Universal entanglement decay of photonic orbital angular momentum qubit states in atmospheric turbulence: an analytical treatment

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1 Universal entanglement decay of photonic orbital angular momentum qubit states in atmospheric turbulence: an analytical treatment David Bachmann Vyacheslav N Shatokhin and Andreas Buchleitner Physikalisches Institut Albert-Ludwigs-Universität Freiburg Hermann-Herder-Str D-794 Freiburg Federal Republic of Germany Dated: March 9) We study the entanglement evolution of photonic orbital angular momentum qubit states with opposite azimuthal indices l in a weakly turbulent atmosphere Using asymptotic methods we deduce analytical expressions for the amplitude of turbulence-induced crosstalk between the modes l and l Furthermore we analytically establish distinct universal entanglement decay laws for Kolmogorov s turbulence model and for two approximations thereof Such states can be generated experimentally [6 8 and large values up to l = were reported [7 The enarxiv:9459v [quant-ph Mar 9 I INTRODUCTION Helical wave fronts of twisted photons allow for encoding high-dimensional qudit) states in the orbital angular momentum OAM) degree of freedom that is characterized by an azimuthal index l = ± ± [ Using qudit states can potentially ensure higher channel capacity [4 5 and enhanced security [6 7 of quantum communication channels as compared to twodimensional polarization-based encoding So far the full-scale use of twisted photons in free space quantum communication has remained elusive due to the sensitivity of the photons wave fronts with respect to intrinsic fluctuations of the refractive index of air atmospheric turbulence) [8 Notwithstanding recent experimental progress where OAM encoding was used in free space quantum key distribution over up to m [9 for entanglement distribution over km [ and for the transmission of classical twisted light over 4 km [ indicates that turbulence-induced phase-front distortions do not in principle preclude the reliable transfer of twisted photons through the atmosphere Yet we still lack a complete understanding of the behavior of photonic OAM states in turbulence even for OAM-encoded qubits Since one of the most prominent quantum communication protocols [ is based on quantum entanglement [4 5 which is readily generated between OAM states in the lab [6 8 experimental [9 and theoretical [4 9 efforts explore how entanglement of twisted biphotons decoheres in the atmosphere In particular it was predicted [ and recently confirmed experimentally [ that entangled qubit states with opposite OAM become more robust in weak turbulence as the azimuthal index l increases Some useful insight into this feature was provided through the introduction of the phase correlation length ξl ) [ that is the characteristic length scale associated with the transverse spatial structure of OAM beams In particular it was numerically shown [ that the individual temporal evolution of the concurrence [4 of initially maximally entangled OAM qubit states with quantum numbers ±l collapses onto an l -independent universal entanglement decay upon rescaling the turbulence strength by ξl ) However no analytical derivation of this universal decay law has so far been available In the present contribution we revisit the entanglement evolution of photonic OAM qubit states in weak turbulence and generalize the results of [ along two directions: First we deduce the entanglement decay law analytically To this end we exploit the universality of the decay and consider the limit l using asymptotic methods Second we derive explicit expessions for the universal decay law for three distinct models of turbulence distinguished by the exponent α of the phase structure function [8 see Sec II) This generalization is inspired by the fact that the universal form of the entanglement decay relies on the weakness of the turbulence The latter assumption implies that the impact of turbulence on the propagated wave can be described by a single phase screen [4 irrespective of the specific turbulence model Since variation of α leads to different types of atmospheric