Abstract
Quantum entanglement is a fundamental resource in quantum information processing and its distribution between distant parties is a key challenge in quantum communications. Increasing the dimensionality of entanglement has been shown to improve robustness and channel capacities in secure quantum communications. Here we report on the distribution of genuine highdimensional entanglement via a 1.2kmlong freespace link across Vienna. We exploit hyperentanglement, that is, simultaneous entanglement in polarization and energytime bases, to encode quantum information, and observe highvisibility interference for successive correlation measurements in each degree of freedom. These visibilities impose lower bounds on entanglement in each subspace individually and certify fourdimensional entanglement for the hyperentangled system. The highfidelity transmission of highdimensional entanglement under realworld atmospheric link conditions represents an important step towards longdistance quantum communications with more complex quantum systems and the implementation of advanced quantum experiments with satellite links.
Introduction
The distribution of quantum entanglement between distant parties is one of the main technological challenges in the pursuit of a globalscale quantum Internet. Several proofofconcept studies have already demonstrated highfidelity transmission of photonic entanglement via terrestrial longdistance freespace links^{1,2,3}, and established the viability of employing optical satellite links for quantum communication on a global scale^{4,5}, and beyond^{6}. However, until very recently, these experimental studies have been focused on bipartite binary photonic systems, that is, the simplest state space that can exhibit quantum entanglement. Specifically, polarization qubits have been the system of choice for freespace quantum communications for over a decade.
Encoding several qubits per transmitted photon increases channel capacity and yields significant benefits in the implementation of advanced quantum information processing protocols, such as improving resilience with respect to noise and eavesdropping in secure quantum communications^{7,8,9,10,11,12,13,14}. Hence, increasing the dimensionality of entangled quantum systems can be considered one of the next key technological steps towards the realization of more practical quantum information processing protocols in realworld scenarios. Furthermore, from a fundamental physics point of view, the more diverse variations of nonclassical correlations that are possible in a large state space also provide a platform for diverse quantum physics experiments^{15,16,17,18}.
Highdimensional quantum information can be encoded in various photonic degrees of freedom (DOF), such as transverse orbital angular momentum (OAM)^{19,20,21,22}, discrete photon arrival time bins^{23} or continuousvariable energy–time modes^{24,25}. The transmission of classical OAM modes through turbulent atmosphere has been studied in several field trials^{26,27} and OAM multiplexing has already been used to achieve record channel capacity in freespace optical communications^{28}. While OAM entanglement has already been successfully demonstrated after atmospheric propagation^{29}, active wavefront correction will be required to fully exploit the potential of OAM encoding. The development of suitable adaptive optics systems is an immensely challenging field of ongoing research. Energy–time entanglement and its discrete analogue timebin entanglement both offer alternatives for highdimensional state encoding. Timebin qubits^{30,31} have been routinely used in fibrebased quantum key distribution systems, which has culminated in the recent demonstration of quantum teleportation over longdistance fibre links^{32,33} but have only recently been considered as a viable option for freespace quantum communications in presence of atmospheric turbulence^{34,35}.
The dimensionality of the state space can also be increased by simultaneously encoding quantum information in several DOF^{36}. This has the significant advantage that singlephoton twoqubit operations can be implemented deterministically between different DOF using only passive linear optics devices^{37,38}. Furthermore, simultaneous entanglement in multiple DOF, known as hyperentanglement^{39}, is readily engineered via the process of spontaneous parametric downconversion (SPDC) in nonlinear crystals^{40}. Hyperentanglement has been exploited in the realization of numerous advanced experiments, such as hyperentanglementassisted Bellstate measurements^{3,41,42,43}, quantum teleportation of multiple DOF of a single photon^{44,45}, robust quantum communications with increased channel capacity^{46} and efficient entanglement purification schemes^{47,48,49}. However, experiments that exploit hyperentanglement have not yet ventured beyond the distance limitations of optical tables and protected laboratory environments.
In this article, we report on the distribution of energy–time and polarization hyperentangled photons via a 1.2kmlong intracity freespace link. We observe highvisibility twophoton interference for successive correlation measurements in the polarization space and a twodimensional energy–time subspace and certify fourdimensional entanglement in the combined system. Our assessment of energy–time entanglement is based on the observation of Franson interference in unbalanced polarization interferometers^{50,51}. This simple approach is highly suitable for the exploitation of such states in future quantum experiments with satellite links.
Results
Experimental setup
The experiment (depicted in Fig. 1) was performed with an ultrabright source of hyperentangled photons and a detection module (Alice) located at the Institute for Quantum Optics and Quantum Information (IQOQI) and a receiver station (Bob) at the University of Natural Resources and Life Sciences (BOKU) in Vienna.
