS to erase any polarization information.

 figure: Fig. 3.

Fig. 3. Intermodal dispersion. The normalized coincidence rate is the ratio between each measured coincidence rate and their sum. The horizontal coordinate represents the optical delay between the signal and idler photons. The blue dots represent the coincidence rates for of $ | {0,0} \rangle $ and the orange dots are for $ | { - 1,1}\rangle $. The intermodal dispersion is estimated by taking the difference in time of the peaks of the Gaussian fits (solid line).

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After precompensation, the signal photon is coupled into a 1-km-long home-designed step-index fiber, which supports OAM modes with $ \ell = 0, \pm 1, \pm 2 $ (see Supplement 1 Section 2 for details on the fiber). The signal photon is then sent to an SLM for analysis (Bob in Fig. 1, yellow region). Because the intermodal dispersion varies as the temperature fluctuates, we put the step-index fiber in a sealed adiabatic box with the temperature fluctuation controlled to within $ \pm 0.01\;{\rm K} $. Finally, the signal photons are detected over a 0.5 nm bandwidth in Bob’s analysis setup.

We obtain the intermodal dispersion by tuning the delay time of the coincidence module and observing the interval between two coincidence count events ($ | {0,0} \rangle $ and $ | { - 1,1}\rangle $), found to be 2.4 ns (see Fig. 3). The accuracy of this is subject only to the temporal resolution of the coincidence module (156 ps), and time jitter only broadens the shape of the coincidence curves but does not alter the interval of their peaks.

3. CHARACTERIZATION AND CERTIFICATION OF HIGH-DIMENSION NONLOCALITY

The reconstructed density matrix of the entangled state after the signal photon goes through the fiber is presented in Figs. 2(c) and 2(d). The 81 measurement settings required for quantum tomography are represented by projectors $ {p_i} \otimes {p_j}(i = 1,2, \ldots ,9) $, where $ {p_k} = | {{\varphi _k}} \rangle \langle {{\varphi _k}} | $. $ | {{\varphi _k}}\rangle $ is selected from the following set: $ \{ | { - 1} \rangle $, $ | 0 \rangle $, $ | 1 \rangle $, $ ( {| 0 \rangle + | { - 1} \rangle } )/\sqrt 2 $, $ ( {| 0\rangle + | 1 \rangle } )/\sqrt 2 $, $ ( {| 0 \rangle + i| { - 1} \rangle } )/\sqrt 2 $, $ ( {| 0 \rangle - i| 1 \rangle } )/\sqrt 2 $, $ ( {| { - 1} \rangle + | 1 \rangle } )/\sqrt 2 $, $ ( {| { - 1}\rangle + i| 1 \rangle } )/\sqrt 2 \} $ [43]. To confirm the quality of the distributed entanglement, we evaluate the fidelity with respect to the three-dimensional MES. The MES can be represented by

$$\left| {{\Psi _{\rm MES}}\left( {\theta ,\varphi } \right)} \right\rangle = \left( {{e^{i\theta }}\left| { - 1} \right\rangle \left| 1 \right\rangle + \left| 0 \right\rangle \left| 0 \right\rangle + {e^{i\varphi }}\left| 1 \right\rangle \left| { - 1} \right\rangle } \right)/\sqrt 3 ,$$
with different values of $ \theta $ and $ \varphi $ corresponding to different MESs. We performed a global search for the MES that would give the highest fidelity. With $ \theta = 0.98 \times 2\pi $ and $ \varphi = 0.07 \times 2\pi $, we obtain a fidelity $ F = \langle {{\Psi _{\rm MES}}( {\theta ,\varphi } )} |\rho | {{\Psi _{\rm MES}}( {\theta ,\varphi } )} \rangle = 0.71 \pm 0.02 $, and purity $ P = {\rm Tr}( {{\rho ^2}} ) = 0.56 \pm 0.03 $.

This fidelity suffices to certify entanglement beyond two dimensions [48], since the overlap of the obtained state and the ideal three-dimensional MES exceeds the lower bound of $ \frac{2}{3} $, and therefore cannot be achieved by an entangled state with lower than three dimensions (see Supplement 1 Section 3). To further illustrate three-dimensional entanglement, we violated a generalized Bell inequality, the Collins–Gisin–Linden–Massar–Popescu (CGLMP) inequality, for qutrits.

