High-dimensional quantum communication: benefits, progress, and future challenges
Daniele Cozzolino, Beatrice Da Lio, Davide Bacco, Leif Katsuo Oxenløwe
HHigh-dimensional quantum communication: benefits, progress andfuture challenges
Daniele Cozzolino, Beatrice Da Lio, Davide Bacco*, and Leif Katsuo Oxenløwe
SPOC, DTU Fotonik, Technical University of Denmark, Kgs. Lyngby 2800, Denmark ∗ [email protected] October 17, 2019
Abstract
In recent years, there has been a rising interest in high-dimensional quantum states and their impact onquantum communication. Indeed, the availability of an enlarged Hilbert space offers multiple advan-tages, from larger information capacity and increased noise resilience, to novel fundamental researchpossibilities in quantum physics. Multiple photonic degrees of freedom have been explored to generatehigh-dimensional quantum states, both with bulk optics and integrated photonics. Furthermore, thesequantum states have been propagated through various channels, e.g. free-space links, single-mode,multicore, and multimode fibers and also aquatic channels, experimentally demonstrating the theoret-ical advantages over two-dimensional systems. Here, we review the state of the art on the generation,the propagation and the detection of high-dimensional quantum states.
The advent of quantum information has strongly influ-enced modern technological progress. Intense researchactivities have been carried out in the last two decadeson such field, producing outstanding results, e.g. inquantum computing [1–5], communication [6–9] andsimulation [10–14]. A qubit , the quantum counterpartof the classical bit, is a two-level quantum system andconstitutes the elementary unit of quantum informa-tion. Qubit manipulation and control were demand-ing tasks at first, but are now routinely used in quan-tum experiments. It is interesting to investigate quan-tum information in larger Hilbert spaces, either by in-creasing the number of qubits or by exploiting d -levelquantum systems, that is qudits . Great advantagesderive from accessing Hilbert spaces of higher dimen-sions, and whether it is better to increase the numberof qubits or to exploit qudits only depends on the par-ticular task to accomplish. In this review, we will dis-cuss the advantages derived by using high-dimensionalstates, i.e. qudits, focusing our attention on those re-lated to quantum communication . Nonetheless, high-dimensional quantum states have shown to yield im-provements in several other fields. Indeed, they allow Note that, throughout this review, entanglement among qu-dit will be discussed by considering only a maximum of 2 parti-cles involved. Moreover, certification of high-dimensional entan-glement is not discussed throughout the text, but an extensivereview on certification methods and experiments is reported inRef. [15] to achieve increased sensitivity in quantum imagingschemes [16], they can boost the transport efficiencyof biological compounds [17], they constitute richer re-sources for quantum simulation [18, 19], they lead tohigher efficiencies in quantum computing [20–23] andclock synchronization [24], and they can be beneficialin quantum metrology applications [25]. Moreover,breakthrough experiments studying quantum informa-tion memories have been performed by using high-dimensional states [26–28].Throughout the review, the physical d -level systems weare going to refer to are photons, but quantum infor-mation processing with other high-dimensional physi-cal systems is also possible [29,30]. In the first part, wediscuss high-dimensional quantum states as a resourceand we summarize the benefit derived by their exploita-tion. The second and third parts constitute the core ofthis work. In the former, we summarize experimentalsources of qudits, either bulk or integrated. In the lat-ter, we present a compendium on quantum communi-cation experiments with high-dimensional states, orga-nized according to the link adopted, that is free-space,fiber or underwater links. Finally, we highlight possibleperspectives for high-dimensional quantum communi-cation and raise several questions for future investiga-tions.1 a r X i v : . [ qu a n t - ph ] O c t Enlarging Hilbert spaces: the morethe better
Qubits are the basic quantum information units andare described by a basis of two orthonormal vec-tor states, {| (cid:105) , | (cid:105)} , corresponding to the classicalbits 0 and 1 respectively. Conversely, a qudit is aquantum system that is not constrained into a two-dimensional space and that, in principle, can have anyinteger number d of levels. However, besides concep-tual limitations– e.g. , how large has to be a quantumstate before exiting the quantum realm?– current ex-perimental devices pose an upper bound on the num-ber of dimensions that can be coherently controlled. Inthis section, we are going to highlight the advantagesoffered by such high-dimensional systems, presentingboth theoretical and experimental results. The first and rather clear advantage offered by quditsis the increased information capacity per quantum sys-tem. For example, by using high-dimensional stateswith d = 4 ( ququarts ), 2 bits of information can beencoded: | (cid:105) = 00 , | (cid:105) = 01 , | (cid:105) = 10 and | (cid:105) = 11 .A quantitative measure of the larger information ca-pacity is given by the relation log d , which returns thenumber of classical bits (or qubits) needed to encodethe same amount of information [31]. Additionally,high-dimensional entangled states yield a larger chan-nel capacity, i.e. the amount of information reliablytransmitted over a communication channel. Entangle-ment was predicted by A. Einstein, B. Y. Podolsky andN. Rosen [32] and causes quantum non-local correla-tions that cannot be devised by any local theory [33].In Refs. [34–36], it has been experimentally demon-strated how bipartite entangled qudit can beat the clas-sical channel capacity, with (Refs. [34, 35]) or without(Ref. [36]) superdense coding schemes. Nonetheless,there exists a capacity limit for direct communicationbetween two parties, the so-called PLOB-repeaterlessbound [37]. However, a recent work by D. Miller [38]shows how the PLOB bound can be surpassed usingerror-corrected qudit repeaters, by analyzing differentparameter regimes. Along with the increased information capacity, high-dimensional quantum states own a very important fea-ture for quantum communication, that is they are morerobust to noise, either if it is environmental or derivedfrom eavesdropping attacks. Indeed, the security ofa quantum channel, which is guaranteed by quantumphysical laws, is the cornerstone for sharing encryptedrandom keys (quantum key distribution– QKD), butalso for general quantum communication protocols.
Distance [km] Q BE R d=2d=4d=8d=16 Figure 1:
Maximum error tolerance for a positivesecret key rate as a function of the distance anddifferent dimensions.
