Self-healing high-dimensional quantum key distribution using hybrid spin-orbit Bessel states

Using spatial modes for quantum key distribution (QKD) has become highly topical due to their infinite dimensionality, promising high information capacity per photon. However, spatial distortions reduce the feasible secret key rates and compromise the security of a quantum channel. In an extreme form such a distortion might be a physical obstacle, impeding line-of-sight for free-space channels. Here, by controlling the radial degree of freedom of a photon's spatial mode, we are able to demonstrate hybrid high-dimensional QKD through obstacles with self-reconstructing single photons. We construct high-dimensional mutually unbiased bases using spin-orbit hybrid states that are radially modulated with a non-diffracting Bessel-Gaussian (BG) profile, and show secure transmission through partially obstructed quantum links. Using a prepare-measure protocol we report higher quantum state self-reconstruction and information retention for the non-diffracting BG modes as compared to Laguerre-Gaussian modes, obtaining a quantum bit error rate (QBER) that is up to 3 times lower. This work highlights the importance of controlling the radial mode of single photons in quantum information processing and communication as well as the advantages of QKD with hybrid states.


Self-healing Bessel modes
Since Bessel modes cannot be realized experimentally, a valid approximation, the Bessel-Gaussian (BG) modes, is commonly used [30]. This approximation inherits from the Bessel modes the ability to self-reconstruct in amplitude, phase [31,32], and polarization [33][34][35], even when considering entangled photon pairs [36] or non-separable vector modes [37][38][39]. Mathematically, they are described by the expression where (r, ϕ, z) represents the position vector in the cylindrical coordinates, ℓ is the azimuthal index (topological charge). Furthermore, J ℓ (·) defines a Bessel function of the first kind , k r and k z are the radial and longitudinal components of the wave number k = k 2 r + k 2 z = 2π/λ. The last factor describes the Gaussian envelope with beam waist w 0 and Rayleigh range z R = πw 2 0 /λ for a certain wavelength λ.
The propagation distance over which the BG modes approximate a non-diffracting mode is given by z max = 2πw 0 /λk r [40]. In the presence of an obstruction of radius R inserted within the non-diffracting distance, a shadow region of length z min ≈ 2π R k r λ is formed [41]. The distance z min determines the minimum distance required for the beam to recover its original form, whereby full reconstruction is achieved at 2z min [31,32].
Here, we exploit this property with single photons that have non-separable polarization and OAM DoF. By carefully selecting a k r value, we show that the information of single photons encoded with a Bessel spatial profile can be recovered after the shadow region of an obstruction. Next, we introduce a high-dimensional self-healing information basis for QKD, constructed from non-orthogonal vector and scalar BG spatial modes. LG radial profiles for ℓ = ±1. The polarization projections on the (c) vector |Ψ and (d) scalar |Φ basis BG modes. The vector modes have spatially varying polarizations which consequently render the polarization and spatial DoF as non-separable. This is easily seen in the variation of the transverse spatial profile when polarization projections are performed (orientation indicated by white arrow) on the |Ψ modes. In contrast, the scalar modes have separable polarization and spatial DoF hence polarization projections only cause fluctuations in the intensity of the transverse profile for the |Φ modes.