disorder it is interesting to explore somewhat in the spirit of [5 how this affects entanglement evolution in turbulence The paper is structured as follows: In the next section we present our general model of turbulence parametrized by the exponent α as well as our method to access the entanglement evolution In Sec III we derive explicit expressions for the relative crosstalk amplitude b in the asymptotic limit l for α = 5/ and compare these asymptotic results to numerically exact data We finally give analytic formulations for the universal entanglement decay law for those different values of α Section IV concludes the manuscript II MODEL We consider a maximally entangled OAM qubit state of two photons encoded in two Laguerre-Gauss LG) modes with radial and azimuthal quantum numbers p = and l = ±l respectively and relative phase ϕ: ψ = l l + e iϕ l l ) )

2 tangled photons are sent in opposite directions along the horizontal) z-axis through the atmosphere and the impact of the latter on the transmitted photons is modelled by two independent phase screens one for each photon This model of atmospheric turbulence ignores turbulence-induced intensity fluctuations Furthermore we neglect diffraction assuming that the widths of the LG beams remain constant along the propagation paths As follows from [6 both assumptions are valid for short propagation distances of about one km for each photon When a pair of twisted photons prepared in state ) is sent across the atmosphere turbulence-induced random phase shifts lead to the coupling or crosstalk of the initially excited LG ±l ) modes to other LG pl modes spreading in general over infinitely many values of p and of l Upon averaging this high-dimensional pure state over independent statistical realizations of those random shifts one ends up with a mixed state Thus the output biphoton state ϱ can be described by a map [ ϱ = Λ Λ )ϱ ) where Λ i is the disorder-averaged turbulence map acting on the state of the ith photon and ϱ = ψ ψ The tensor product expresses the independence of the action of the two screens on either photon Since we are only interested in the OAM entanglement evolution in the subspace of the initially populated OAM modes we trace over the radial quantum number p and project the output state onto the subspace spanned by the four product vectors: l l l l l l l l [ Due to the projection the norm of the resulting state is reduced and the state needs to be renormalized by its trace [7 The entanglement evolution of the thereby obtained mixed bipartite qubit state can be evaluated using Wootter s concurrence [ The elements of the map Λ i required for the evaluation of the output state ϱ are given by Λ ll l±l = π δ l l l±l π rdrdϑ R l r)r l r)e i ϑ [l±l) l+l ) { exp [ )} ϑ D φ r sin ) where upper and lower indices of Λ ll l±l correspond to the input and output states respectively δ l l l±l is the Kronecker delta ) l R l r) = r e r /w 4) l! w w is the radial part of the LG ±l mode at z = with p = l = ±l and w is the mode s width Furthermore ) α x D φ x) = γ 5) r is the phase structure function of turbulence which determines the statistics of the spatial wavefront deformations [8 Here γ = 688 and r = 4 C nk L) /5 6) is the Fried parameter [ 9 with Cn the index-ofrefraction structure constant k the optical wave number and L the propagation distance Throughout this work we assume that both photons traverse equal-length paths in the atmosphere characterized by the same Cn Hence r coincides for both screens The exponent α in Eq 5) specifies the turbulence model In particular α = 5/ characterizes Kolmogorov s description [8 but we subsequently also consider the entanglement evolution for α = which can be regarded as integer-value approximations of α = 5/ In fact the phase structure function with α = is oftentimes used in the literature [ 8 with Eq 5) then called the quadratic approximation [4 of the Kolmogorov phase structure function by analogy we refer to the case α = as the linear approximation ) Apart from the fact that the quadratic approximation allows for obtaining analytical results for entanglement decay of OAM qubit states even under conditions of strong