The curves have been derived byconsidering a single-photon d -dimensional BB84 protocol,using ideal detectors affected only by the dark count prob-ability ( p d = 10 − per detector) and in case of coherentattacks [39, 40]. As attenuation, we consider the standardsingle-mode fiber parameter α = 0 . dB/km and we as-sumed to have d detectors to measure d states simultane-ously. Each curve identifies a region within which a positivesecret key rate can be extracted. The maximum attainabletransmission distance decreases by increasing the state di-mension d , indeed the greater d is the more the states aresensitive to the dark counts of the detectors. The security of a settled quantum link is ensured byhaving the quantum bit error rate (QBER), i.e. the ra-tio of an error rate to the overall received rate, below acertain threshold. In the case of qubit-based protocols,the threshold value has been proven to be 11% againstthe more general coherent attacks and by using twomutually unbiased bases (MUBs) in one-way reconcili-ation [41]. The higher resilience to noise sources ownedby qudits has been shown in Refs. [39, 40, 42], wherethe information gained by a potential eavesdropper,Eve, performing coherent attacks is calculated bothconsidering the use of and d + 1 MUBs. As a re-sult, it has been demonstrated that the robustness tonoise of qudits increases with their dimension d , thatis QBERs threshold values that ensure secure commu-nication increase. For instance, for d = 4 and d = 8 ,the thresholds are 18.93% and 24.70% respectively, byusing MUBs [39, 40]. Such higher noise tolerance hasalso implications on the final secret key rate. Indeed,for fixed noise level the secret key rate increase withthe Hilbert space dimensions. In Figure 1, we showthe maximum acceptable error rate to generate a pos-itive secret key rate as a function of the distance andfor different qudit dimensions. The curves are derivedfrom Refs. [39,40] and refer to an ideal system perform-ing a single-photon d -dimensional BB84 protocol [43],2sing ideal detectors affected only by the dark countprobability ( p d ) and assuming coherent attacks. Foreach dimension, a region within which a positive secretkey rate can be extracted is identified. Besides, theachievable transmission distance decreases by increas-ing dimensions and suggests qubit protocols to reachthe longest distance. However, in a practical scenario,the actual advantages of high-dimensional states overqubits strongly depend on the particular physical im-plementation, which varies the operational constraintsrequired. Thus, there might be cases where high-dimensional states perform better than qubits also interms of transmission distance [44, 45].The higher noise resilience of qudits also has advan-tages if they are entangled. Indeed, it has been demon-strated that the security of the E91 protocol [46], gener-alized to the qudit case, is ensured with increased errorthresholds if the dimensions are increased as well. Sucha conclusion implies that the robustness of the quan-tum correlations is influenced by the dimension of theHilbert space. In Ref. [46] this result has been theoret-ically proven. In particular, the authors analyze howthe number of dimensions d and the number of entan-gled particles N affect the entanglement robustness fora phase damping and depolarizing channel. Althoughone might expect that the dependence of the entan-glement robustness on N and d could be similar (inboth cases the overall Hilbert space is enlarged), theyshow that the entanglement becomes more fragile byincreasing N (with d fixed), whereas it is more robustby increasing d (with N fixed) [47]. This conclusioncan be intuitively understood by considering an exam-ple. If the dimension d is fixed, by increasing N , thecomponents of the state, i.e. the number of photonsconstituting the final system ρ , are increasing. Duringthe transmission through a channel, the noise sourceswill act locally on every system, thus making the en-tanglement more fragile with the growth of N . Con-versely, if N is fixed, the influence of the noise sourcesin the channel on ρ will be more and more negligibleby increasing d , which leads to a more robust entangle-ment [47]. Hitherto, the only experiment demonstrat-ing the robustness of high-dimensional entangled stateshas been carried out in Vienna by S. Ecker and co-authors [48]. They certified entanglement either withtime-energy or with photonic orbital angular momen-tum (OAM) degrees of freedom (see Section 3) in differ-ent noise conditions, by changing the noise level duringthe measurements. As theoretically predicted, in bothcases the noise level threshold increases with the quditdimension. In particular, time-energy entangled pairswith dimension d = 10 can tolerate of noise inthe channel before the entanglement breaks and suchthreshold increases to when d = 80 . For photonpairs entangled in the OAM degree of freedom, instead,entanglement breaks with of noise in the channelwhen the system has dimension d = 2 , whereas with d = 7 value of of noise can be tolerated [48]. The backbone for the security of quantum communica-tion protocols is the no-cloning theorem, which statesthat an unknown quantum state cannot be perfectlycopied [49]. Although creating a perfect copy of anunknown quantum state is forbidden, it is possible tomake imperfect clones, each with fidelity– i.e. the over-lap between the initial state to be cloned and the clonedcopy– lower than one, where one corresponds to theperfect determination of the initial state [50,51]. If thecloning scheme maximizes the attainable fidelity of thecopied state, it is called optimal quantum cloning andif it does not depend on the initial state, it is said to beuniversal [52–54]. The most common cloning schemeproduces cloned systems- i.e. , all output states of thecloning machine- characterized by the same clonedstate and thus is called symmetric [52, 53]. Nonethe-less, asymmetric quantum cloning is also possible [55].For the symmetric cloning, given N copies of the initialstate and M > N imperfect cloned copies ( N → M ),the optimal cloning fidelity in a d -dimensional Hilbertspace is given by: F dclon ( N, M ) = M − N + N ( M + d ) M ( N + d ) (1)which reduces to F dclon = + d in the case → ,that is one input state and two imperfect copies [56,57].Thus, by increasing the input state dimension, thecloning fidelity decreases from the upper bound F clon =0 . for qubits to F ∞ clon = 0 . when d approaches in-finity. Such a feature clearly shows the benefit of high-dimensional states for quantum cryptography.To our knowledge, only two experiments testing opti-mal quantum cloning fidelities with qudits have beenperformed [58, 59]. They are both based on the sym-metrization method described in Refs. [60–62]. InRef. [58] an optimal quantum cloning → is car-ried out for ququarts, i.e. four-dimensional states, en-coded in the polarization and OAM degrees of freedomof a single-photon. In Ref. [59] instead, → uni-versal optimal quantum cloning is performed for arbi-trary input states with dimensions up to d = 7 and, toshow the enhanced robustness of the high-dimensionalstates, cloning attacks are performed on the BB84 pro-tocol in d = 7 dimensions. In 1935, the main concern Einstein, Podolsky andRosen had on quantum mechanics was the violationof the local realism principle, which the three scien-tists considered necessary [32]. Local realism assumesthat every object has physical properties that are priorto and independent from any possible measurementcarried out by an observer and that the causality isbounded by the speed of light, i.e. the special relativity3olds. However, in 1964, in his seminal work, John Bellshowed how specific quantum measurements on qubitscould not be explained by any local theory [33]. Thus,a violation of the notorious Bell’s inequality impliesthe impossibility to explain with local theories the cor-relations under investigation. Many experiments ob-taining such violation have been carried out and, veryrecently, also in a loophole-free manner [63–65]. Thegeneralization of the Bell’s inequality to a system withhigher dimensions was firstly studied by the pioneer-ing works of N. D. Mermin and A. Grag [66, 67], anda few years later by A. Peres [68]. In the early 2000sthe advantages of high-dimensional states for violationof local-realism were shown by D. Kaszlikowski et al. and T. Durt et al. [69, 70]. Indeed they showed, with ≤ d ≤ first [69] and with d up to later [70], howthe violation increases with the dimensions, indicatingenhanced robustness of the violation against the noise.Collins et al. [71] generalized the concept for a sys-tem of any dimensions d , obtaining an inequality oftenreferred to as CGLMP inequality. Moreover, a very im-portant result for fundamental research is reported inRef. [72]. Indeed, Vértesi and co-authors showed that,due to the higher local realism violation given by qu-dits, the detector efficiency required to close the detec-tion loophole decreases with dimensions. Very recently,W. Weiss et al. challenged the robustness of high-dimensional states to violate Bell-like inequalities in amore practical scenario [73]. In particular, they studiedthe impact of imperfections in state-preparation andmeasurement settings on the violation of generalizednonlocality tests. Interestingly, these imperfections af-fect the violation in a dimension-dependent manner.Thus, it is possible to identify noise thresholds, for eachdimension d , such that if exceeded the quantum-to-classical transition will emerge, making large systemsbehaving classically under Bell-like tests. Experimen-tal violation of generalized Bell inequalities have beenachieved with energy-time entangled qutrits ( d = 3 qu-dit) [74, 75], with entangled radial and angular degreesof freedom of Laguerre-Gauss (LG) modes for quditswith dimension ≤ d ≤ [76] and up to d = 12 withentangled qudits encoded in the OAM degree of free-dom [77]. The larger violation of Bell inequalities givesalso benefit on entanglement-based device-independentQKD protocols. Indeed, to establish secure quantumkey distribution, the randomness of the Bell measure-ments is needed. However, if the randomness is weak,above a certain threshold the communication cannotbe considered secure. In Ref. [78], the authors showhow the acceptable loss of randomness is significantlylarger for qudit systems.Performing typical communication protocols, likequantum teleportation or entanglement swapping, us-ing high-dimensional states is still an open experimen-tal challenge since they require a complete Bell statemeasurement. Indeed, it has been demonstrated inRefs. [79–82] that projection onto a high-dimensional Bell state, such that it would be unequivocally identi-fied, are unattainable with only linear optics elements.One way to work around this limitation is to use an-cillary photons. Two experiments proved the principlevery lately [83, 84]. In Ref. [83] two auxiliary entan-gled photons are exploited to carry out a deterministicthree-dimensional Bell state measurement, obtainingteleportation fidelities above 0.63 (surpassing the clas-sical limit 0.5). In Ref. [84], the authors draw a schemewhere high-dimensional quantum teleportation can berealized by using d − ancillary photons, with d beingthe Hilbert space dimension of the photon pair to beteleported. In their work, they realize qutrit telepor-tation by using only one ancillary photon, yielding ateleportation fidelity of 0.75. Communication complexity addresses problems on theamount of information that distributed parties needto share to accomplish a specific task [85, 86]. For in-stance, two parties, Alice and Bob, receive two inputs x and y respectively. They have to evaluate a cer-tain function f ( x, y ) without knowing which data thepartner received. To improve the success of the proto-col, before starting it, they are allowed to share classi-cally correlated random strings or any other local data.Communication complexity problems can be mainly di-vided into two branches: one investigates the amountof communication required to all the possible partiesto determine with certainty the value of f ( x, y ) ; theother investigate which is the highest probability thatthe parties get to the correct value of f ( x, y ) if onlya limited amount of communication is allowed. Froma quantum information perspective, the question thatarises is whether there are some advantages in termsof complexity by using quantum correlations, that isentangled resources, in place of the classically corre-lated data. The issue has been studied by the pioneer-ing works of H. Buhrman, H. Cleve and W. Van Dam[87, 88], who showed that the highest success probabil-ity to determine f ( x, y ) is given by P C = 0 . by usingclassically correlated resources and P Q = 0 . if Aliceand Bob share a maximally entangled pair of qubits.Thus, in a classical protocol 3 bits of information areneeded to compute f at least with the probability P C = P Q , whereas 2 bits are sufficient if the protocolis supported by non-classical correlations [87]. Lateron, this result has been generalized by Č. Brukner etal. [89, 90]. They showed that for every Bell’s inequal-ity, also in the case of high-dimensional systems, therealways exists a communication complexity problem forwhich an entangled assisted protocol is more efficientthan any classical one.A second approach to deal with communication com-plexity problems is by using quantum communicationinstead of classical communication. Indeed, many com-4unication tasks can be successfully performed, out-performing classical constraints in terms of communi-cation complexity problems, either by entangled as-sisted classical protocol or by sending single quan-tum systems [91–94]. In particular, experimental ev-idences in which the former performs better than thelatter [95,96] and vice-versa [97,98] can be found. How-ever, D. Martínez et al. have shown, both theoreti-cally and experimentally, that high-dimensional quan-tum communication outperforms classical protocols as-sisted by nonlocal correlation whenever dimensions are d ≥ [99]. Indeed, they demonstrated how dimen-sion six acts as a threshold to reveal the benefits ofquantum communication over implementations basedon the violations of CGLMP inequalities. For dimen-sions below six, both communication complexity prob-lem strategies are equally efficient, whereas for d ≥ they are not. Experimentally, they proved the state-ment by implementing qudits encoded in the lineartransverse momentum of single-photons up to dimen-sion d = 10 . Very recently, W. Kejin et al. demon-strated a communication complexity advantage givenby high-dimensional protocols based on the quantumswitch (a novel quantum resource which creates a co-herent superposition of the causal order of events) forcasually ordered protocols [100]. After having declaimed the advantages gained in quan-tum communication by using larger Hilbert spaces, aquestion naturally arises: how can a qudit be practi-cally implemented? The goal is to increase the avail-able dimensions to send more than 1 bit/photon fromone party (Alice) to another (Bob). To expand theHilbert space, different photonic degrees of freedom orcombination of them can be used. Within this sec-tion, we are going to discuss techniques and methodsadopted to control those degrees of freedom and gen-erate qudits.
Orbital angular momentum – The orbital angularmomentum (OAM) of light is one of the most fre-quently exploited photonic properties to generate high-dimensional quantum states. Indeed, photons carryingan OAM different from zero are characterized by a heli-cal phase factor e i(cid:96)φ , with φ being the azimuthal angleand (cid:96) the quantum number indicating the amount oforbital angular momentum (cid:96) (cid:126) carried by them. Since (cid:96) takes integer values and is unbounded, an arbitrarilylarge Hilbert space can be spanned. Thus, by prop-erly controlling (cid:96) , OAM offers a discrete basis to de-vise high-dimensional states. The optical fields de-scribing such photons present a topological phase sin- gularity at the beam axis, resulting in a characteristicring-shaped intensity pattern in the case of classicallight. Optical modes like LG modes [101] or circularbeams [102] carry a non-zero OAM, so progress on itsmanipulation is inevitably related to technological ad-vances on waveshaping devices. Several apparatuses,like cylindrical lenses [103], specially designed lasercavities [104], spiral phase plates [105] or integrateddevices [106, 107], can be used to reshape the wave-front of an initial Gaussian photon and thus creatingqudits encoded in OAM. More frequently experimen-tal realizations make use of two other devices: holo-grams or q -plates [108, 109]. Holograms can be con-sidered as diffracting gratings such that the first-orderdiffracted beam acquires a unique phase and amplitudepattern. Holographic patterns can be easily createdwith commercially available devices called spatial lightmodulators (SLMs). A very different concept under-lies q -plates, which apply suitable transformations onthe local polarization state of light to generate phaseshifts [109]. They are made of liquid crystals havingan azimuthal pattern around a central point and con-fined within two slabs. The topological charge q ofthe central singularity adds an OAM of (cid:126) q per pho-ton and it can be an integer or half-integer. Q -platesare responsible for the spin-to-orbital angular momen-tum conversion, namely the exact conversion withinthe same quantum system of the spin angular momen-tum (SAM), i.e. polarization, into OAM. Many ex-periments have been implemented either with SLMs or q -plates achieving quite important results [25,110–114].Notwithstanding, it is worth mentioning for our pur-poses the experiment carried out recently in Rome byT. Giordani et al. [115], which combines both SLM and q -plates to generate arbitrary qudits through quantumwalks [116]. Indeed, by controlling the walk’s dynamicsthrough convenient step-dependent coin operations, itis possible to steer the state of the walker towards thedesired final state. The experimental setup built forsuch a task is reported in Figure 2. They implementeda discrete-time quantum walk with n = 5 steps by us-ing the OAM degree of freedom with (cid:96) = ± , ± , ± to encode the walker state and the circular-polarizationstates {| R (cid:105) , | L (cid:105)} for the coin state. Five sets of half-and quarter-wave plates (HWP, QWP) are used to per-form arbitrary coin operators, while five q -plates areused to implement the shift operator, which movesthe walker conditionally to the coin state. A type-II periodically poled potassium titanyl phosphate (pp-KTP) crystal generates photons via parametric down-conversion. The two photons emitted are separatedby a polarizing beam-splitter (PBS) and then coupledto a single-mode fiber (SMF): only one photon under-goes the quantum walk dynamics, whereas the otheracts as a trigger. A first PBS sets the initial stateof the walker and coin as | ψ (cid:105) = | (cid:105) w ⊗ | + (cid:105) c . Theprotocol requires a final projection on the state | + (cid:105) c of the coin, which can be performed by a final PBS.5 Coin 2 Coin 3 Coin 4Coin 1 Coin 5Step 1 Step 2 Step 3 Step 4 Step 5
Input
Output -5 -3 -1 1 3 5 b APD1SMF SLMS1 S2 S3 SMF
Coin projection ۦ+|
S4 S5C1 C2 C3 C4 C5
Projection on arbitrary walker state ۦ𝜓|
QWP
HWP
Q-plate
PBSLens Iris
Optical elements
Laser APD2PPKTP c …… Figure 2:
Setup of the arbitrary qudits generation through quantum walks. (a) Schematic of the protocol:at each step, the coin operator is changed to have the desired walker state at the output. (b) Experimental setup. AppKTP source generates pairs of photons, which are then coupled to SMF. One photon acts as a trigger, while the otherone is prepared in the initial | ψ (cid:105) with a PBS and a polarization controller. Coin and shift operators are implementedwith a set of wave plates and q -plates respectively. The detection consists of a PBS, followed by an SLM, an SMF andavalanche photodiode detectors (APD). (c) Pictures of the OAM modes after the PBS for (from right to left) OAMeigenstate corresponding to (cid:96) = 5 ; balanced superposition of (cid:96) = ± ; balanced superposition of all OAM componentsinvolved in the quantum walk dynamics, i.e. (cid:96) = ± , ± , ± . Reprinted figure with permission from T. Giordani et al. ,Physical review letters, vol. 122, no. 2, p. 020503, 2019 [107]. Copyright (2019) by the American Physical Society. The OAM analysis is carried out by an SLM, allowingfor an arbitrary OAM superposition detection. Fiveclasses of qudit have been implemented: superpositionof large OAM states, spin-coherent states, balancedstates forming computational and Fourier bases, andrandom states. The average quantum fidelity obtainedis ¯ F = 0 . ± . , showing the correct implementa-tion of all the desired multilevel quantum states.Another very common method to generate photonswith non-zero OAM is to get photons directly from aspontaneous parametric down-conversion (SPDC) pro-cess in χ (2) materials, e.g. ppKTP or β -barium bo-rate (BBO) crystals. This technique is very suitableif photon pairs entangled in the OAM are intended tobe used [111]. Indeed, the conservation of the angu-lar momentum in an SPDC process (cid:126) (cid:96) p = (cid:126) (cid:96) + (cid:126) (cid:96) implies the generation of photon pairs with oppositeOAM quantum numbers (cid:126) (cid:96) = − (cid:126) (cid:96) , if the pumpingphotons have (cid:126) (cid:96) p = 0 , that is if they are in a Gaussianmode. Thus, the theoretical states produced by theSPDC process are expressed as: | Ψ (cid:105) = + ∞ (cid:88) (cid:96) = −∞ c (cid:96) | (cid:96) (cid:105) |− (cid:96) (cid:105) (2)where |± (cid:96) (cid:105) i are the photon states with OAM ± (cid:96) and c (cid:96) are complex probability amplitudes. Experimen-tal conditions impose boundaries on (cid:96) , i.e. on theHilbert space spanned, so that (cid:96) ∈ {− d, . . . , d } . Suit-able engineering of such photon sources allows the gen- eration of very interesting states as four-dimensionalBell states [117], qutrits Greenberger-Horne-Zeilinger(GHZ) states [118] or multi-photon entanglement inhigh dimensions [119]. In Ref. [120], a very novel ap-proach to generate OAM high-dimensional states isproposed. The authors carefully design an experimen-tal setup in such a way that high-dimensional entangledstates could be post-selected through Hong-Ou-Mandelinterference [121]. Time – The process of SPDC allows the generation ofqudits by also considering the time of photons emis-sion as a degree of freedom. In this way, time-energyand time-bin entangled qudits can be produced. Inthe former case, the emission times of the photon pairsare undetermined with an uncertainty ∆ t given by theHeisenberg uncertainty relation. The uncertainty ∆ t can be described in terms of the coherence time ofthe pumping laser t p , which is inversely related to therespective linewidth in the spectral domain. So, bypumping a crystal with a narrowband laser, longer co-herence time is achieved, thus allowing for an increaseduncertainty of the emission time of the photon pairsand giving rise to entanglement in the time-domain.The states generated by this process can be written as: | Ψ (cid:105) = d (cid:88) k =1 α k | k (cid:105) | k (cid:105) (3)where | k (cid:105) refers to a photon in the k -th time slot withinthe coherence time of the pump and α i is a complex6 igure 3: Time encoding schemes. (a) Time-energy en-tanglement. Within the coherence time of the pump t p ,photon pairs are generated with a temporal uncertainty ∆ t . Thus, by pumping with narrowband light, and in turnwidening the coherence time, it is possible to define timeslots, t , . . . , t n , in which the photon pairs are generated.The number of time slots define the Hilbert space dimen-sion and can be scanned with Franson interferometers. (b)Time-bin entanglement. In this case, the pump is pulsedand each pulse corresponds to a different time slot. (c)Time-bin qudit. Most frequently prepared by using attenu-ated pulses, time-bin qudits are very easily generated withoff-the-shelf equipment as intensity and phase modulators.In terms of bases, the states of the computational one areidentified by the pulse position, while those in the Fourierbasis are identified by the relative phases among the pulses. probability amplitude [48]. Entangled qudits gener-ated in this way can be measured by using unbalancedMach-Zehnder interferometers, also known as Fransoninterferometers [122].The generation of a time-bin entangled qudit differsonly slightly from the time-energy one. Indeed, in thelatter case, the discretization in different time stateshas to happen within the coherence length, whereas inthe former this condition is not required. To generatetime-bin entangled states, a pulsed pump is needed,so that each time state is identified by the pulse re-sponsible for the photon pair generation [123]. Thehigher the number of pump pulses taken into account,the larger is the Hilbert space spanned. In this case,as before, Franson interferometers or combination ofthem are needed to reveal the entanglement among thestates.Interestingly, time-bin qudits can be easily created byusing attenuated laser pulses (also called weak coher- ent pulses – WCPs). Indeed, standard off-the-shelfequipment is required to generate them, making themwidely adopted especially for QKD. Pulses in differenttime slots form the computational basis and, as be-fore, the higher the number of time slots, the higherthe dimension d is. The superposition among thesestates is devised by controlling the relative phase ofthe pulses. Figure 3 summarizes the time-encodingschemes we have discussed. Time encoding has beenexploited in a large number of experiments, mostly fo-cused on quantum communication and quantum cryp-tography [48, 123–126]. Frequency – Generating high-dimensional quantumstates in the frequency domain is also possible, al-though demanding with bulk optics instruments. In-deed, to our knowledge, not many experiments havebeen carried out, but many times this degree of free-dom has been used in integrated photonic devices, aswe are going to see in the following. Frequency entan-gled qudits can be generated starting from a parametricdown-conversion process. The two-photon state froman SPDC process can be described as: | Ψ (cid:105) = (cid:90) + ∞−∞ d s d i f ( ω s , ω i )ˆ a † s ( ω s )ˆ a † i ( ω s ) | (cid:105) (4)with ˆ a † ( ω ) being the creation operator at angular fre-quency ω , the subscripts s and i indicating the sig-nal and idler photons, respectively, and f ( ω s , ω i ) be-ing their joint spectral amplitude (JSA), which de-pends on the crystal and the pumping light. A cleverway to create frequency entangled qudits is proposedin Ref. [127]. The authors combined the photon pairstate as described in eq. (4) together with the Hong-Ou-Mandel interference. The obtained final state isexpressed by: | Φ( τ ) (cid:105) = 1 √N (cid:90) + ∞−∞ d s d i h ( ω s + ω i − ω p ) × δ ( ω s + ω i − ω p )(1 − e − i ( ω s − ω i ) τ ) × ˆ a † s ( ω s )ˆ a † i ( ω s ) | (cid:105) (5)where N is a normalization factor, τ is the adjustabletime delay between the two photons, h ( x ) is a func-tion dependent on the phase-matching condition, δ ( x ) is the Dirac delta function. As we can see from (5),the frequencies oscillate with peaks at ω s − ω i = 2 π/τ ,thus generating a frequency entangled qudit state. Path – One of the first degrees of freedom to be ex-ploited and manipulated for the generation of multi-level quantum systems has been the path. In Ref. [128], ˙Z ukowski and co-authors showed how combinations ofmultiport beam-splitters (BS) can be suitably engi-neered to have non-classical correlations in higher di-mension in path. Such a seminal study can be consid-ered of great importance, especially for integrated pho-tonic devices. Indeed, BSs implementation on a chip,either silicon or silica-based, is straightforward and this7ade it possible to develop integrated sources emit-ting entangled photon pairs up to dimension d = 15 ,as we are going to see in the next subsection [129].An innovative technique to generate entanglement inhigher dimensions by using path has been presentedin Ref. [130]. The authors show how two-photon ar-bitrary high-dimensional entanglement can be gener-ated by path identity. In particular, starting withseparable (non-entangled) photons, photon pairs arecreated in different crystals and their paths are over-lapped, producing several types of entanglement inhigh-dimensions. The authors achieve proper controlon which state generate by using modes and phaseshifters, showing the great flexibility of the method. Degrees of freedom composition – Every degree of free-dom analyzed offers practical advantages and con-straints. Nonetheless, a combination of them some-times can help to explore larger Hilbert spaces withfewer difficulties. For instance, although polariza-tion bases live in a two-dimensional Hilbert space,they can be combined with almost all the other de-grees of freedom. Indeed, noteworthy and funda-mental results have been achieved by considering hy-brid high-dimensional states. Some examples of hy-brid qudits are spin-orbit states (combination of po-larization and OAM) [58, 132–134], path-polarizationstates [135, 136], polarization-time states [124] andfrequency-path states [137].