The Bell expression in three dimension is [16]

$$\begin{split}{I_3} &\equiv + [P\left( {{A_1} = {B_1}} \right) + P\left( {{B_1} = {A_2} + 1} \right) \\ &\quad + P\left( {{A_2} = {B_2}} \right) + P\left( {{B_2} = {A_1}} \right)] \\ &\quad - [P\left( {{A_1} = {B_1} - 1} \right) + P\left( {{B_1} = {A_2}} \right) \\ &\quad + P\left( {{A_2} = {B_2} - 1} \right) + P\left( {{B_2} = {A_1} - 1} \right)],\end{split}$$
where
$$P\left( {{A_a} = {B_b} + k} \right) \equiv \sum\limits_{j = 0}^2 P\left( {{A_a} = j,{B_b} = j + k \;{\rm mod}\; 2} \right).$$
We searched for the specific measurement settings $ {A_1} $, $ {A_2} $ and $ {B_1} $, $ {B_2} $ to maximally violate the inequality, $ {I_3} \le 2 $, imposed by local realism. We obtained $ {I_3} = 2.12 \pm 0.04 $. The violation exceeds the classical bound by about three standard deviations. Bell-type inequalities such as the CGLMP inequality we use can be considered as entanglement witnesses [49]. Under the locality and fair-sampling assumptions, the violation that we show sufficiently proves the existence of three-dimensional entanglement.

When the photon propagates through the fiber, phase shifts may arise among different OAM modes. This is mainly due to the change in the effective refractive index for different OAM modes in the fiber (see Supplement 1, Section 2). Consequently, the entangled state may be different before and after transmission through the fiber. However, this is irrelevant to the violation of the Bell inequality—which is what we really care about—and can be easily fixed by simply redefining the local basis: the phase shift in the fiber can be accounted for by redefining the qutrit basis. This can be seen in the Supplement 1 (Section 4), where we perform the process tomography. Ideally, the state transfer process consisting of propagation through the precompensation unit and fiber should be an identity matrix $ {\chi _0} = {\cal I} $. We show in Fig. S2 of Supplement 1 that the process matrix consists largely of the identity operator, thus reflecting the satisfactory performance of our precompensation. We note there still exist some deviations from identity due to experimental imperfection. More details are presented in Supplement 1 (Section 4).

4. DISCUSSION AND CONCLUSION

In summary, we have distributed OAM entanglement over 1 km of fiber, 3 orders of magnitude over previous work [19,20]. The challenging task of maintaining a stable phase relation in the entangled state when one photon undergoes evolution in an optical fiber was overcome by (1) keeping the temperature of the fiber stable to avoid temperature-induced phase fluctuations in the fiber, and (2) precompensating for the intermodal dispersion. For the latter, we redesigned an OAM sorter [47] into a Sagnac configuration for better phase stability, and used an unbalanced MZ interferometer to introduce different delays between the $ |0\rangle $ and $ | \pm 1\rangle $ states. The dispersion between $ |1\rangle $ and $ | - 1\rangle $ is extremely small even for 1-km-length fiber and made even more negligible by narrowing the bandwidth of the measurement such that this dispersion becomes negligible compared to the long coherence time. With our measures, we are able to certify three-dimensional entanglement via a fidelity to the MES of 0.71, and a violation of a CGLMP inequality.

Our work is a significant step forward for distributing high-dimensional entanglement in the transverse spatial modes of photons. Further improvements would utilize a higher-brightness source of OAM-entangled photons and more efficient telecom single photon detectors such as those afforded by superconducting nanowire systems. The dimensionality can also be increased by introducing different delays to OAM modes that were sorted simultaneously via more efficient OAM sorters [50], different from the odd–even mode sorter we used here. Note that although we did not use the radial index of the LG modes, modes with nonzero radial indices are also supported by the fiber. Intensity-flattening enables high-quality measurement of modes varying in both $ \ell $ and $ p $ indices [51], providing a pathway for extending dimensionality. Preserving the wavefront is challenging in free-space propagation, preventing further manipulation of the information carried by the photon. We show that preserving the wavefront is possible with precompensation, potentially enabling further information processing after the fiber. We hope that together with recent results on the noise resilience exploiting higher dimensions [52], our work will motivate further experimental research into novel protocols that involve long-distance high-dimensional quantum communications through fiber.