The source of hyperentangled photons (see ‘Methods’ section) was based on type0 SPDC in a polarization Sagnac interferometer^{52,53} with a continouswave 405nm pump laser. It produced fibrecoupled polarizationentangled photon pairs with a twophoton coherence time and centre wavelengths λ_{A}∼840 nm and λ_{B}∼780 nm, where subscripts A and B label the respective singlemode fibre for Alice and Bob. Since the emission time of a photon pair is uncertain within the significantly longer coherence time of the pump laser ≳100 ns , photons A and B were entangled in energy–time^{50}. In our proofofconcept demonstration, we focused on a twodimensional subspace of the highdimensional energy–time space (see ‘Methods’ section). The total state space considered in our proofofconcept experiment is thus a fourdimensional hyperentangled state in polarization and energy–time:
where H and V represent horizontally and vertically polarized photon states, whereas t and t+τ denote photonpair emission times with a delay τ with
Photon A was guided to a local measurement module and photon B was guided to a transmitter telescope on the roof of the institute via a 15mlong singlemode fibre. The photons emanating from the transmitter telescope were made to overlap with a 532nm beacon laser for pointing, acquisition and tracking and sent to a receiver telescope at BOKU via a 1.2kmlong optical freespace link. The receiver telescope consisted of a telephoto objective (Nikkor f=400 mm f/2.8) and an additional collimation lens. Note that the same type of objective is currently installed in the ISS Cupola module, and was recently proposed as a receiver in a quantum uplink scenario^{4}. The beacon laser was transmitted through a dichroic mirror and focused onto a CCD (chargecoupled device) image sensor while the collimated singlephoton beam was guided to Bob’s measurement module.
The measurement modules for Alice and Bob each featured a polarization analyser and an optional transfer setup that coupled the energy–time DOF to the polarization DOF (see also Supplementary Fig. 2). Alice’s polarization analyser consisted of a variable phase shifter, a halfwave plate and a polarizing beam splitter (PBS) with multimode fibrecoupled singlephoton avalanche diodes (SPAD) in each of its two output ports. A variable phase shift φ(θ) could be introduced between the computational basis states by tilting a birefringent YVO_{4} crystal by an angle θ about its optical axis using a stepper motor. With the halfwave plate set to 22.5°, this configuration corresponds to a polarization measurement in a superposition basis +φ/−φ, where . Bob’s polarization analyser module used a motorized halfwave plate and a PBS with a SPAD (active area of 180 μm) in each of its two output ports. To reduce background counts from the city, longpass filters and interference filters were added and the optical system was engineered such that the detectors had a small field of view (225 μrad). Bob’s analysis setup allowed for measurements in any linear polarization basis, in particular the basis +45°/−45°, where
For the analysis of energy–time entanglement, we employed a variant of the original Franson interferometer^{50} that uses polarizationdependent delays to map an energy–time subspace spanned by early and late pair emissions to the polarization state space^{51}. This variant has the advantage that the polarization entanglement acts as a pair of synchronized switches, such that there is no need for detection post selection^{54}. These unbalanced polarization interferometers at Alice and Bob were implemented with 3mmlong calcite crystals, which could be inserted before the polarization analysers. The calcite crystal introduced a birefringent delay of τ∼2 ps, which exceeded the coherence time of the SPDC photons but was significantly shorter than the coherence time of the pump laser. Note that the particular choice of delay restricts our considerations to a twodimensional subspace of the intrinsically continuousvariable energy–time space. Hence, after introducing the polarizationdependent delay, polarization measurements in a superposition basis correspond to measurements of energy–time superpositions of the form . A more detailed discussion is provided in the Supplementary Discussion.
The arrival times of singlephoton detection events at Alice and Bob were recorded relative to local 10 MHz global positioning system (GPS)disciplined clocks and stored on local hard drives for post processing of twophoton detection events. Bob’s measurement data were also streamed to Alice via a 5 GHz directional WiFi antenna, where all combinations of twophoton detection events within a coincidence window of 2 ns were monitored onthefly, while compensating for relative clock drifts (see Fig. 2)^{55}.
Link performance
Directly at the source, we detected a total coincidence rate of R^{(2)}∼84 kcps and singles rates of and . Of the single photons sent via the freespace link, we measured an average of 100 kcps in Bob’s two detector channels, and an average rate of ∼20 kcps twophoton detection events per second. For nighttime operation, the background counts were and for Bob’s two detector channels, whereby 200 cps and 50 cps were due to intrinsic detector dark counts.