The route towards the full deployment of quantumtechnologies resides in the capacity of creating iden-tical and replicable quantum devices. Integrated opti-cal sources offer immense advantages due to their in-trinsic scalability, high stability, and repeatable pro-duction process. However, not all degrees of freedomcan be efficiently manipulated on integrated circuits.For instance, the polarization of photons needs to be properly addressed in order for integrated chips to beable to carry two orthogonally polarized fields. Fur-thermore, the generation and propagation of OAMstates through waveguides is very demanding as well,but small steps towards the reliable on-chip transmis-sion and source integration of such states have beenproved [107, 138]. As previously mentioned, frequencyand path degrees of freedom are the two that can bemore easily controlled and manipulated on integrateddevices [139–141]. In the following, we are going tofocus and discuss in detail two very important and im-pressive experiments, which use path and frequency en-coded high-dimensional states respectively. The resultsobtained are fundamental to speed up the developmentof quantum technologies and to bridge the gap betweenthe classical ones.The first experiment involves path encoded qudits insilicon integrated platform [129]. Silicon quantum pho-tonics is a promising candidate to further develop in-tegrated quantum devices, as it offers intrinsic stabil-ity, high precision and integration with other classi-cal devices. The device allows the generation of high-dimensional entangled states with a controllable degreeof entanglement. Figure 4 shows a schematic of thechip design. Photons entangled over d spatial modesare generated by coherently pumping d different single-photon sources. In particular, 16 photon sources, emit-ting photon pairs by spontaneous four-wave mixingprocess (SFWM), are integrated [142]. Thus, an high-dimensional entangled state is created: | Ψ (cid:105) d = d − (cid:88) k =0 c k | k (cid:105) s | k (cid:105) i (6)where the qudit state | k (cid:105) is associated to a photon inthe k -th optical mode, the subscripts s and i standsfor signal and idler and the coefficients c k are com-plex probability amplitudes. Such coefficients can bechosen arbitrarily by changing the pump distributionof the sources and the relative phase of the mode. qudit analysisphoton routingpump divison generation a) b) Figure 4:
Diagram and picture of the PIC. (a) Circuit diagram. By coherently pumping all the 16 sources a photonpair is generated in superposition across 16 optical modes, producing a multidimensional bipartite entangled state. Thetwo photons, signal and idler, are routed through the chip by using asymmetric Mach-Zehnder interferometer (MZI)filters. By using triangular networks of MZIs, arbitrary local projective measurements are feasible. Photons are coupledoff-chip and detected by two superconducting nanowire detectors. (b) Picture of the device. igure 5: Experimental setup of frequency high-dimensional chip.
A mode-locked laser coupled to the integratedmicro-ring excites precisely a single resonance. Spontaneous four-wave mixing (SFWM) (see left inset) process generatesphoton pairs (signal and idler) spectrally symmetric in a quantum superposition of the frequency modes. Programmablefilters and a modulator were used for manipulating the quantum states, before the single-photon detector measurement.Reproduced with permission [131]. Copyright 2017, Nature.
This precise control is achieved with cascaded Mach-Zehnder interferometers (MZIs) at the input and phaseshifters on each optical mode. By uniformly pump-ing the sources, maximally entangled states can be ob-tained. On the same device, linear optical circuits al-low for the implementation of any local unitary trans-formation in d dimensions. The authors estimate theindistinguishability of the 16 sources by performing areverse Hong-Ou-Mandel interference and by calculat-ing the visibility of the fringes. All the visibilities ob-tained are higher than 0.90, being higher than 0.98 inmore than 80% of the cases. Quantum state tomogra-phies and certification on the system dimensionalityas well as the violation of generalized Bell’s inequal-ities (CGLMP) are performed. In addition, the au-thors studied unexplored quantum applications, whichare quantum randomness expansion and self-testing onmultidimensional states, thus showing exhaustively thepotentials of such an integrated device.The second experiment by M. Kues and colleagues [131]demonstrates the generation of high-dimensional fre-quency entangled states up to dimension d =
10. Thestates are obtained by pumping a micro-ring resonatorto provoke the SFWM process and hence generate pairsof photons in a superposition of multiple frequencymodes. In particular, a spectrally filtered mode-lockedlaser excites a single resonance of the micro-ring, pro-ducing pairs of correlated signal and idler photons spec-trally symmetric to the excitation field, as reportedin Figure 5. Thus, the quantum states are selectedand manipulated using commercially available telecom-munication programmable filters. The joint spectralintensity, describing the two-photon state’s frequencydistribution, can therefore be measured, and Bell-testmeasurements and quantum state tomography can becarried out [131]. Also, the authors have sent a two-dimensional frequency-entangled state through a 24.2 km long fiber, and they prove the correct propagationof such states through Bell’s inequality test.In this section, we have reviewed all the possibleplatforms and schemes capable to generate high-dimensional states to our knowledge. We have dividedthem into two different classes: bulk and integratedplatforms. The former constitutes the backbone of op-tics experiments and it is a very good approach forproof-of-principle experiments on quantum informa-tion and fundamental physics. However, due to its lackof scalability, it is not a good platform for advances inquantum technology, whereas the latter is more appro-priate. Integrated optics limitations are mostly relatedto the degrees of freedom that can be exploited andproperly controlled. Indeed, frequency and path arevery well suited for integrated platforms, but devicesable to manipulate and control with the same preci-sion other degrees of freedom, e.g. polarization andspatial modes, are still lacking. In terms of degrees offreedom, using the time to generate qudit states is aclever approach and it is also suitable for integrateddevices. However, increasing dimensions by using timelowers the repetition rate of the generated states andthis could be a non-trivial issue for technological ap-plications. Finally, although the OAM of light consti-tutes a natural basis for high-dimensional states, it isvery challenging to manipulate on integrated devices,thus it is mainly exploited in bulk optics experiments.Nonetheless, noteworthy results that might open thedoors to integrated devices exploiting OAM have beenachieved [106, 107, 138, 143].
Subsequently the generation of high-dimensional quan-tum states, this section regards the propagation of suchstates through a communication channel, e.g. , optical9 igure 6:
Setup of the intracity OAM distribution.