Funding

National Natural Science Foundation of China (11704371, 11734015, 11774335, 11821404, 11874345, 61327901, 61490711, 61525502, 11804330); National Key Research and Development Program of China (2017YFA0304100); Anhui Initiative in Quantum Information Technologies (AHY070000); Westpac Research Fellowship; Local Innovative and Research Teams Project of Guangdong Pearl River Talents Program (2017BT01X121); Key Research Program of Frontier Sciences, CAS (QYZDY-SSW-SLH003); National Youth Top Talent Support Program of National High level Personnel of Special Support Program; Fundamental Research Funds for the Central Universities (WK2030020019); Australian Research Council Centre of Excellence for Engineered Quantum Systems (EQUS, CE170100009).

Acknowledgment

We thank Zongquan Zhou, Zhiyuan Zhou, Xiao Liu, and Zhaodi Liu for beneficial discussion for the experiment, and Andrew White for careful reading of the paper.

 

See Supplement 1 for supporting content.

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31. R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. USA 113, 13642–13647 (2016). [CrossRef]  

32. F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017). [CrossRef]  

33. L.-J. Kong, R. Liu, Z.-X. Wang, W.-R. Qi, S.-Y. Huang, Q. Wang, C. Tu, Y. Li, and H.-T. Wang, “Manipulation of eight-dimensional Bell-like states,” Sci. Adv. 5, eeat9206 (2019). [CrossRef]  

34. A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-dimensional single-photon quantum gates: concepts and experiments,” Phys. Rev. Lett. 119, 180510 (2017). [CrossRef]  

35. M. Erhard, M. Malik, M. Krenn, and A. Zeilinger, “Experimental GHZ entanglement beyond qubits,” arXiv:1708.03881 (2017).

36. M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” Proc. Natl. Acad. Sci. USA 112, 14197–14201 (2015). [CrossRef]  

37. D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019). [CrossRef]  

38. J. Capenter, C. Xiong, M. J. Collins, J. Li, T. F. Krauss, and B. J. Eggleton, “Mode multiplexed single-photon and classical channels in a few-mode fiber,” Opt. Express 21, 28794–28800 (2013). [CrossRef]  

39. J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003). [CrossRef]  

40. F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66, 1–6 (2012). [CrossRef]  

41. F. M. Miatto, H. D. L. Pires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012). [CrossRef]  

42. F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011). [CrossRef]  

43. R. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002). [CrossRef]  

44. J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003). [CrossRef]  

45. Z.-Q. Zhou, Y.-L. Hua, X. Liu, G. Chen, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115, 070502 (2015). [CrossRef]  

46. H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20–A23 (2014). [CrossRef]  

47. W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112, 153601 (2014). [CrossRef]  

48. R. Fickler, R. Lapkiewicz, M. Huber, M. P. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5, 4502 (2014). [CrossRef]  

49. P. Hyllus, O. Gühne, D. Bruß, and M. Lewenstein, “Relations between entanglement witnesses and Bell inequalities,” Phys. Rev. A 72, 012321 (2005). [CrossRef]  

50. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013). [CrossRef]  

51. F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018). [CrossRef]  

52. S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019). [CrossRef]  

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    [Crossref]
  49. P. Hyllus, O. Gühne, D. Bruß, and M. Lewenstein, “Relations between entanglement witnesses and Bell inequalities,” Phys. Rev. A 72, 012321 (2005).
    [Crossref]
  50. M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
    [Crossref]
  51. F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
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  52. S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
    [Crossref]

2020 (1)

J. Liu, I. Nape, Q. Wang, A. Vallés, J. Wang, and A. Forbes, “Multidimensional entanglement transport through single-mode fiber,” Sci. Adv. 6, eaay0837 (2020).
[Crossref]

2019 (5)

D. Cozzolino, E. Polino, M. Valeri, G. Carvacho, D. Bacco, N. Spagnolo, L. K. Oxenløwe, and F. Sciarrino, “Air-core fiber distribution of hybrid vector vortex-polarization entangled states,” Adv. Photon. 1, 046005 (2019).
[Crossref]

L.-J. Kong, R. Liu, Z.-X. Wang, W.-R. Qi, S.-Y. Huang, Q. Wang, C. Tu, Y. Li, and H.-T. Wang, “Manipulation of eight-dimensional Bell-like states,” Sci. Adv. 5, eeat9206 (2019).
[Crossref]

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
[Crossref]

Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, A. Zeilinger, and J.-W. Pan, “Quantum teleportation in high dimensions,” Phys. Rev. Lett. 123, 070505 (2019).
[Crossref]

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
[Crossref]

2018 (3)

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
[Crossref]