Self-healing information basis
In the standard BB84 protocol, Alice and Bob unanimously agree on two information basis. The first basis can be arbitrarily chosen in d dimensions as {|ψ i , i = 1..d}. However, the second basis must fulfill the condition making |ψ and |φ mutually unbiased. Various QKD protocols were first implemented using polarization states, spanned by the canonical right |R and left |L circular polarization states constituting a two-dimensional Hilbert space, i.e., H σ = span{|L , |R }. More dimensions where later realized with the spatial DoF of photons [9,15], using the OAM DoF spanning the infinite dimensional space, i.e. H ∞ = H ℓ , such that H ℓ = {|ℓ , |−ℓ } is qubit space characterized by a topological charge ±ℓ ∈ Z.
Here we exploit an even larger encoding state space by combining polarization and OAM, is a qu-quart space spanned by the states {|L |ℓ , |R |ℓ , |L |−ℓ , |R |−ℓ }, described by the so-called higher-order Poincaré spheres (HOPs) [42,43]. These modes feature a coupling between polarization and OAM DoF (see Fig. 1(a) for BG and (b) for LG profiles). A high-dimensional information basis with a radially concentric ring structure of BG modes can be constructed ( Fig. 1(a)). Amongst these are the non-separable vector BG modes ( Fig. 1(a), first row), known to posses self-reconstruction after encountering obstructions [44].
For this experiment, we chose a mode basis on the H 4 subspace with ℓ = ±1, although, in principle more modes can be used. Our encoding basis is constructed as follows: we define the radial profile J ℓ,k r (r) representing the radial component of the BG mode in Eq. (1). Our first mode set is comprised of a self-healing vector BG mode basis, mapped as being R and L the circular polarization states while ℓ is the topological charge, (see Fig. 1(c) for polarization projections). The second set of orthogonal modes is given by where D and A are the diagonal and anti-diagonal polarization states (see Fig. 1(d) for polarization projections). Conversely, we contrast these MUB modes with an LG vector and scalar MUB by forming an equivalent mode set, i.e, with the same OAM and polarization information but an LG radial profile. The two mode sets are contrasted in Fig. 1(a) and (b). The set |Ψ ij and |Φ ij are mutually unbiased and, therefore, form a reputable information basis for QKD in high-dimensions.

Single photon heralding
For unconditional security to be achieved in QKD, a single photon source must by used. However, such sources are yet to be realized, although it is possible to use correlated pairs of photons, where the detection of one photon, the heralding photon, conditions the existence of its correlated pair [45]. Such sources, of heralded single photons, can be produced by means of SPDC. In this process, the statistics of the heralded photon have low multi-photon probabilities. Thus, we herald a single photon via SPDC where a high frequency photon (λ = 405 nm) is absorbed with low probability in a nonlinear crystal, generating a signal (s) and idler (i) correlated paired photons at λ = 810 nm. In the case of a collinear emission of s and i, the probability amplitude of detecting mode functions |m s and |m i , respectively, is given by [46] where m p (x) is the field profile of the pump (p) beam which best approximates the phasematching condition in the thin crystal limit; the Rayleigh range of the pump beam is much The physical obstruction is placed at a distance L from the rightmost SLM, which decodes the radial information of Bob's photon. The optics are within z max = 54 cm distance of the BG modes depicted as the rhombus shape. The propagation of the post-selected BG mode can be determined via back-projection. (b) Optical elements required by Alice and Bob to prepare and measure the spin-coupled states of the heralded photons (cf. Table 1). (c) Numerical scattering probability matrix for the vector and scalar modes sets in free-space. The channels correspond to the probabilities |C ij | 2 calculated from Eq. (22).
larger than the crystal length. The probabilities amplitudes c s,i can be calculated using the Bessel basis, where exp(iℓϕ) corresponds to the characteristic azimuthal phase mapping onto the state vector |ℓ . Taking into account a SPDC type-I process and a Gaussian pump beam, the quantum state used to encode and decode the shared key can be written in the Bessel basis as being |ℓ, k r s ∼ J ℓ,k r (r) |ℓ and H the horizontal polarization state. The probability amplitudes c ℓ,k r,1 ,k r,2 can be calculated using the overlap integral in Eq. (11). Experimentally |c ℓ,k r,1 ,k r,2 | 2 is proportional to the probability of detecting a coincidence when the state |ℓ, k r,1 s |−ℓ, k r,2 i is selected. Coincidences are optimal when |k r,1 | and |k r,2 | are equivalent. In this experiment, the idler photon (i) is projected into the state |0, k r i , heralding only the signal photons (s) with the same spatial state |0, k r s , as can be seen in the sketch of Fig. 2(a). Therefore, a prepare-measure protocol can be carried out by using the same s photon. In otherwords, Alice remotely prepares her single photon with a desired radial profile from the SPDC before encoding the polarization and OAM information.