turbulence [8 9 it is sometimes considered to be slightly pessimistic [4 as compared to the Kolmogorov model Indeed as follows from Eqs ) and 5) the matrix elements of the map Λ i governing the evolution of the density operator will decay faster with x for α = than for α = 5/ Consequently the first-order spatial correlation function of the propagated field also known as the mutual coherence function [8 ) decreases more rapidly within the square-law approximation which is considered [4 to be harmful for optical communication However as for the entanglement evolution of OAM states a smaller magnitude of those matrix elements of Λ i that reflect the coupling between OAM modes with l and l is actually useful because it is the intermodal crosstalk that causes entanglement decay [ As a result as we show in this work the robustness of entanglement is most pronounced for α = From this viewpoint the quadratic approximation appears as overly optimistic Consistently the linear approximation α = yields pessimistic results as compared to those derived from the Kolmogorov model III UNIVERSAL ENTANGLEMENT DECAY LAWS A General expressions Upon application of the turbulence map ) the output state s concurrence is given analytically by [ [ b Cϱ) = max + b) 7)

3 where b = b/a is the reduced or relative crosstalk amplitude with = Λll l l 8a) 8b) - t := w /r γ l ) t and ) is the turbulence strength The asymptotic solutions ) and ) essentially accomplish our target: Indeed ) already has an explicit analytical form Equation ) involves a highly oscillatory integral which in turn can be evaluated by asymptotic methods Though for any α = l/k with l k positive integers Eq ) can be evaluated exactly and is given in terms of either elementary α = ) or of special α = 5/) functions Next we analyze these three cases in more detail t= -4 b) - t=5 t= - t= - t=5-4 t= -5-6 c) - t=5 t= - t= - ~ where A = α/ α t=5 - ~ Thus the entanglement evolution within the Kolmogorov and single phase screen model is determined by the ratio of the only two characteristic length scales of which ξl ) refers to the OAM beam and r to the turbulent medium In the following we analytically show that the universality of the entanglement decay persists also for α = and α = though with a different functional form which we identify for α = 5/ By virtue of Eq 7) the universality of the concurrence decay law originates from that of the relative crosstalk amplitude b b ξl )/r ) Therefore we need to infer the explicit form of the latter To that end we consider the limit l and apply asymptotic analysis Using the method of steepest descent [4 we obtain the following expressions see Appendix A): /α a A Γ + ) π α Z b Re dx exp A xα il x) ) π t= t= ~ expressed in terms of the phase screen map s matrix elements ) we drop their upper indices p = for brevity) We note that Eq 7) is valid for a single phase screen description of turbulence whereas the specific turbulence model is included via the dependence of the parameters a and b on α In the framework of the Kolmogorov model of turbulence by numerical assessment of a and b we established earlier that the behavior of C%) is universal in the sense that the a priori l -dependent entanglement evolution with the penetration depth into the turbulent medium collapses onto one universal curve for arbitrary OAM l > provided C%) is considered as a function of the rescaled turbulence strength ξl )/r [ Here ξl ) is the phase correlation length of an LGl beam of width w which reads [ w Γ l + /) π 9) ξl ) = sin l Γ l + ) α t=5 - b b= = l Λ l l l = l Λl l l b = l Λ l l l l Λ l l l b a= Λll ll a) t=5-4 t= l FIG Color online) Relative crosstalk amplitude b versus the azimuthal index l on a log-log scale) for t w /r = 5 5 and for three values of α [see Eq 5): a) α = b) α = 5/ c) α = Solid lines indicate exact numerical integration of Eq ) dashed lines correspond to asymptotic expressions a) ) b) 5) c) 8) Thin dotted lines in panel b) represent the asymptotic expansion 6) B Asymptotic expressions for survival and crosstalk amplitudes Linear approximation α = ) For α = ) and ) yield the following asymptotic results l ): γ l t 4 b ) a πγ l t 4π 4l + γ t l /8 In Fig a) we show the function b = b/a as obtained by exact numerical evaluation of the integrals in Eq )