T. Ikuta and H. Takesue, “Four-dimensional entanglement distribution over 100 km,” Sci. Rep. 8, 817 (2018).
[Crossref]

X.-M. Hu, B.-H. Liu, Y. Guo, G.-Y. Xiang, Y.-F. Huang, C.-F. Li, G.-C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
[Crossref]

2017 (5)

F. Steinlechner, S. Ecker, M. Fink, B. Liu, J. Bavaresco, M. Huber, T. Scheidl, and R. Ursin, “Distribution of high-dimensional entanglement via an intra-city free-space link,” Nat. Commun. 8, 15971 (2017).
[Crossref]

H. J. Lee, S.-K. Choi, and H. S. Park, “Experimental demonstration of four-dimensional photonic spatial entanglement between multi-core optical fibers,” Sci. Rep. 7, 4302 (2017).
[Crossref]

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-dimensional single-photon quantum gates: concepts and experiments,” Phys. Rev. Lett. 119, 180510 (2017).
[Crossref]

F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017).
[Crossref]

2016 (3)

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. USA 113, 13642–13647 (2016).
[Crossref]

M. Kleinmann and A. Cabello, “Quantum correlations are stronger than all nonsignaling correlations produced by n-outcome measurements,” Phys. Rev. Lett. 117, 150401 (2016).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

2015 (3)

T. M. Graham, H. J. Bernstein, T.-C. Wei, M. Junge, and P. G. Kwiat, “Superdense teleportation using hyperentangled photons,” Nat. Commun. 6, 7185 (2015).
[Crossref]

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” Proc. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

Z.-Q. Zhou, Y.-L. Hua, X. Liu, G. Chen, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115, 070502 (2015).
[Crossref]

2014 (4)

H. Qassim, F. M. Miatto, J. P. Torres, M. J. Padgett, E. Karimi, and R. W. Boyd, “Limitations to the determination of a Laguerre–Gauss spectrum via projective, phase-flattening measurement,” J. Opt. Soc. Am. B 31, A20–A23 (2014).
[Crossref]

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112, 153601 (2014).
[Crossref]

R. Fickler, R. Lapkiewicz, M. Huber, M. P. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5, 4502 (2014).
[Crossref]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

2013 (5)

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref]

J. Nunn, L. Wright, C. Söller, L. Zhang, I. Walmsley, and B. Smith, “Large-alphabet time-frequency entangled quantum key distribution by means of time-to-frequency conversion,” Opt. Express 21, 15959–15973 (2013).
[Crossref]

J. Mower, Z. Zhang, P. Desjardins, C. Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” Phys. Rev. A 87, 062322 (2013).
[Crossref]

J. Capenter, C. Xiong, M. J. Collins, J. Li, T. F. Krauss, and B. J. Eggleton, “Mode multiplexed single-photon and classical channels in a few-mode fiber,” Opt. Express 21, 28794–28800 (2013).
[Crossref]

M. Mirhosseini, M. Malik, Z. Shi, and R. W. Boyd, “Efficient separation of the orbital angular momentum eigenstates of light,” Nat. Commun. 4, 2781 (2013).
[Crossref]

2012 (5)

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66, 1–6 (2012).
[Crossref]

F. M. Miatto, H. D. L. Pires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
[Crossref]

Y. Kang, J. Ko, S. M. Lee, S.-K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref]

V. D’Ambrosio, E. Nagali, S. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961 (2012).
[Crossref]

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86, 012334 (2012).
[Crossref]

2011 (3)

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

W. Löffler, T. Euser, E. Eliel, M. Scharrer, P. St. J. Russell, and J. Woerdman, “Fiber transport of spatially entangled photons,” Phys. Rev. Lett. 106, 240505 (2011).
[Crossref]

F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011).
[Crossref]

2007 (2)

L. Aolita and S. Walborn, “Quantum communication without alignment using multiple-qubit single-photon states,” Phys. Rev. Lett. 98, 100501 (2007).
[Crossref]

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref]

2005 (1)

P. Hyllus, O. Gühne, D. Bruß, and M. Lewenstein, “Relations between entanglement witnesses and Bell inequalities,” Phys. Rev. A 72, 012321 (2005).
[Crossref]

2004 (1)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref]

2003 (3)

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref]

J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003).
[Crossref]

J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
[Crossref]

2002 (4)

R. Thew, K. Nemoto, A. G. White, and W. J. Munro, “Qudit quantum-state tomography,” Phys. Rev. A 66, 012303 (2002).
[Crossref]

A. O’neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref]

2000 (1)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Alexandrescu, A.

J. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003).
[Crossref]

Ali-Khan, I.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref]

Allen, L.

A. O’neil, I. MacVicar, L. Allen, and M. Padgett, “Intrinsic and extrinsic nature of the orbital angular momentum of a light beam,” Phys. Rev. Lett. 88, 053601 (2002).
[Crossref]

Andersson, E.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Aolita, L.

V. D’Ambrosio, E. Nagali, S. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961 (2012).
[Crossref]

L. Aolita and S. Walborn, “Quantum communication without alignment using multiple-qubit single-photon states,” Phys. Rev. Lett. 98, 100501 (2007).
[Crossref]

Azaña, J.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Babazadeh, A.

F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017).
[Crossref]

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-dimensional single-photon quantum gates: concepts and experiments,” Phys. Rev. Lett. 119, 180510 (2017).
[Crossref]

Bacco, D.

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
[Crossref]

D. Cozzolino, E. Polino, M. Valeri, G. Carvacho, D. Bacco, N. Spagnolo, L. K. Oxenløwe, and F. Sciarrino, “Air-core fiber distribution of hybrid vector vortex-polarization entangled states,” Adv. Photon. 1, 046005 (2019).
[Crossref]

Barnett, S.

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66, 1–6 (2012).
[Crossref]

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86, 012334 (2012).
[Crossref]

Barnett, S. M.

F. M. Miatto, H. D. L. Pires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
[Crossref]

F. M. Miatto, A. M. Yao, and S. M. Barnett, “Full characterization of the quantum spiral bandwidth of entangled biphotons,” Phys. Rev. A 83, 033816 (2011).
[Crossref]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref]

Bavaresco, J.

F. Steinlechner, S. Ecker, M. Fink, B. Liu, J. Bavaresco, M. Huber, T. Scheidl, and R. Ursin, “Distribution of high-dimensional entanglement via an intra-city free-space link,” Nat. Commun. 8, 15971 (2017).
[Crossref]

Bechmann-Pasquinucci, H.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Bernstein, H. J.

T. M. Graham, H. J. Bernstein, T.-C. Wei, M. Junge, and P. G. Kwiat, “Superdense teleportation using hyperentangled photons,” Nat. Commun. 6, 7185 (2015).
[Crossref]

Bouchard, F.

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
[Crossref]

Boyd, R. W.

Brandt, F.

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
[Crossref]

Broadbent, C. J.

I. Ali-Khan, C. J. Broadbent, and J. C. Howell, “Large-alphabet quantum key distribution using energy-time entangled bipartite states,” Phys. Rev. Lett. 98, 060503 (2007).
[Crossref]

Bruß, D.

P. Hyllus, O. Gühne, D. Bruß, and M. Lewenstein, “Relations between entanglement witnesses and Bell inequalities,” Phys. Rev. A 72, 012321 (2005).
[Crossref]

Buchler, B.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. USA 113, 13642–13647 (2016).
[Crossref]

Bulla, L.

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
[Crossref]

Buller, G. S.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Cabello, A.

X.-M. Hu, B.-H. Liu, Y. Guo, G.-Y. Xiang, Y.-F. Huang, C.-F. Li, G.-C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
[Crossref]

M. Kleinmann and A. Cabello, “Quantum correlations are stronger than all nonsignaling correlations produced by n-outcome measurements,” Phys. Rev. Lett. 117, 150401 (2016).
[Crossref]

Campbell, G.

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. USA 113, 13642–13647 (2016).
[Crossref]

Capenter, J.

Carvacho, G.

D. Cozzolino, E. Polino, M. Valeri, G. Carvacho, D. Bacco, N. Spagnolo, L. K. Oxenløwe, and F. Sciarrino, “Air-core fiber distribution of hybrid vector vortex-polarization entangled states,” Adv. Photon. 1, 046005 (2019).
[Crossref]

Caspani, L.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Chen, G.

Z.-Q. Zhou, Y.-L. Hua, X. Liu, G. Chen, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115, 070502 (2015).
[Crossref]

Chen, L.

W. Zhang, Q. Qi, J. Zhou, and L. Chen, “Mimicking Faraday rotation to sort the orbital angular momentum of light,” Phys. Rev. Lett. 112, 153601 (2014).
[Crossref]

Choi, S.-K.