Spatial profile post-selection
Spatial light modulators (SLMs) are a ubiquitous tool for generating and detecting spatial modes [47,48]. We exploit their on-demand dynamic modulation via computer generated holograms to post-select the spatial profiles of our desired modes (see hologram inset in Fig. 2(a)). For the detection of BG modes, we choose a binary Bessel function as phase-only hologram, defined by the transmission function with the sign function sign{·} [49,50]. Classically, this approach has the advantage of generating a BG beam immediately after the SLM and, reciprocally, detects the mode efficiently [36]. Importantly, a blazed grating is used to encode the hologram, with the desired mode being detected in the first diffraction order [51] and spatial filtered with a single mode fiber (SMF).
Here, we set k r = 18 rad/mm and ℓ = 0 for the fundamental Bessel mode and, conversely, k r = 0 to eliminate the multi-ringed Bessel structure.

Mode generation and detection
Liquid crystals q-plates represent a convenient and versatile way to engineer several types of vector beams [52]. In our setup, vector and scalar modes, described in Fig. 1, are either generated or detected by letting signal photons pass through a combination of these devices and standard wave plates (see Fig. 2(c)). A q-plate consists of a thin layer of liquid crystals (sandwiched between glass plates) whose optic axes are arranged so that they form a singular pattern with topological charge q [29]. By adjusting the voltage applied to the plate it is possible to tune its retardation to the optimal value δ = π [53]. In such a configuration indeed the plate behaves like a standard half-wave plate (with an inhomogeneous orientation of its fast axis) and can be used to change the OAM of circularly polarized light by ±2q, depending on the associated handedness being left or right, respectively. In the Jones matrix formalism, the q-plate is represented by the operatorQ = cos(2qϕ) sin(2qϕ) where ϕ is the azimuthal coordinate. The matrix is then written in the following linear basis In our experiment we use q-plates with q = 1/2, and half-( λ 2 ) as well as quarter-wave plates ( λ 4 ) for polarization control, represented by the Jones matriceŝ Here, θ represents the rotation angle of the wave plates fast axis with respect to the horizontal polarization. The operator associated with the generation of the vector mode iŝ where α 1 and α 2 are the rotation angles for the half-wave plates andP H = 1 0 0 0 represents the operator for a horizontal linear polarizer. Similarly, the operator for the scalar modes iŝ where β 1 and β 2 are the rotation angles for the quarter-wave plates.
.4} be associated with the generation of vector modes fromV(α 1 , α 2 ), and M 2 = {Ŝ j |Ŝ j → |φ j , j = 1..4} for the scalar modes fromŜ(β 1 , β 2 ). The orientation of the angles required to obtain them is given in Table 1 for the vector and scalar modes (see also schematics of wave plates arrangement in Fig. 2(c)). Table 1. Generation of vector and scalar modes from a horizontally polarized BG mode (ℓ = 0) at the input. The angles α 1,2 and β 1,2 are defined with respect to the horizontal polarization. For eachV i andŜ i we present the angles needed to perform the mapping of M 1 → {|ψ i } and M 2 → {|φ i } with a one-to-one correspondence.

Scattering probability
LetÂ i ∈ M 1 ∪ M 2 represent operators selected by Alice and Bob, respectively. Alice first obtains a heralded photon from the SPDC with the input state |ψ in = J 0,k r |H . Then, Alice prepares the photon in a desired state from the MUB with Bob similarly measures the state whereB j ∈ M 1 ∪ M 2 . The probability of Bob's detection is The theoretical probabilities |C ij | 2 are presented in Fig. 2(b). is spatially filtered to clean the beam profile obtaining 40 mW of a 337 µm diameter Gaussian beam. This pump beam traverses the temperature controlled 2-mm-long PPKTP nonlinear crystal (Raicol) where photons are absorbed, with low probability, generating two lowerfrequency photons (both centered at λ = 810 nm) by means of the spontaneous parametric down-conversion (SPDC) process. This particular SPDC process is collinear type-I, meaning that the two lower-frequency photons exiting the nonlinear crystal, signal and idler, have the same linear momentum and horizontal polarization, possessing the temporal correlations needed for the BB84 protocol implementation. The nonlinear crystal generates the paired photons being able to recreate the non-diffracting length (z max ), generating BG modes given a particular post-selected radial wave number. Note that the (z max ) distance can be verified by back propagating through the system. We postselected a wave number of k r = 18 rad/mm in our case. That is why the crystal plane is relayed with a 4 f -system, also relaying the non-diffracting length, having the spatial and polarization projections within the z max = 54 cm, as schematically shown in Fig. 2(a).