Because of atmospheric turbulence, the link transmission varied on the timescale of ms (see Fig. 2). The timeaveraged beam diameter at the receiver was of the same order as the receiver aperture (14.5 cm). We observed an average total link transmission of ∼18%, including all optical losses from source to receiver, where approximately half of the transmission loss was due to absorption in optical components.
Besides being used for pointing, acquisition and tracking, the CCD image sensor also monitored angle of arrival fluctuations caused by atmospheric turbulence^{56}. The fullwidth at halfmaximum of the angular variation at the telescope was estimated with a series of short exposure images and was in the order of ∼25 μrad, which corresponds to an atmospheric Fried parameter of ∼2 cm at 532 mm. This is similar to that experienced in a freespace link over 144 km on the Canary islands^{1} and represents a worst case scenario in a satellite communication experiment through the atmosphere. Note that the angle of arrival fluctuations were significantly smaller than the detector’s field of view; the fluctuation of detected count rates visible in Fig. 2 stem from beam wander at the aperture of the receiver telescope.
Experimental visibilities
To verify the integrity of the atmospheric quantum communication channel for hyperentangled photons, we first assessed experimental twophoton polarization correlation:
where N_{i,j} denotes the number of coincidence counts between Alice and Bob’s SPAD detectors (i, j∈{1, 0}). We define the experimental visibility V in the superposition basis as the maximum correlation observed while scanning the phase of Alice’s measurement basis +φ/−φ and keeping Bob’s measurement setup in the linear +45°/−45° polarization basis, that is, V=max_{θ}(E(φ(θ))). The scan over the polarization correlations depicted in Fig. 3 exhibited a maximum value of . The correlation in the linear H/V measurement basis was . Note that the H/V visibility is limited almost exclusively due to accidental coincidences.
Similarly, with the transfer setup inserted in both measurement modules, we observed Franson interference with a visibility of (Fig. 3). To verify that the high visibility is due to twophoton energy–time entanglement, and not singlephoton interference of photons A and B independently, we removed the transfer setup in Bob’s detection module. In this case, the measurement outcomes were completely uncorrelated, irrespective of φ(θ), since the polarizationdependent delay exceeded the coherence time of the SPDC photons. This is indicated by the straight line in Fig. 3.
Lower bounds on entanglement
The experimental visibilities establish lower bounds of 0.978±0.0015 and 0.912±0.006 on the concurrence^{57} in the polarization space and energy–time subspace, respectively (see ‘Methods’ section). These values correspond to respective minimum values of 0.940±0.004 and 0.776±0.014 ebits of entanglement of formation.
In the ‘Methods’ section, we use these values to establish a lower bound for the Bellstate fidelity of the hyperentangled state of the combined system (see also Supplementary Discussion). We achieve this by formulating this lower bound as a semidefinite programming (SDP) problem, in which we minimize the fourdimensional concurrence and fidelity to a fourdimensional Bell state over all possible states in the combined Hilbert space that satisfy the experimentally observed subspace concurrences. We obtain lower bounds of 1.4671 ebits of entanglement of formation and a Bellstate fidelity of 0.9419, thus certifying fourdimensional entanglement^{58}.
Discussion
We have distributed hyperentangled photon pairs via an intracity freespace link under conditions of strong atmospheric turbulence. Despite the severe wavefront distortions, we observed a high twophoton detection rate of ∼20 kcps over a link distance of 1.2 km. In a series of experiments, we independently observed highvisibility twophoton interference in the twodimensional polarization state space and a twodimensional energy–time subspace. These visibilities are sufficient to certify entanglement in both subspaces individually, and, for the first time, the coherent transmission of genuine highdimensional quantum entanglement via a realworld freespace link. While the transmission of polarizationentangled photons has been studied in a number of previous field trials, our results now demonstrate the feasibility of exploiting energy–time/polarization hyperentanglement in realworld link conditions with strong atmospheric turbulence.
Our analysis of interference in the energy–time DOF relies on an unbalanced polarization interferometer that coherently couples the polarization space with a twodimensional energy–time subspace. The current approach of mapping the timebin entanglement to the polarization DOF is of course intrinsically limited to accessing twodimensional subspaces of the highdimensional energy–time space. As recent experiments have clearly shown^{59,60}, the potential dimensionality of energy–time entanglement is orders of magnitudes larger. In fact, theoretically, it should only be limited by the achievable number of time bins within the coherence time of the pump laser. The main challenge remains the implementation of superposition measurements, where a single calcite is inherently limited to two dimensions. Future setups for freespace experiments could use several delay lines, or a variable delay line^{25}, to greatly increase the dimensionality and with it the resistance to inevitable background noise.