4 4 together with the asymptotic solution derived from Eq ) We notice that the asymptotic value of the ratio b/a provides excellent agreement with the exact numerical solution already for l As follows from Eq ) at leading order the relative crosstalk amplitude reads b γ t l 4) ie is proportional to t /l It is noteworthy that as in the linear-approximationscenario see Sec III B ) the relative crosstalk amplitude is represented as a power series in t /l At leading order b exhibits a power-law behavior t /l ) 4/ which is faster than the linear dependence 4) characteristic of the linear approximation For instance the values of the relative crosstalk at l = 5 and at the minimum turbulence strength t = presented in Fig are 4 and 6 for α = and α = 5/ respectively Kolmogorov turbulence α = 5/) After a suitable coordinate transformation see Appendix A) the integral in Eq ) reduces to a tabulated integral [4 Explicitly the amplitudes a and b for α = 5/ read: ) γ/) /5 8 a πt Γ 5a) l 5 5 b π 4 5b) [ e iπ/5 Re G q 5 s s ) 4 5 where s = il e iπ/5 q = γ /6 t 5/ l 5/6 e iπ/ and G 5 5 x ) is the Meijer G-function which can be expressed through generalized hypergeometric functions [44 As in the case α = the numerically exact and the asymptotic results merge for l see Fig b)) However the hypergeometric functions are a representation of an infinite series [45 A transparent analytical form of the dependence of b on l can be deduced from the asymptotic series expansion of b which can be derived directly from ) For a large value of a parameter in our case for l ) the subsequent terms of an asymptotic series first decrease in magnitude but then start progressively increasing [46 Thus the asymptotic series are divergent Nonetheless these series are useful since a finite number of terms of an asymptotic series expansion yields a very accurate representation of the function at large parameter values [46 As regards the relative crosstalk amplitude b its best approximation for t 5 and l 5 is obtained by the first three terms of the asymptotic series see Eqs A)-A6) in Appendix A): b 8 t l t + )4 l 6 t +75 ) l ) +O[ t l ) 6 6) This contrasts the behavior of convergent series representing an analytic function fz) around its regular point z = z Quadratic approximation α = ) In contrast to the above cases α = 5/ for α = the relevant elements of the map Λ i defined by Eq )- 5) can be found in tables of integrals [47 Setting the upper and lower indices in Eq ) according to Eqs 8a) and 8b) we obtain the following exact results for the amplitudes a and b in ) respectively: a = l+ + τ) l l + F l ) ) + τ ; ; 7a) + τ b = l+ + τ) l τ ) ) l l l l + F l ) ) + τ ; l + ; + τ 7b) where F β δ; η; z) is the hypergeometric function [45 and τ γt Since solutions in terms of hypergeometric functions provide poor insight we inspect the approximate forms of Eqs 7a 7b) using the asymptotic formulas ) ) For α = these produce the compact expressions a t πγl b t πγl exp [ 4l γt 8) We remark that according to Eq 8) and Fig c) both the asymptotic and the exact solutions for b exponentially decay versus l ; furthermore they tend to coalesce for l 5 and t > For t the decay of the asymptotic b versus l /t in Eq 8) is faster than the exact one given by 7) Note however that the deviations occur at exact and approximate values which are fairly small For instance at l = 5 the approximate and the exact results at t = are b 4 and 9 respectively and can be neglected At t < the exact and the approximate values of b become even tinier The exponential dependence of the relative crosstalk amplitude on the ratio l /t see Eq 8) indicates a qualitative difference between the present and the previous α = 5/ cases both characterized by slow powerlaw dependencies of b on t /l Indeed owing to this