Experimental set-up
The two paired photons, signal and idler, are spatially separated by a 50:50 beam splitter (BS). The distance between the BS and the SLM are exactly the same for both photon paths. The signal photon, transmitted from the BS, traverses the preparation section (A), where Alice would choose a particular vector or scalar state from the MUB alphabet. The photon encoded with a secure bit is then propagated through the non-diffracting length with self-healing properties, where it encounters a variable sized obstacle. This mimics a real building-to-building quantum channel. The state measurement (B) is implemented right after the obstacle. This simulates Bob's station where particular projections are chosen, from the same MUB that Alice used. The SLM is acting as a horizontal polarization filter, post-selecting the radial wave number defining the z max distance. The idler photon, reflected from the BS, is heralding the whole process enabling for the prepare-measure BB84 protocol to be performed.
Subsequently, both photons are properly coupled into single mode fibres (SMF), after being resized by another 4 f -system, thus post-selecting the spatial profile by spatial filtering them with the SLM and SMF, and also after being spectrally filtered by a band-pass filter (BPF) with 10 nm bandwidth at full-width at half-maximum (FWHM). The single photon detectors (D 1&2 ; Perkin-Elmer) output pulses are then synchronized with a coincidence counter (C.C.), discarding also the cases where the two photons exit the same output port from the BS.

Procedure and analysis
We measure the scattering matrix for the BG and, for comparison reasons, the LG profiles under three conditions: (I) in free-space, (II) with a R 1 = 600µm obstruction placed strategically such that the complete decoding is performed after L > z min (L: distance between obstruction and decoding SLM). Subsequently, (III) with a R 2 = 800µm obstruction, placed at the same position, however, in this case, the shadow region overlaps the detection system (L < z min ). Thus, in the latter case, the mode is not able to self-reconstruct completely before being projected. We measure the quantum bit error ratio (QBER) in each of these cases and computed the mutual information between Alice and Bob in d = 4 dimensions by [11] Here, e denotes the QBER. Lastly, we measured the practical secure key rate per signal state emitted by Alice, using the Gottesman-Lo-Lütkenhaus-Preskill (GLLP) method [54,55] for practical implementations with BB84 states, given by where Q µ ∼ 10 −4 is the photon yield for an average intensity µ, H d (·) is the high-dimensional Shannon entropy and f EC is a factor that accounts for error correction and is nominally f EC = 1.2 for error correction systems that are currently in practice. Furthermore, ∆ is the multi photon rate computed as 1−P 0 −P 1 Q µ where P 0,1 are the vacuum and single photon emission probabilities. Moreover the term, 1 − ∆ accounts for photon splitting attacks [55] and can be reduced by using ideal photon sources since ∆ → 0. In our experiment, we measure the photon intensity to be µ = 10 −3 in free-space and decreases accordingly with the varied obstructions.

Experimental results
We perform a prepare-measure protocol in four dimensions using heralded single photons. The photons are encoded using states from a secure information basis of vector |ψ and scalar |φ MUB modes. We change between BG and LG (with a radial index p = 0, i.e. with no concentric rings) spatial profiles and compare their performance under the influence of varying sized obstructions. Accordingly, the reconstruction distance for both radial profiles, varied the levels of noise at the SLM plane. We subsequently measure the QBER (e) and security parameters.
We present the measured detection probability matrices for three tested cases in Fig. 4: (a) free-space; (b) and (c) when the single photon is perturbed and subsequently self-constructed (i.e. with R 1 = 600µm, L > z min ) for BG and LG modes, respectively; and (d) and (e) when reconstruction is not yet complete (with R 2 = 800µm, L < z min ), i.e., the shadow of the obstruction overlaps the detection system for BG and LG modes, respectively. r the In the freespace case, we measure QBERs of e = 0.04 ± 0.004 for the BG and LG spatial profiles (see Fig.  4(a) and Table 2). We compute a mutual information of I AB = 1.69 bits/photon and a secure key rate of R ∆ /Q µ = (1.38) bits/s per photon for both radial profiles. LG 0 1 2