H. J. Lee, S.-K. Choi, and H. S. Park, “Experimental demonstration of four-dimensional photonic spatial entanglement between multi-core optical fibers,” Sci. Rep. 7, 4302 (2017).
[Crossref]

Y. Kang, J. Ko, S. M. Lee, S.-K. Choi, B. Y. Kim, and H. S. Park, “Measurement of the entanglement between photonic spatial modes in optical fibers,” Phys. Rev. Lett. 109, 020502 (2012).
[Crossref]

Chu, S. T.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Cino, A.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Collins, D.

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref]

Collins, M. J.

Cortés, L. R.

M. Kues, C. Reimer, P. Roztocki, L. R. Cortés, S. Sciara, B. Wetzel, Y. Zhang, A. Cino, S. T. Chu, B. E. Little, D. J. Moss, L. Caspani, J. Azaña, and R. Morandotti, “On-chip generation of high-dimensional entangled quantum states and their coherent control,” Nature 546, 622–626 (2017).
[Crossref]

Cozzolino, D.

D. Cozzolino, E. Polino, M. Valeri, G. Carvacho, D. Bacco, N. Spagnolo, L. K. Oxenløwe, and F. Sciarrino, “Air-core fiber distribution of hybrid vector vortex-polarization entangled states,” Adv. Photon. 1, 046005 (2019).
[Crossref]

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
[Crossref]

D’Ambrosio, V.

V. D’Ambrosio, E. Nagali, S. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961 (2012).
[Crossref]

Da Lio, B.

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
[Crossref]

Dada, A. C.

A. C. Dada, J. Leach, G. S. Buller, M. J. Padgett, and E. Andersson, “Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,” Nat. Phys. 7, 677–680 (2011).
[Crossref]

Dalgaard, K.

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
[Crossref]

Dalton, R. B.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref]

Defienne, H.

N. H. Valencia, S. Goel, W. McCutcheon, H. Defienne, and M. Malik, “Unscrambling entanglement through a complex medium,” arXiv:1910.04490 (2019).

Desjardins, P.

J. Mower, Z. Zhang, P. Desjardins, C. Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” Phys. Rev. A 87, 062322 (2013).
[Crossref]

Deyanova, Y.

J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
[Crossref]

Ding, Y.

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
[Crossref]

Dudley, A.

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref]

Ecker, S.

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
[Crossref]

F. Steinlechner, S. Ecker, M. Fink, B. Liu, J. Bavaresco, M. Huber, T. Scheidl, and R. Ursin, “Distribution of high-dimensional entanglement via an intra-city free-space link,” Nat. Commun. 8, 15971 (2017).
[Crossref]

Eggleton, B. J.

Eliel, E.

W. Löffler, T. Euser, E. Eliel, M. Scharrer, P. St. J. Russell, and J. Woerdman, “Fiber transport of spatially entangled photons,” Phys. Rev. Lett. 106, 240505 (2011).
[Crossref]

Englund, D.

J. Mower, Z. Zhang, P. Desjardins, C. Lee, J. H. Shapiro, and D. Englund, “High-dimensional quantum key distribution using dispersive optics,” Phys. Rev. A 87, 062322 (2013).
[Crossref]

Erhard, M.

Y.-H. Luo, H.-S. Zhong, M. Erhard, X.-L. Wang, L.-C. Peng, M. Krenn, X. Jiang, L. Li, N.-L. Liu, C.-Y. Lu, A. Zeilinger, and J.-W. Pan, “Quantum teleportation in high dimensions,” Phys. Rev. Lett. 123, 070505 (2019).
[Crossref]

A. Babazadeh, M. Erhard, F. Wang, M. Malik, R. Nouroozi, M. Krenn, and A. Zeilinger, “High-dimensional single-photon quantum gates: concepts and experiments,” Phys. Rev. Lett. 119, 180510 (2017).
[Crossref]

F. Wang, M. Erhard, A. Babazadeh, M. Malik, M. Krenn, and A. Zeilinger, “Generation of the complete four-dimensional Bell basis,” Optica 4, 1462–1467 (2017).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

M. Erhard, M. Malik, M. Krenn, and A. Zeilinger, “Experimental GHZ entanglement beyond qubits,” arXiv:1708.03881 (2017).

Euser, T.

W. Löffler, T. Euser, E. Eliel, M. Scharrer, P. St. J. Russell, and J. Woerdman, “Fiber transport of spatially entangled photons,” Phys. Rev. Lett. 106, 240505 (2011).
[Crossref]

Fickler, R.