5 5 a= a=5/ a=5/ fit) a= 8 C%) ~ b a= a=5/ a=5/ as series) a= x/r x/r FIG Color online) Universal relative crosstalk amplitude b versus ξ/r for α = [Eq ) α = 5/ [the universal function Eqs ) and the asymptotic series Eq A6) and α = [Eq 4) The shaded area highlights the parameter range b / and ξ/r where the concurrence in Fig remains finite due to Eq 7) slow decay for α = 5/ a finite crosstalk amplitude between widely separated OAM modes l and l is induced already for weak turbulence strength t ' In contrast for α = such a coupling is exponentially suppressed even for moderate t ' As we will see in Sec III C the consequence of this suppression is that photonic OAM-entangled qubit states are most robust under turbulence characterized by α = FIG Color online) Universal decay of C%) versus ξ/r for α = 5/ obtained from 7) with b given respectively by Eqs ) ) and 4) The thin dashed line that is nearly overlapping with the solid line represents the fitting function gξ/r ) = exp[ 46ξ/r )4 employed in [ for α = 5/ see Appendix A) r /5 b = 5 π) Γ 58 " iγ 55 ξ 5 5 Re G5 5 π 5 6 r5 5 9ξ/r )8/ ) !# ) with the bottom line of ) valid in the limit ξ/r see Eqs 6) and ) and for α = π r b = exp 4) γξ C Rescaled relative crosstalk amplitude and concurrence In the limit l the expression for the phase correlation length given in Eq 9) can be simplified by recalling the asymptotics of the sine sin π/l ) π/l and of the gamma-function [45: Γl + x) l +x / πe l l 9) One obtains π w ξ = ξl ) l ) which can be employed in Eqs 5 8) to express the rescaled crosstalk in terms of ξ/r yielding the soughtafter universal behavior We find for α = b = γξ/r ) ; π + γξ/r ) It is easy to check that in all three cases Eqs 4) b ) = and b ) = where the latter value indicates the turbulence-induced saturation of the relative crosstalk amplitude However the behavior of b ξ/r ) is rather distinct in the range < ξ/r see Fig ) for the three models of turbulence what implies different forms of the entanglement decay see Fig ) ) Figure shows that the rescaled relative crosstalk amplitude b is maximal for α = and minimal for α = in the range < ξ/r Moreover for α = the amplitude b remains vanishingly small for ξ/r This is a consequence of the exponential suppression of the crosstalk in the neighbourhood of ξ/r = [see Eq 4) In contrast for α = 5/ and especially for α = b exhibits a power-law increase with ξ/r by Eqs ) and attains finite values in the interval ξ/r The behavior of the universal functions b at leading order in ξ/r is summarized in Table I below for the three models of turbulence considered here

6 6 TABLE I Relative crosstalk amplitude b in the limit ξ/r as a function of ξ/r for the three models of turbulence here considered α = α = 5/ α = linear) Kolmogorov) quadratic) ) b ξ/r) b 9 ξ/r) 8/ b = exp π r 76 ξ Finally inserting 4) into 7) we obtain the universal entanglement decay laws for the three considered models of turbulence which we present in Fig As anticipated concurrence is most least) robust against the rescaled turbulence strength for α = α = ) with an intermediate behavior for the Kolmogorov model Apart from the three universal results here considered we also plot the fitting function gx) = exp 46 x 4 ) which was obtained for α = 5/ in [ Although gx) is almost indistinguishable from the analytical result for finite values of ξ/r our above discussion of the asymptotic behavior of b anticipates its limitations: Since gξ/r ) vanishes exponentially as ξ/r Eq 7) implies b ) = / in disagreement with the actual asymptotic limit b ) = IV CONCLUSION We studied the entanglement evolution of photonic orbital angular momentum bipartite qubit states in a weakly turbulent atmosphere for three models of turbulence which are distinguished by the exponent α = 5/ of the phase structure function D φ x) = γ x/r ) α The entanglement evolution is entirely determined by the relative crosstalk amplitude b such that an increase of b entails a further loss of entanglement b is an α-dependent universal function of the ratio of the phase correlation length ξl ) of an OAM beam with azimuthal index l and radial index p = ) to the atmospheric transverse correlation length r Using asymptotic methods we obtained explicit analytical expressions for bξl )/r ) In particular for small ξl )/r the relative crosstalk amplitude exhibits power-law dependencies ξl )/r ) and ξl )/r ) 8/ for α = and α = 5/ respectively whereas it is exponentially suppressed