BG
LG 0 Under the perturbation of the R 1 = 600µm obstruction, we measure a QBER of e = 0.05 for both spatial profiles, indicative of information retention, i.e. high fidelity. The intensity fields from the back-projected classical beam (see insets of Fig. 4(b) and (c)), show self-healing of the BG mode at the SLM plane (see Fig. 4(b)), although the LG is not completely reconstructed (see Fig. 4(c)). The photons encoded with the LG profile may have a large component of the input mode which is undisturbed in polarization and phase. Furthermore, the photon counts decrease to 49% for the LG profile relative to the counts in free-space, as highlighted in Fig.  5(a). In comparison, the BG modes retain the same intensity thanks to the multiple concentric rings.
Lastly, we investigate the security when the R 2 = 800µm obstruction is used. Remarkably, as illustrated in Fig. 5(a), the signal decreased by almost four orders of magnitude, remaining only the 0.07% of the signal for the LG set, but up to 71% for the BG self-healing mode set, owing to an earlier reconstruction of the BG radial profile in comparison to the LG radial profile. Based on the measurement results shown in Fig. 4(d) and (e), we determine a QBER of e = 0.15±0.01 and e = 0.51 ± 0.00 for the BG and LG modes, respectively. The mutual information (I AB ) and secure key rates are higher for the BG basis than the LG, even though the BG MUB has not fully reconstructed (see Fig. 5(b)). In total, see Table 2 for the summarized measured security parameters of the BG and LG mode sets.

Security of high-dimensional self-reconstructing modes
We have presented a proof-of-concept experiment showing the advantage of using BG selfhealing, high-dimensional states, to encode information in both, the OAM and polarization DoFs. Our scheme shows that with high-dimensional encoding and self-reconstruction, high information transmission rates in quantum communication coupled with the added improvement to the security [11] are achievable.
Our scheme exploits the radial DoF which has previously not been used in high-dimensional QKD implementations with spatial modes. All previous reports only exploited either the OAM or spin-orbit coupled DoF. Our scheme outperforms these previous implementations since no provision for channel losses due to obstructions have been made and with the aid of selfreconstruction, channel perturbations can be alleviated.
Furthermore, we stress that although there are reported benefits with using HD encoding, not all protocols have been generalized to high-dimensions, for example, the SARG04 protocol [56] which is designed for robustness against the photon number splitting attacks or the B92 protocol which is a simpler version of the BB84 protocol [57]. Alternatively, decoy states can be implemented with the BB84 states and there have been various reports with HD encoding using spatial modes in free-space and fiber [58,59]. Another alternative is the KMB09 protocol [60][61][62]. The security of the scheme is due to a minimum index transmission error rate (ITER) and quantum bit error rate (QBER) introduced by an eavesdropper. The ITER increase significantly for higher dimensional photon states allowing for more noise in the transmission line. Using HD self-healing spatial modes could improve the performance of such protocols in free-space and overcome some of the practical difficulties that are faced when using the transverse spatial modes of photons. Although the scheme we present is filter based, i.e. filtering states one at a time, we stress that the experiment can be performed robustly and more efficiently using a deterministic detector for spin-orbit coupled states, sorting the modes in position [19]. This ensures high detection rates. Obtaining high switching between modes during generation may require fast modulators which is a serious experimental challenge when implementing HD QKD [63].

Conclusion
The self-healing property of the Bessel-Gaussian modes opens an important research field, being able to securely share the cryptographic key despite any possible obstruction partially blocking the quantum channel. We have shown in this manuscript the experimental results of the scattering probabilities, mutual information and secret key rates in a prepare-measure protocol, comparing two different modes forming the QKD quantum state alphabet: Bessel-Gaussian (BG) and Laguerre-Gaussian (LG). Our results clearly show lower quantum bit error rate (QBER) by using BG modes when transmitting the shared key through a mostly blocked quantum channel. Concretely, we measured a QBER of 0.15 ± 0.01 and 0.51 ± 0.00 for the BG and LG modes, respectively. Furthermore, when almost completely blocking the channel, the mutual information for the BG modes only drops due to the increase of the noise with respect of the signal. The quantum state information can be reconstructed even when having barely any photons after the obstacle.