Abstract

High-dimensional entanglement enriches our understanding of entanglement and shows many advantages in quantum communications tasks, so its long-distance distribution is a key technology. Here, we use multicore fiber to distribute four-dimensional (4D) entanglement over 11 km. The observed fidelity is $F = 0.921 \pm 0.001$, which is feasible for high-dimensional quantum networks. We observed the violation of high-dimensional Bell inequality (${I_4} = 2.635 \pm 0.014$) and steering inequality (${S_4} = 1.900 \pm 0.003$) for a 4D system over a distance of 11 km and finally demonstrated 4D quantum key distribution based on entanglement in proof of principle. Our work provides an experimental platform for high-dimensional quantum technologies and enables the intracity high-dimensional quantum network.

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

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2020 (2)

2019 (6)

H. J. Lee and H. S. Park, “Generation and measurement of arbitrary four-dimensional spatial entanglement between photons in multicore fibers,” Photon. Res. 7, 19–27 (2019).
[Crossref]

Y. Guo, B. H. Liu, C. F. Li, and G. C. Guo, “Advances in quantum dense coding,” Adv. Quantum Technol. 2, 1900011 (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]

Y. Guo, S. M. Cheng, X. M. Hu, B. H. Liu, E. M. Huang, Y. F. Huang, C. F. Li, G. C. Guo, and G. E. Cavalcanti, “Experimental measurement-device-independent quantum steering and randomness generation beyond qubits,” Phys. Rev. Lett. 123, 170402 (2019).
[Crossref]

F. X. Wang, W. Chen, Z. Q. Yin, S. Wang, G. C. Guo, and Z. F. Han, “Characterizing high-quality high-dimensional quantum key distribution by state mapping between different degrees of freedom,” Phys. Rev. A 11, 024070 (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]

2018 (8)

D. Martinez, A. Tavakoli, M. Casanova, G. Caas, B. Marques, and G. Lima, “High-dimensional quantum communication complexity beyond strategies based on Bell’s theorem,” Phys. Rev. Lett. 121, 150504 (2018).
[Crossref]

J. W. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360, 285–291 (2018).
[Crossref]

X. M. Hu, Y. Guo, B. H. Liu, Y. F. Huang, C. F. Li, and G. C. Guo, “Beating the channel capacity limit for superdense coding with entangled ququarts,” Sci. Adv. 4, eaat9304 (2018).
[Crossref]

Q. Zeng, B. Wang, P. Y. Li, and X. D. Zhang, “Experimental high-dimensional Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 120, 030401 (2018).
[Crossref]

Y. Guo, X. M. Hu, B. H. Liu, Y. F. Huang, C. F. Li, and G. C. Guo, “Experimental witness of genuine high-dimensional entanglement,” Phys. Rev. A 97, 062309 (2018).
[Crossref]

Y. Guo, X. M. Hu, B. H. Liu, Y. F. Huang, C. F. Li, and G. C. Guo, “Experimental realization of path-polarization hybrid high-dimensional pure state,” Opt. Express 26, 28918–28926 (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 (7)

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]

G. Canas, N. Vera, J. Cariñe, P. González, J. Cardenas, P. W. R. Connolly, A. Przysiezna, E. S. Gómez, M. Figueroa, G. Vallone, P. Villoresi, T. F. da Silva, G. B. Xavier, and G. Lima, “High-dimensional decoy-state quantum key distribution over multicore telecommunication fibers,” Phys. Rev. A 96, 022317 (2017).
[Crossref]

J. Yin, Y. Cao, Y.-H. Li, S.-K. Liao, L. Zhang, J.-G. Ren, W.-Q. Cai, W.-Y. Liu, B. Li, H. Dai, G.-B. Li, Q.-M. Lu, Y.-H. Gong, Y. Xu, S.-L. Li, F.-Z. Li, Y.-Y. Yin, Z.-Q. Jiang, M. Li, J.-J. Jia, G. Ren, D. He, Y.-L. Zhou, X.-X. Zhang, N. Wang, X. Chang, Z.-C. Zhu, N.-L. Liu, Y.-A. Chen, C.-Y. Lu, R. Shu, C.-Z. Peng, J.-Y. Wang, and J.-W. Pan, “Satellite-based entanglement distribution over 1200 kilometers,” Science 356, 1140–1144 (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. Martin, T. Guerreiro, A. Tiranov, S. Designolle, F. Fröwis, N. Brunner, M. Huber, and N. Gisin, “Quantifying photonic high-dimensional entanglement,” Phys. Rev. Lett. 118, 110501 (2017).
[Crossref]

A. Sit, F. Bouchard, R. Fickler, J. Gagnon-Bischoff, H. Larocque, K. Heshami, D. Elser, C. Peuntinger, K. Günthner, B. Heim, C. Marquardt, G. Leuchs, R. W. Boyd, and E. Karimi, “High-dimensional intracity quantum cryptography with structured photons,” Optica 4, 1006–1010 (2017).
[Crossref]