Critically, the energy–time to polarization transfer setup can be understood as an implementation of a singlephoton twoqubit operation^{37}, which can be exploited in, for example, hyperentanglementassisted Bellstate measurements and efficient entanglement purification schemes^{47,48,49,61}. To fully benefit from hyperentanglement in such applications, the delay between early and late photon arrival times will have to be directly resolved by the detectors. The main challenge therein lies in maintaining a constant phase relation between the long and short arms of the unbalanced interferometers for distorted input beams with a wide range of angles of incidence. However, such freespace compatible timebin analysers have recently been demonstrated in refs 34, 35, where the issue was ingeniously tackled via the implementation of a 4f imaging system in the long arm of the interferometer.
The coherent transmission of quantum information embedded in a genuine highdimensional state space under realworld link conditions represents an important step towards longdistance quantum communications with more complex quantum systems and could play a key role in the implementation of advanced quantum information processing protocols in the future. A large quantum state space not only allows for larger information capacity in quantum communication links, as well as devising quantum communication schemes with more resilience against noise and improved security against eavesdroppers, but also allows for more diverse types of nonclassical correlations, which could prove vital in addressing technological challenges on the path towards globalscale quantum networks, as well as fundamental physics experiments.
Since polarizationentangled photon sources based on SPDC quite naturally exhibit energy–time entanglement when pumped with a continuouswave pump laser, the approach can readily be implemented with existing sources and proposals for satellitelink experiments with polarizationentangled photons without need for additional critical hardware^{4,62,63,64}. The additional possibility of analysing energy–time entanglement could provide a platform for entirely new fundamental physics experiments with longdistance satellite links, such as the evaluation of models for gravityinduced wave function collapse^{65} or quantum information processing in a relativistic framework^{66}. Highdimensional energy–time entangled states can also be considered as a natural candidate for applications in quantumenhanced clock synchronization protocols^{67}, and could allow for significant gains in performance by employing other quantum features, such as nonlocal cancellation of dispersion^{68}. We also believe that our results will motivate both further theoretical research into energy–time entanglement experiments conceivable at relativistic scenarios with satellite links, as well as experimental research into the exploitation of hyperentanglement in longdistance quantum communications.
Methods
Hyperentangled photon source
The hyperentangled photon source was based on type0 SPDC in a periodically poled KTiOPO_{4} (ppKTP) crystal. The ppKTP crystal was bidirectionally pumped inside a polarization Sagnac interferometer^{52,53} and generated polarizationentangled photon pairs with centre wavelengths λ_{A}∼840 nm and λ_{B}∼780 nm. Photons A and B were separated using a dichroic mirror and coupled into optical singlemode fibres. For a pump power of 400 μW incident on the crystal, we detected a pair rate of of R^{(2)}∼84 kcps and singles rates of and directly after the source’s singlemode fibres. This corresponds to a normalized detected pair rate of 200 kcps mW^{−1} and a detected spectral brightness of 100 kcps mW^{−1} nm^{−1}. Without correcting for background counts, losses or detection inefficiency, we measure an average coincidencetosingles ratio .
The quasiphase matching condition in the 20mmlong ppKTP crystal^{69} resulted in a spectral bandwidth of Δλ∼2 nm, which corresponds to a twophoton coherence time of . The emission time of a photon pair is uncertain within the significantly longer coherence time of the continuouswave grating stabilized pump laser diode ≳100 ns, such that the biphoton state is in a superposition of possible pairemission times (see also Supplementary Fig. 1), that is, entangled in the energy–time DOF^{50}.
Energy–time visibility measurement
We employed a variant of the original Franson scheme^{50,51} with unbalanced polarization interferometers to assess the coherence of the energy–time state. The polarization interferometers were implemented with birefringent calcite crystals, which introduced a polarizationdependent time shift τ (Fig. 4). The particular choice of delay defines a twodimensional subspace (of the intrinsically continuousvariable energy–time space) spanned by the timedelayed basis states and . Since this delay is significantly shorter than the timing resolution of the detectors, our experimental results can be understood as averages over a larger state space in the energy–time domain. The maximally entangled Bell state in this subspace reads:
In the Supplementary Discussion (see also ref. 54), we show how the transfer setup in combination with polarization entanglement is used to probe the experimental density matrix in the energy–time subspace. After introducing a polarizationdependent time shift for Alice and Bob, the visibility of polarization measurements in the superposition basis is determined by the offdiagonal coherence terms via:
The total state space accessed in our experiment thus comprises the twodimensional polarization space and an effectively twodimensional energy–time subspace. The hyperentangled state of the total system can be expressed as a maximally entangled state in four dimensions:
with basis vectors . For more details, refer to the the Supplementary Discussion.