Schematic of the transmitter (left) with a heralded single-photon source and state preparation. Alice prepares the quantum states using a polarizing beam splitter (PBS), waveplates (WP), and a q-plate (QP). The signal and idler photons are recombined on a dichroic mirror (DM) before thepropagation on the free-space channel. Two telescopes comprised of lenses with focal lengths of f = 75 mm, f and f =400 mm (diameter of mm), and f = 50 mm are used to enlarge and collect the beam, minimizing itsdivergence through the 0.3 km link. Bob, receiver (right), performs projective measurements on the quantum states andrecords the coincidences between signal and idler photons with detectors D and D and time-tagger unit. Examples ofexperimentally reconstructed polarization distributions of a structured mode using a continuous wave laser prepared byAlice (top left) and measured by Bob (bottom right) are shown in the insets. Legend: ppKTP, periodically poled KTPcrystal; LP: long-pass filter; BP: band-pass filter. Map data: Google Maps, 2016. Reprinted figure with permission fromRef. [140], The Optical Society. fiber, free-space or underwater links. Even though pro-found improvements have been made to generate andmanipulate high dimensional quantum states, their re-liable transmission, the cornerstone for future quantumnetworks, remains an open challenge. The distribution of quantum states between distantparties, connected by a free-space link, is one of themain technological challenges towards a global-scalequantum Internet. Several proof-of-concept studieshave already demonstrated the high-fidelity transmis-sion of entangled photons up to 143 km for a groundlink [144], 1200 km with a satellite link for quantumcommunication on a global scale [145], and transmis-sion of attenuate laser up to Global Navigation Satel-lite Orbit [146,147]. However, until very recently, mostof the demonstrations used a bipartite binary photonicsystem while only a few took advantage of qudit en-coding. As described in the previous sections, thegain offered by high-dimensional systems can be ap-plied to multiple areas. In particular, for communica-tion purposes, the ability to encode more informationin a single-photon is a peculiar characteristic for push-ing the entire field.As introduced in the generation section, a straightfor-ward way to generate high-dimensional quantum statesis to use space encoding. Spatial modes, e.g.
LG, canbe adopted to implement high-dimensional quantumstates without any constraint on the Hilbert space size. A first example is represented by the correct genera-tion and detection of maximally entangled qutrits forquantum key distribution accomplished in an opticaltable [148]. S. Gröblacher et al. used a parametricdown-conversion scheme to generate pairs of photonsentangled in their orbital angular momentum. Thequantum key is then encoded in different LG modescreated by tunable phase holograms. The detection isaccomplished with multiple beam splitters and holo-grams which allow projecting the quantum states indifferent bases.Furthermore, thanks to their small divergence angleand intrinsic rotational symmetry, LG modes are suit-able for long-distance free-space optical communica-tion. However, there are practical limitations on thefinite size of apertures in a realistic system, which lim-its the dimensions of the Hilbert space that can be usedfor communication; indeed, in free-space links, beamdivergence must be taken into account [149].The main example of spatial encoded qudits over a free-space channel is represented by the work of A. Sit andcolleagues [150], where they proved the correct trans-mission of LG modes (of dimension four) in a 300 me-ters intracity air link in Ottawa, as reported in Fig-ure 6. Their transmitter unit is composed of a paramet-ric down-conversion single-photon source, where non-degenerate wavelengths are selected for the signal andidler photons. The signal photon is subsequently usedfor key encoding by employing a q-plate combined withwave plates to prepare the ququarts in the mutuallyunbiased bases. At the receiver side, the signal pho-10 igure 7:
Setup of the hyper-entangled distribution in Vienna.
A hyper-entangled photon source was locatedin a laboratory at the IQOQI Vienna. The source utilized SPDC crystal, which was placed at the center of a Sagnacinterferometer and pumped with a continuous-wave laser diode (LD) to obtain polarization/energy–time hyper-entangledphoton pairs. Photon A was sent to Alice at IQOQI using a short fiber link, while photon B was guided to a transmittertelescope on the roof of the institute and sent to Bob at the BOKU via a 1.2-km-long free-space link. At Bob, thephotons were collected using a large-aperture telephoto objective with a focal length of 400 mm. The beacon laser wasseparated from the hyper-entangled photons using a dichroic mirror and focused onto a CCD image sensor to maintainlink alignment and to monitor atmospheric turbulence. Alice’s and Bob’s analyzer modules allowed for measurements inthe polarization or energy–time basis. Single-photon detection events were recorded with a GPS-disciplined time taggingunit (TTU) and stored on local hard drives for post-processing. Bob’s measurement data were streamed to Alice via aclassical WiFi link to identify photon pairs in real time. Map data: Google Maps, 2017. Reprinted with permission fromRef. [124], Nature Publishing group. ton is projected in one of the states, while the idlerphoton is measured by a single-photon detector. Co-incidence counts are registered and a key is extractedbetween Alice and Bob. To be noted that the receiverallows measuring only one of the quantum states atthe time, limiting both the receiver efficiency and fu-ture applications. In other words, since Alice can pre-pare one of the four quantum states in each basis andBob does not implement an optimal quantum receiver,the number of actual sifted bits is decreased by theprobability of choosing the same symbol both for Aliceand for Bob. This setup configuration ends up lim-iting the range of applications: some quantum proto-cols, such as complete device-independent demonstra-tions and loophole-free measurements for non-localitytests, require to measure all the possible outcomes atthe same time. Indeed, for a D dimensional Hilbertspace detection loophole-free test, D + 1 outcomes arerequired to strictly violate Bell’s inequalities. How-ever, projecting on N < D outcomes, only a subsetof all emitted pairs are measured, introducing possibleclassical correlation [129].Other examples, from the same authors, of high-dimensional protocols based on OAM modes are re-ported in [151], where many quantum protocols (BB84,Chau15, Singapore) are studied in different dimen- Chau15 [152] is a new proposal for qubit-like qudit of proto-cols. In particular, it requires fewer resources, in terms of statepreparation, compared to a full high-dimensional protocol; Sin-gapore protocol instead, implements a specific POVM operatorallowing a full tomography of the quantum states [153]. sions, from 2 to 8, demonstrating an ideal range ofapplication depending on the noise and on the systemenvironment.Besides LG modes, other spatial modes can be used forqudit encoding. S. Etcheverry et al. [154] uses lineartransverse momentum of weak coherent pulses as thedegree of freedom for encoding a 16-dimensional quditstate. At Alice and Bob’s sites, the quantum statesspanning the mutually unbiased bases are randomlyproduced with the help of a spatial light modulator,dynamically introducing relative phase shifts betweenthe paths. The stability of the system, over a few me-ter link, is measured for several hours [154]. Again,the receiver implemented in this experiment allows pro-jection one quantum state at the time, penalizing theoverall efficiency as discussed above.Other degrees of freedom commonly adopted in free-space links are time-energy and polarization encod-ing. Both time-energy and polarization are not muchaffected by the effects of free-space propagation, i.e. beam wandering and scintillation. In this directionF. Steinlechner et al. have demonstrated the cor-rect propagation of high-dimensional entangled pho-tons, exploiting hyper-entanglement between polariza-tion and time-energy, over a 1.2 km free-space link inVienna [124]. The source of hyper-entangled photonswas based on the SPDC process using standard opti-cal components and a long coherence time of the opti-cal pump. Both transmitter and receiver are equippedwith a polarization analyzer module and an unbalanced11 igure 8:
Setup of the high-dimensional chip to chip experiment. (a) , (b) Schematic of the integrated compo-nents for the high-dimensional QKD protocol based on the multicore fiber (MCF). Labels are: variable optical attenuator(VOA); Mach-Zehnder interferometer (MZI), phases ( ϕ ); single photon detectors (SPD). (c) , (d) pictures of the inte-grated photonic chips used in the experiment. (e) Cross section of the MCF showing the four cores used. polarization interferometer (based on calcite crystal)to convert the energy-time degree of freedom to polar-ization. Further details on the setup are reported inFigure 7. The experimental data showed lower boundvisibilities of 98% and 91% for polarization and energy-time respectively, corresponding to a minimum valueof 0.94 and 0.77 ebits (entangled bits) of entanglementof formation [124]. Furthermore, by considering thecombined hyper-entangled state, they obtained 1.46 ofebits of entanglement of formation [124] and a Bell-state fidelity of 0.94, certifying the dimensionality ofthe system to d = 4 .These experiments certify the capability of employinghigh-dimensional quantum states, encoded in multipledegrees of freedom, for free-space links. Thus, qu-dits could be useful for future quantum communica-tion links, such as satellites to Earth connections andsatellite to satellite communication. Besides the transmission of high-dimensional quantumstates in a free-space link, fiber links are the most at-tractive channel since the infrastructure is already inplace and furthermore optical fiber communication iscommonly used in our lives, e.g. for the Internet back-bone. Different fibers can be exploited for the prop-agation of high-dimensional quantum states: single-mode fibers (SMFs- most used and deployed), multi-mode fibers (MMFs- including few-mode fibers, andhigher-order modes fibers) and multicore fibers (MCFs-special fibers with more than one core within the samecladding). Depending on the application, an optimalsolution can be found in different fiber types. As anexample, in data centers, where space is limited and aremarkably high number of connections are required,the footprint of the fibers is very important and hencethe use of MCFs is a very attractive solution.