Y. Ding, D. Bacco, K. Dalgaard, X. Cai, X. Zhou, K. Rottwitt, and L. K. Oxenløwe, “High-dimensional quantum key distribution based on multicore fiber using silicon photonic integrated circuits,” NPJ Quantum Inf. 3, 25 (2017).
[Crossref]

2016 (3)

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]

X. M. Hu, J. S. Chen, B. H. Liu, Y. Guo, Y. F. Huang, Z. Q. Zhou, Y. J. Han, C. F. Li, and G. C. Guo, “Experimental test of compatibility-loophole-free contextuality with spatially separated entangled qutrits,” Phys. Rev. Lett. 117, 170403 (2016).
[Crossref]

M. Pant and D. Englund, “High-dimensional unitary transformations and boson sampling on temporal modes using dispersive optics,” Phys. Rev. A 93, 043803 (2016).
[Crossref]

2015 (6)

C. Schaeff, R. Polster, M. Huber, S. Ramelow, and A. Zeilinger, “Experimental access to higher-dimensional entangled quantum systems using integrated optics,” Optica 2, 523–529 (2015).
[Crossref]

C. M. Li, Y. N. Chen, N. Lambert, C. Y. Chiu, and F. Nori, “Certifying single-system steering for quantum-information processing,” Phys. Rev. A 92, 062310 (2015).
[Crossref]

D. Y. Cao, B. H. Liu, Z. Wang, Y. F. Huang, C. F. Li, and G. C. Guo, “Multiuser-to-multiuser entanglement distribution based on 1550 nm polarization-entangled photons,” Sci. Bull. 60(12), 1128–1132 (2015).
[Crossref]

S. Pirandola, J. Eisert, C. Weedbrook, A. Furusawa, and S. L. Braunstein, “Advances in quantum teleportation,” Nat. Photonics 9, 641–652 (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]

C. Schwemmer, L. Knips, D. Richart, H. Weinfurter, T. Moroder, M. Kleinmann, and O. Gühne, “Systematic errors in current quantum state tomography tools,” Phys. Rev. Lett. 114, 080403 (2015).
[Crossref]

2014 (2)

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 (2014).
[Crossref]

2013 (3)

T. Inagaki, N. Matsuda, O. Tadanaga, M. Asobe, and H. Takesue, “Entanglement distribution over 300 km of fiber,” Opt. Express 21, 23241–23249 (2013).
[Crossref]

M. Huber and J. I. de Vicente, “Structure of multidimensional entanglement in multipartite systems,” Phys. Rev. Lett. 110, 030501 (2013).
[Crossref]

D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013).
[Crossref]

2010 (1)

T. Vértesi, S. Pironio, and N. Brunner, “Closing the detection loophole in Bell experiments using qudits,” Phys. Rev. Lett. 104, 060401 (2010).
[Crossref]

2009 (3)

R. Inoue, T. Yonehara, Y. Miyamoto, M. Koashi, and M. Kozuma, “Measuring qutrit-qutrit entanglement of orbital angular momentum states of an atomic ensemble and a photon,” Phys. Rev. Lett. 103, 110503 (2009).
[Crossref]

V. Scarani, H. B. Pasquinucci, N. J. Cerf, M. Dusek, N. Lütkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81, 1301–1350 (2009).
[Crossref]

A. Rossi, G. Vallone, A. Chiuri, F. De Martini, and P. Mataloni, “Multipath entanglement of two photons,” Phys. Rev. Lett. 102, 153902 (2009).
[Crossref]

2007 (3)

A. Fedrizzi, T. Herbst, A. Poppe, T. Jennewein, and A. Zeilinger, “A wavelength-tunable fiber-coupled source of narrowband entangled photons,” Opt. Express 15, 15377–15386 (2007).
[Crossref]

C. Z. Peng, J. Zhang, D. Yang, W. B. Gao, H. X. Ma, H. Yin, H. P. Zeng, T. Yang, X. B. Wang, and J. W. Pan, “Experimental long-distance decoy-state quantum key distribution based on polarization encoding,” Phys. Rev. Lett. 98, 010505 (2007).
[Crossref]

A. Acín, N. Brunner, N. Gisin, S. Massar, S. Pironio, and V. Scarani, “Device-independent security of quantum cryptography against collective attacks,” Phys. Rev. Lett. 98, 230501 (2007).
[Crossref]

2002 (3)

Č. Brukner, M. Żukowski, and A. Zeilinger, “Quantum communication complexity protocol with two entangled qutrits,” Phys. Rev. Lett. 89, 197901 (2002).
[Crossref]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (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. B. Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

1994 (1)

M. Reck, A. Zeilinger, H. J. Bernstein, and P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58–61 (1994).
[Crossref]

1991 (1)

A. K. Ekert, “Quantum cryptography based on Bell’s theorem,” Phys. Rev. Lett. 67, 661–663 (1991).
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Supplementary Material (1)

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» Supplement 1       details of the experiment

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Figures (4)