Certification of entanglement
In ref. 57, easily computable lower bounds for the concurrence of mixed states that have an experimental implementation were derived:
where ρ is the density matrix in the twodimensional subspace. In the Supplementary Discussion, we show the concurrence can be related to the experimental polarization space and energy–time visibilities via:
Note that the bound on the energy–time concurrence involves the additional assumption that there is no phase relationship between accidental coincidence that occur in time bins separated by more than the coherence time. We believe that, while this assumption precludes a certification of entanglement that meets the requirements for quantum cryptography, it is completely justified for our proofofconcept experiment. This also agrees with our experimental observation that scanning the phase of the entangled state in the source had no effect on the singlephoton coherence.
With the experimentally obtained lower bounds for and at hand, we calculate a lower bound for the concurrence of the global state by solving the following convex optimization problem: a minimization of the function that defines a lower bound for the concurrence, over all states ρ acting on a fourdimensional Hilbert space such that the concurrence of the reduced states in twodimensional subspaces satisfy the constraints of being lower bounded by the values and . As demonstrated in the Supplementary Discussion, this convex optimization problem has a SDP characterization and satisfies the condition of strong duality. Hence, the obtained lower bound of has an analytical character.
Another useful measure of entanglement is the entanglement of formation E_{oF}(ρ), which represents the minimal number of maximally entangled bits (ebits) required to produce ρ via an arbitrary local operations and classical communication procedure. It can be shown^{70} that the entanglement of formation is lower bounded by the concurrence according to:
Hence, from the lower bound for the concurrence it is possible to calculate a lower bound of for the entanglement of formation, which is sufficient to certify threedimensional bipartite entanglement^{70}.
By adapting the objective function of our SDP from the concurrence to the fidelity to the maximally entangled fourdimensional state, it is possible to lower bound the latter quantity by performing a minimization over the same variable and same constraints. As shown in the Supplementary Discussion, this second SDP also satisfies strong duality and provides the analytical bound of , which certifies fourdimensional bipartite entanglement^{58}.
Data availability
Data supporting our experimental findings are available from the corresponding authors on reasonable request.
Additional information
How to cite this article: Steinlechner, F. et al. Distribution of highdimensional entanglement via an intracity freespace link. Nat. Commun. 8, 15971 doi: 10.1038/ncomms15971 (2017).
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
We thank Johannes Handsteiner, Dominik Rauch and Sören Wengerowsky for their support in setting up the experiment. We also thank Mario Krenn and Sven Ramelow for helpful conversations and comments on the initial draft of the manuscript. We also thank the Bundesimmobiliengesellschaft (BIG) for providing the room for our receiving station. Financial support from the Austrian Research Promotion Agency (FFG)—Agentur für Luft—und Raumfahrt (FFGALR contract 844360), the European Space Agency (ESA contract 4000112591/14/NL/US), the Austrian Science Fund (FWF) through (P24621N27) and the START project (Y879N27), as well as the Austrian Academy of Sciences is gratefully acknowledged.
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Author notes
 Fabian Steinlechner
 & Sebastian Ecker
These authors contributed equally to this work.
Affiliations
Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A1090 Vienna, Austria
 Fabian Steinlechner
 , Sebastian Ecker
 , Matthias Fink
 , Bo Liu
 , Jessica Bavaresco
 , Marcus Huber
 , Thomas Scheidl
 & Rupert Ursin
School of Computer, NUDT, Changsha 410073, China
 Bo Liu
Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, A1090 Vienna, Austria
 Rupert Ursin
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Contributions
F.S. conceived the experiment. S.E. designed and developed the entangled photon source and local detection module under the supervision of F.S. M.F. designed and developed the freespace link and receiver optics with help from T.S. F.S., S.E. and M.F. performed the experiment under the guidance of R.U. M.F. and B.L. designed the coincidence tracking software and processed detection events. F.S., M.F. and S.E. analysed the experimental data. F.S., M.F., T.S., S.E. and R.U. discussed and evaluated the experimental results. M.H. and J.B. established the bounds on highdimensional entanglement and provided theory support. F.S. wrote a first draft and all authors contributed to the final version of the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding authors
Correspondence to Fabian Steinlechner or Rupert Ursin.
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