Single-mode fibers – Regarding the distribution of high-dimensional quantum states through SMFs, a first ex- ample is represented by high-dimensional time-bin en-coding for high rate quantum key distribution proto-cols. N. T. Islam et al. in Ref. [126], demonstratedhow high-dimensional quantum states can be used togenerate a very high secret key rate (26 Mbit/s), ina one-way protocol with 4 dB channel loss (emulatedwith a variable attenuator) under the general coherentattacks scenario.However, one-way protocols and technological imper-fections of QKD devices open the possibility of suc-cessful eavesdropping methods, like side-channel at-tacks [155]. Device-independent (DI) protocols ormeasurement-device-independent protocols (MDI) canovercome these limitations [156]. In this direction, twoproof-of-concept of high-dimensional QKD in an MDIscheme has been proposed [157] and proved [158].Another appropriate approach to distributing quditsthrough fibers is represented by a time-energy encod-ing. T. Zhong et al. demonstrated in Ref. [159] thecorrect propagation over a 20 km link of time-energyqudits up to dimension d = Multicore fibers – Multicore fibers present well-performant characteristics: they offer low losses (com-parable with the standard single-mode fibers) andlow cross-talk between cores (fundamental for reliabletransmission of the qudits) [161]. Previous experi-ments already demonstrated the capability of trans-ferring spatial modes of light with high-fidelity up to adimension equal to four [162–164].In particular, Y. Ding and coauthors used two sili-con photonic platforms, connected by a 3 m MCF, forpreparing and measuring the quantum states [162], asreported in Figure 8. The qudits are path-encodedin the cores of the MCF and prepared by usingintegrated Mach-Zehnder interferometers and phaseshifters, which allow creating at least two mutually un-biased bases necessary for a QKD protocol. Weak co-herent pulses are injected into Alice’s chip, and decoy-state technique is applied to avoid the photon numbersplitting attack. The QBER is measured to be belowthe coherent attacks limit for several minutes, provingthe correct propagation of a four-dimensional quan-tum state over the MCF. This work plays an impor-tant role in future quantum networks since it combinestwo very attractive solutions for the generation and thetransmission of high-dimensional quantum states: sili-con photonics and multicore fibers. However, the mainchallenge in these fibers is to maintain the phase sta-bility between different cores, required to preserve thecoherence of the superposition states– when the infor-mation is encoded in the relative phase between thecores the result is a long fiber interferometer. A possi-ble solution is the use of phase-locked loop systems ableto compensate for phase drifts in real-time, as provedin [164], or the use of reference-frame independent pro-tocols [165].
Multimode fibers – A final approach is represented bythe use of multimode fibers. Traditionally, fibers havebeen engineered to be single-mode, since this is gener-ally advantageous in optical communication. Despitethis, other interesting applications are enabled by mul-timode fibers [166].For instance, recently D. Cozzolino and colleagues re-ported the first demonstration of a high-dimensionalquantum state, encoded in a superposition of orbitalangular momentum modes ( l = ±
6, and l = ± ),transmitted over a 1.2 km air-core fiber [44]. Weakcoherent pulses are prepared in a four-dimensionalHilbert space by utilizing bulk and fiber optics. Quditsare then propagated through the OAM-carrying fiberand measured in two mutually unbiased bases. Themeasurement of the OAM states is realized by imple-menting a free-space OAM sorter followed by projectivemeasurements, allowing simultaneous measurements ofall the states within the same basis. Different QKDprotocols are implemented to test the correct propaga-tion of the quantum states. Furthermore, H. Cao et al. recently showed the correctpropagation and detection of three-dimensional entan-gled states [167] encoded in the OAM degree of free-dom. They used lower-order OAM modes ( l = 0 , ± )prepared with an SPDC and free-space optics to real-ize a three-dimensional entangled source with 88% offidelity. After the transmission over 1 km of fiber, thequdits are measured using an SLM and single-photondetectors. A fidelity measurement of 71% and high-dimensional Bell inequalities violation proved the cor-rect propagation of the qudit entangle states.Other works have investigated these special fibersproving the transmission of hybrid vector vortex-polarization entanglement over an air-core fiber [134],the distribution of bidimensional structured photons ina vortex fiber [168] and the transmission of spatially en-coded qudits over few meters of multimode fiber [169].Despite these proof-of-concept experiments, the propa-gation of qudits encoded in OAM through special fibersis still challenging. The main limitations are the phaseinstability between the modes and intermodal disper-sion. Theoretical work, with a more appropriate designand simulation, is necessary to engineer new fibers withless intermodal dispersion and crosstalk, with valuessuitable for future quantum communications.Another interesting application worth to be consid-ered is the use of multimode fiber for programminglinear quantum networks. Linear optical networks aregood candidates for future realization of quantum com-puting. However, limitations in terms of scalabilityand performance arise from current implementations.S. Leedumrongwatthanakun and colleagues report inRef. [170] the implementation of a fully programmablehigh-dimensional linear optical network by using spa-tial and polarization mixing processes in a multimodefiber. The experimental implementation of practical quan-tum communication systems has so far been mainlylimited to fibers and free-space links. However, dur-ing the last few years, the community has started toinvestigate quantum communication in an underwaterenvironment.In 2012, M. Lanzagorta proposed the idea of bringingthe technology of free-space quantum communicationin the water [171] by performing a feasibility analy-sis on the BB84 protocol in point-to-point communi-cation. During the last years, the paper was followedby other theoretical investigation [172, 173], in whichnot only point-to-point links were considered, but alsonon-line of sight underwater communication was stud-ied. Besides, few experimental demonstrations investi-gated the propagation of polarization-based quantumstates [174, 175].However, optical communication in an aquatic envi-ronment is subjected to multiple degradation factors:13igh losses, strong turbulence effects and external noise(sun or moon radiation). These factors can be directlytranslated to higher noise in the communication sys-tem, which will influence the final performances of thecommunication, limiting the total distance and the keyrate. However, since qudits are intrinsically more ro-bust to the noise they can be used for underwater chan-nels.Similarly to free-space links, qudits encoded in spa-tial modes are suitable for the generation and thetransmission of large dimensional quantum states. Inthis direction, a recent study from F. Bouchard andcolleagues [176] has investigated the effects of turbu-lence on an underwater quantum channel with high-dimensional quantum states encoded in spatial modes.Photon pairs are generated via SPDC and, by usinga spatial light modulator, Alice prepared the quantumstates to propagate over a 3 m link. Bob projectedthe quantum states into the different bases using anSLM, while single-photon detectors were used to mea-sure coincidences. To prove the correct propagationof the quantum states, three proof-of-concept QKDprotocols have been demonstrated. A two-dimensionalBB84 protocol, exploiting OAM modes, proved the cor-rectness of the quantum states with a QBER around6.57%. A six-state protocol, instead, allowed a lowerQBER of 6.35% generating 0.395 bit per sifted photons.Qudits were also investigated (qutrits and ququarts)employing different OAM modes l = 0 , ± , ± . Theresults report a QBER of 11.73% (below the thresh-old of 15.95%) for the qutrit system and 29.77% forthe ququart case, which is above the 18.93% thresh-old of the collective