Fig. 1.
Fig. 1. High-dimensional entanglement distribution setup. (a) Preparation of the 1550 nm 4D entanglement source. A CW laser (power is 200 mW and wavelength is 775 nm) is focused by two lenses, and the waist radius is approximately 0.7 mm. Then, the light beam is separated into two paths by beam-displacer 1 (BD1). These two beams are injected into a Sagnac interferometer to pump a type-II cut periodically poled potassium titanyl phosphate (PPKTP) crystal ($1\;{\rm{mm}} \times 7\;{\rm{mm}} \times 10\;{\rm{mm}}$, the polling period is 46.2 µm, and the temperature is set at 35ºC. In our experiment, the full width at half maximum of the down-conversion source is approximately 2 nm.) and generates two-photon polarization entanglement $1/\sqrt 2 (|HV\rangle + |VH\rangle)$ in each path [36,37]. After HWP4, which set at 45°, the states are rotated to $1/\sqrt 2 (|HH\rangle + |VV\rangle)$ in both paths. We encode horizontally polarized (H) photons in path $a1(a3)$ as $|0\rangle$, vertically polarized (V) photons in path $a1(a3)$ as $|1\rangle$, H photons in path $a2(a4)$ as $|2\rangle$, and V photons in path $a2(a4)$ as $|3\rangle$. The state is prepared on a 4D maximally entangled two-photon state $|{\psi _4}\rangle = (|00\rangle + |11\rangle + |22\rangle + |33\rangle)/2$. (b) The multicore fiber (MCF) is 11 km long and has seven cores. The diameter of each core is 8 µm. The interval between two cores is 41.5 µm. In our system, we use MCF to transmit not only the path information, but also the polarization information. To maintain polarization stability, we lay all fibers on the optical table [38]. One also can use active feedback systems to calibrate the polarization [39] in the outdoor environment. Together with our fiber-locking system, this feedback system does not affect the transmission of system light. The relative phase between different paths is locked by a fiber-locking system. (c) Fiber-locking system (FLS). A weak 1570 nm CW laser is used as reference light. To reduce the influence of the reference light on the quantum system, two techniques have been employed here. First, the propagation direction of the reference light (from right to left) is opposite to that of the system light. Second, we use single photons as the reference. The FLS is established using a Mach–Zehnder interferometer comprised of elements BD2, BD3, and PZT. Noticed that we use mirrors worked at 1550 nm as the dichromatic mirrors, which do not decrease the coupling efficiency of the system light. For the reference light, we use a superconducting detector to record the photon count and the total photon count is approximate ${10^6}$. (d) and (e) Alice’s and Bob’s measurement setup. Each measurement setup consists of liquid crystal (LC), half-wave plates (HWPs) and polarization beam splitters (PBSs). By adjusting the voltages loaded on LCs and the angles of the HWPs, we can complete the construction of various measurement bases. All single-photon signals are detected using superconducting detectors optimized at 1550 nm.
Fig. 2.
Fig. 2. Results of 4D entangled state tomography after entanglement distribution. (a) and (b) are the real part and imaginary part of the reconstructed density matrix. The density matrix of the two ququarts is reconstructed from a set of 256 measurements represented by the operators ${u_i} \otimes {u_j}$ (with $i,j = 1,2,\ldots,16$) and ${u_k} = | {{\Psi _k}} \rangle \langle {{\Psi _k}} |$. Detailed descriptions of the tomographic measurements are presented in [40].
Fig. 3.
Fig. 3. (a) and (b) Violation of Bell’s inequality and the steering inequality, respectively. Red (block) points represent the theoretical values obtained for the maximally entangled states; blue (circular) points represent the experimental values. The blue line represents the classical upper bound from the local hidden variables (LHV) model. Error bars are smaller than the marker size.
Fig. 4.
Fig. 4. (a), (b), and (c) Correlations matrices for 2, 3, and 4-dimensional entanglement-based QKD, respectively. Alice and Bob measure in the mutually unbiased bases. The label Z indicates the computational basis and F indicates the Fourier basis.

Equations (5)

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| ψ 4 = 1 2 ( | 00 + | 11 + | 22 + | 33 ) .
I d k = 0 [ d / 2 ] 1 ( 1 2 k d 1 ) { [ P ( A 1 = B 1 + k ) + P ( B 1 = A 2 + k + 1 ) + P ( A 2 = B 2 + k ) + P ( B 2 = A 1 + k ) ] [ P ( A 1 = B 1 k 1 ) + P ( B 1 = A 2 k ) + P ( A 2 = B 2 k 1 ) + P ( B 2 = A 1 k 1 ) ] } ,
| k A , a = 1 d j = 0 d 1 exp [ i 2 π d j ( k + α a ) ] | j A , | l B , b = 1 d j = 0 d 1 exp [ i 2 π d j ( l + β b ) ] | j B ,
S d i = j = 0 d 1 P ( i A , j B ) + i + j = 0 d 1 P ( L i A , L j B ) ,
R d = l o g 2 d 2 H d ( 1 F Q K D ) ,

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