S. Ecker, F. Bouchard, L. Bulla, F. Brandt, O. Kohout, F. Steinlechner, R. Fickler, M. Malik, Y. Guryanova, R. Ursin, and M. Huber, “Overcoming noise in entanglement distribution,” Phys. Rev. X 9, 041042 (2019).
[Crossref]

F. Bouchard, N. H. Valencia, F. Brandt, R. Fickler, M. Huber, and M. Malik, “Measuring azimuthal and radial modes of photons,” Opt. Express 26, 31925–31941 (2018).
[Crossref]

M. Malik, M. Erhard, M. Huber, M. Krenn, R. Fickler, and A. Zeilinger, “Multi-photon entanglement in high dimensions,” Nat. Photonics 10, 248–252 (2016).
[Crossref]

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10,010,” Proc. Natl. Acad. Sci. USA 113, 13642–13647 (2016).
[Crossref]

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” Proc. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

R. Fickler, R. Lapkiewicz, M. Huber, M. P. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5, 4502 (2014).
[Crossref]

M. Krenn, M. Huber, R. Fickler, R. Lapkiewicz, S. Ramelow, and A. Zeilinger, “Generation and confirmation of a (100 × 100)-dimensional entangled quantum system,” Proc. Natl. Acad. Sci. USA 111, 6243–6247 (2014).
[Crossref]

Fink, M.

F. Steinlechner, S. Ecker, M. Fink, B. Liu, J. Bavaresco, M. Huber, T. Scheidl, and R. Ursin, “Distribution of high-dimensional entanglement via an intra-city free-space link,” Nat. Commun. 8, 15971 (2017).
[Crossref]

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, and A. Zeilinger, “Twisted photon entanglement through turbulent air across Vienna,” Proc. Natl. Acad. Sci. USA 112, 14197–14201 (2015).
[Crossref]

Forbes, A.

J. Liu, I. Nape, Q. Wang, A. Vallés, J. Wang, and A. Forbes, “Multidimensional entanglement transport through single-mode fiber,” Sci. Adv. 6, eaay0837 (2020).
[Crossref]

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref]

Franke-Arnold, S.

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86, 012334 (2012).
[Crossref]

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66, 1–6 (2012).
[Crossref]

Fujiwara, M.

M. Fujiwara, M. Takeoka, J. Mizuno, and M. Sasaki, “Exceeding the classical capacity limit in a quantum optical channel,” Phys. Rev. Lett. 90, 167906 (2003).
[Crossref]

Galili, M.

D. Cozzolino, D. Bacco, B. Da Lio, K. Ingerslev, Y. Ding, K. Dalgaard, P. Kristensen, M. Galili, K. Rottwitt, S. Ramachandran, and L. K. Oxenløwe, “Orbital angular momentum states enabling fiber-based high-dimensional quantum communication,” Phys. Rev. Appl. 11, 064058 (2019).
[Crossref]

Gilchrist, A.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O’Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[Crossref]

Giovannini, D.

D. Giovannini, J. Romero, J. Leach, A. Dudley, A. Forbes, and M. J. Padgett, “Characterization of high-dimensional entangled systems via mutually unbiased measurements,” Phys. Rev. Lett. 110, 143601 (2013).
[Crossref]

J. Romero, D. Giovannini, S. Franke-Arnold, S. Barnett, and M. Padgett, “Increasing the dimension in high-dimensional two-photon orbital angular momentum entanglement,” Phys. Rev. A 86, 012334 (2012).
[Crossref]

F. Miatto, D. Giovannini, J. Romero, S. Franke-Arnold, S. Barnett, and M. Padgett, “Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,” Eur. Phys. J. D 66, 1–6 (2012).
[Crossref]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

D. Collins, N. Gisin, N. Linden, S. Massar, and S. Popescu, “Bell inequalities for arbitrarily high-dimensional systems,” Phys. Rev. Lett. 88, 040404 (2002).
[Crossref]

Goel, S.

N. H. Valencia, S. Goel, W. McCutcheon, H. Defienne, and M. Malik, “Unscrambling entanglement through a complex medium,” arXiv:1910.04490 (2019).

Graham, T. M.

T. M. Graham, H. J. Bernstein, T.-C. Wei, M. Junge, and P. G. Kwiat, “Superdense teleportation using hyperentangled photons,” Nat. Commun. 6, 7185 (2015).
[Crossref]

Gühne, O.