attacks. The errors are attributedto aberrations induced by the underwater turbulenceintroducing crosstalk between OAM modes. Since theoscillations introduced by the water turbulence are ofthe order of tens of Hertz, an adaptive optics systemmight be used to correct the wavefront [177] and todecrease the inter-modal crosstalk.In these sections, experimental demonstrations of high-dimensional quantum state transmission through dif-ferent types of communication channels have been re-ported. Many of the experiments are proof of conceptrealizations and, compared to ordinary qubit imple-mentations, they required a more advanced setup bothfor the transmitter and for the receiver, not alwayspractical with the current technology. Also, the max-imum dimensionality explored and successfully trans-ported over a communication channel is four (ququart).However, such limitations can be overcome by the useof integrated photonics, which allows exceptional con-trol on the generation and manipulation of the quan-tum states, and by further improvement in the realiza-tion of multicore/multimode optical fibers. Summa-rizing, it is currently difficult to foresee an imminentuse of high-dimensional quantum schemes in quantumnetworks, but depending on the channel characteristics(noise, distance, hardware availability) qudits can play a very important role in the future quantum systems. The quantum internet represents the final goal of quan-tum communication. It can disclose a whole universeof new applications that can enlighten new fundamen-tal physics questions or boost secure communicationand remote quantum computing [178, 179]. The keyingredient of the quantum internet is the capability todistribute and store entanglement between separatedusers. Despite the big efforts over the last decade, long-distance transmission and long-time storage of entan-gled states remain open challenges. In this direction,high-dimensional quantum states can play a promi-nent role, due to their enhanced robustness to noiseand higher information capacity. Nonetheless, their ex-ploitation is not straightforward, mainly due to experi-mental limitations and theoretical problems still open.Indeed, generalizing and experimentally proving pro-tocols like entanglement swapping or quantum telepor-tation (primary tasks in quantum communication) byusing qudits is not trivial [79]. Exploiting non-linearoptics devices can be an approach [180], but the highlyprobabilistic processes involved can limit the advan-tages of the higher dimensionality. Ancillary photonsor hyper-entanglement between two or more degreesof freedom can represent a solution to implement suchprotocols, as some proof-of-concept experiments haveproven [181–184]. From a general perspective, the re-alization of a quantum network cannot happen regard-less of quantum memories. The quantum communityhas already proved the capacity of storing multidimen-sional states in quantum memories, but very little hasbeen done to prove the compatibility between exter-nal qudit sources with quantum memories for high-dimensional states [185]. Hence, we think that in thenext years more research needs to be done towards theconjunction of these two branches.In terms of quantum foundations, an open problem re-lated to high-dimensional entangled states is the certi-fication of the actual entangled dimensions of a quan-tum system. In fact, despite the theoretical debateswe have not addressed in this review, the experimentalcertification of high-dimensional entanglement requiresfull state tomographies of bipartite d -dimensional sys-tems. This implies that ( d + 1) global product basesmeasurements are needed, which quickly becomes im-practical for high dimensions. Recently, it has beenproposed by J. Bavaresco et al. [186] a new way to cer-tify the dimensions of entangled qudits with carefullyconstructed measurements in two bases. This resultwill be of great advantage and will boost the researchon entangled qudits towards unexplored applications.As final a remark, we would like to raise a questionwhich hitherto has not a conclusive answer: how much nformation can be encoded into a single qudit? Whilein general quantum theory does not impose any limiton the mass or dimension of a quantum system, aquantum-to-classical transition can be expected. Manygroups are working to squeeze the boundaries betweenthe microscopic and macroscopic description of theworld, and thus trying to understand the limits, if thereare, between quantum and classical states. A way toaddress these questions is represented by quantum op-tomechanics, which, by coupling mechanical oscillatorsto optical fields, not only can study quantum infor-mation applications but also it has great potential totest quantum physics at the microscopic edge [187].In this sense, qudits can be key players by helping inthe boundary definition process between classical andquantum world, and thus bearing to a deeper under-standing of the physical world.
During the last few years, an increasing numberof reports on theoretical and experimental advancesin the generation, propagation, and measurement ofhigh-dimensional quantum states have been published.These include the generation of high-dimensional quan-tum states, up to dimension 15, in a silicon photonicplatform [129], the transmission of high-dimensionalquantum states through multicore [162] and multimodefibers [44] and the distribution of OAM modes [150]and hyper-entangled photons [124] via a free-space in-tracity channel. These achievements are possible owingto technological progress and a better understanding ofthe physical principles underlying larger Hilbert spaces.We believe that high-dimensional quantum states willplay a fundamental role in the next quantum techno-logical leap.
Biography
Daniele Cozzolino is a Ph.D. candidate of the SPOCcenter at the Department of Photonics Engineering atthe Technical University of Denmark (DTU). He ob-tained his B.Sc and M.Sc in Physics at the Universityof Naples Federico II. His research interests are focusedon quantum information and fundamental physics.
Beatrice Da Lio is a Ph.D. candidate of the SPOCcenter at the Department of Photonics Engineering at the Technical University of Denmark (DTU). She ob-tained her B.Sc at the University of Padova. She holdsa double M.Sc degree in Engineering Telecommunica-tion at the University of Padova and at the TechnicalUniversity of Denmark. Her research interests are fo-cused on quantum cryptography and quantum commu-nication.
Davide Bacco is an Assistant Professor at the De-partment of Photonics Engineering at the TechnicalUniversity of Denmark (DTU). He received his degreein Engineering Telecommunication in 2011 at the Uni-versity of Padova, Italy. In 2015 he finished in the sameUniversity the Ph.D. degree on Science Technology andSpatial Measures (CISAS). His research interests re-gard quantum communication and silicon photonics forquantum communications.
Leif K. Oxenløwe is the group leader of the High-Speed Optical Communications group at DTU Fotonik,at the Technical University of Denmark (DTU), andhe is leader of the Centre of Excellence SPOC (Sili-con Photonics for Optical Communications). He re-ceived his B.Sc. and M.Sc. degrees in physics andastronomy from the Niels Bohr Institute, University ofCopenhagen, in 1996 and 1998, respectively. He re-ceived his Ph.D. degree in 2002 from DTU and since2009 he is Professor of Photonic Communication Tech-nologies. His research interests are focused on siliconphotonics for optical processing and high-speed opticalcommunication.15 unding Information
This work is supported by the Centre of Excel-lence SPOC - Silicon Photonics for Optical Commu-nications (ref DNRF123), by the People Programme(Marie Curie Actions) of the European Union’s SeventhFramework Programme (FP7/2007-2013) under REAgrant agreement n ◦ (COFUNDPostdocDTU). Acknowledgments
We thank Y. Ding, K. Rottwitt and M. Galili for theirhelp and contribution to our works. We would like toshow our gratitude to our collaborators from Univer-sity of Bristol, Sapienza University of Rome, BostonUniversity, University of Copenhagen and OFS Den-mark.
Conflict of interest
The authors declare no conflict of interest.
Keywords high-dimensional, qudit, quantum communications
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