P. Hyllus, O. Gühne, D. Bruß, and M. Lewenstein, “Relations between entanglement witnesses and Bell inequalities,” Phys. Rev. A 72, 012321 (2005).
[Crossref]

Guo, G.-C.

X.-M. Hu, B.-H. Liu, Y. Guo, G.-Y. Xiang, Y.-F. Huang, C.-F. Li, G.-C. Guo, M. Kleinmann, T. Vértesi, and A. Cabello, “Observation of stronger-than-binary correlations with entangled photonic qutrits,” Phys. Rev. Lett. 120, 180402 (2018).
[Crossref]

Z.-Q. Zhou, Y.-L. Hua, X. Liu, G. Chen, J.-S. Xu, Y.-J. Han, C.-F. Li, and G.-C. Guo, “Quantum storage of three-dimensional orbital-angular-momentum entanglement in a crystal,” Phys. Rev. Lett. 115, 070502 (2015).
[Crossref]

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Adv. Photon. (1)

D. Cozzolino, E. Polino, M. Valeri, G. Carvacho, D. Bacco, N. Spagnolo, L. K. Oxenløwe, and F. Sciarrino, “Air-core fiber distribution of hybrid vector vortex-polarization entangled states,” Adv. Photon. 1, 046005 (2019).
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Eur. Phys. J. D (2)

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F. M. Miatto, H. D. L. Pires, S. M. Barnett, and M. P. van Exter, “Spatial Schmidt modes generated in parametric down-conversion,” Eur. Phys. J. D 66, 263 (2012).
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J. Opt. Soc. Am. B (1)

Nat. Commun. (5)

R. Fickler, R. Lapkiewicz, M. Huber, M. P. Lavery, M. J. Padgett, and A. Zeilinger, “Interface between path and orbital angular momentum entanglement for high-dimensional photonic quantum information,” Nat. Commun. 5, 4502 (2014).
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[Crossref]

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Supplementary Material (1)

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Figures (3)

Fig. 1.
Fig. 1. Schematic of the experimental setup. The entangled photons are generated by Type-II SPDC in a PPKTP crystal. The idler photon is directly measured (Alice) while the signal photon is fed to a precompensation module and coupled into a 1-km-long few-mode fiber and finally analyzed (Bob). The single photon detectors we used are InGaAs detectors. Here the dotted box is an actively locked unbalanced MZ interferometer (more details are shown in the inset). Both the 1550 nm single photon and classical 775 nm reference beam go through the Mach–Zehnder interferometer.
Fig. 2.
Fig. 2. Reconstructed density matrices (a), (b) before and (c), (d) after distribution. The real parts of the reconstructed density matrix are shown in (a) and (c), and the imaginary parts are shown in (b) and (d). For comparison, the closest maximally entangled state (MES) and the difference of the density matrix elements between the experimental result and theoretical MES are shown together in the orange-outlined insets.
Fig. 3.
Fig. 3. Intermodal dispersion. The normalized coincidence rate is the ratio between each measured coincidence rate and their sum. The horizontal coordinate represents the optical delay between the signal and idler photons. The blue dots represent the coincidence rates for of $ | {0,0} \rangle $ and the orange dots are for $ | { - 1,1}\rangle $. The intermodal dispersion is estimated by taking the difference in time of the peaks of the Gaussian fits (solid line).

Equations (6)

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| Ψ S P D C = , p s , p i C , p s , p i | , p s | , p i ,
| Ψ S P D C = C 0 , 0 | 0 | 0 + C 1 , 1 | 1 | 1 + C 1 , 1 | 1 | 1 .
| Ψ M E S ( θ , φ ) = ( e i θ | 1 | 1 + | 0 | 0 + e i φ | 1 | 1 ) / 3 ,
| Ψ M E S ( θ , φ ) = ( e i θ | 1 | 1 + | 0 | 0 + e i φ | 1 | 1 ) / 3 ,
I 3 + [ P ( A 1 = B 1 ) + P ( B 1 = A 2 + 1 ) + P ( A 2 = B 2 ) + P ( B 2 = A 1 ) ] [ P ( A 1 = B 1 1 ) + P ( B 1 = A 2 ) + P ( A 2 = B 2 1 ) + P ( B 2 = A 1 1 ) ] ,
P ( A a = B b + k ) j = 0 2 P ( A a = j , B b = j + k m o d 2 ) .

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