for α = Our work thereby provides an explanation of the recent experimental [ and theoretical [48 results where it was shown that decoherence of entanglement is slower within the quadratic as compared to the Kolmogorov model of turbulence Secondly our results suggest that the statistics of the wavefront distortions depends via D φ x) on the turbulence model Therefore a relevant future research topic is to identify the type of wavefront distortions resulting for distinct α especially for α = wherein the entanglement decay is fastest This information will potentially be useful when designing adaptive optics systems aimed at entanglement protection of twisted photons under weak to moderate turbulence [49 ACKNOWLEDGMENTS Enjoyable and helpful discussions with Giacomo Sorelli are gratefully acknowledged This work was supported by Deutsche Forschungsgemeinschaft under Grant DFG BU 7/7- Appendix A: Asymptotic evaluation of Eq ) Equation ) can be rewritten in the form: Λ ll l l = l+ πl! π dρdϑe iϑl l) exp [ l + ) lnρ) ρ dϑ)ρ α A) where ρ r/w α and ) α ) dϑ) = α w ϑ γ sin α A) r We first note that for arbitrary l l α w /r and for any fixed ρ the integral over ϑ in Eq A) is real The corresponding integrand exp{ dϑ)ρ α +iϑl l )} attains maximum values at the end points of the integration region and is symmetric under reflection with respect to the ϑ = π axis Therefore when evaluating the integral over ϑ we can consider only the values ϑ We use this consideration in our subsequent analysis of the integral over ρ in Eq A) which is a Laplace integral [5 with dρ exp[λfρ) A) fρ) = ln ρ ρ dϑ)ρ α A4) l + and λ = l + For l the asymptotics of the integral A) can be evaluated using the method of steepest descent [4 5 according to which the main contribution to A) comes from the neighborhood of the saddle points of the exponential fρ) The function fρ) has a non-degenerate saddle point at ρ m l / A5) where we neglected the term proportional to dϑ) in A4) above Inserting the expansion fρ) fρ m ) + f ρ m )ρ ρ m ) into A) extending the lower limit of the integral over ρ to and evaluating the resulting Gaussian integral we arrive at the intermediate expression Λ ll l l π Re [ π dϑ exp{ Aϑ α + iϑl l )} A6)

7 7 where A = γ α/ l α/ w /r ) α A7) For l = l and l = l Eq A6) yields the amplitudes a and b respectively For the survival amplitude a one immediately obtains a π A /α Γ + α ) A8) Since the main contribution to the crosstalk amplitude b [ π π Re dϑ exp Aϑ α il ϑ) A9) originates from ϑ we can replace the upper integration limit by infinity to obtain b π Re [ dϑ exp Aϑ α il ϑ) =: π Re[J A) For α = and α = the oscillatory integral J in Eq A) reduces to the Fourier transform of respectively an exponential and a Gaussian function The parameter b can be then easily calculated to yield Eqs ) and 8) for α = and α = respectively For α = 5/ Eq A) can be transformed to a tabulated integral by a rotation of the real semi-axis by an angle β into the complex plane [4 Then the integral J transforms into J = e iβ dx exp qx α sx) A) where q = A expiαβ) and s = il expiβ) The integral on the right hand side of Eq A) converges if Req) > Res) > To ensure that both inequalities are satisfied for α = 5/ and for convenience we choose β = π/5 The exact value of Eq A) can be found in [4; it is expressed in terms of the Meijer G-function [44 see Eq 5b) We mention that the same method can also be used to find b for other α = l/k where l k are positive integers and in particular for α = and α = In the latter case the Meijer G-function simplifies to the expressions for b given by Eqs ) and 8) respectively The relative crosstalk amplitude derived directly from Eqs 5a) and 5b) reads where 5 b = 5 π) Γ ) 8 5 [ ) Re z 5 G 5 5 z A) z = i 5 5 t 5 γ / l 5/ To show that b is a function of the sole parameter ξ/r we use the identity [44 ) ) z σ G mn a pq z r = G b mn a pq z r + σ A) r b r + σ where a r b r ) represent the upper lower) vectors of the Meijer G-function eg in Eq A) a r = { /5 /5 /5 4/5}) and the summation in the right-hand side of A) is to be understood elementwise Applying the identity A) to Eq A) we arrive at the result /5 b = 5 π) Γ ) 8 A4) 5 [ ) Re G 5 iγ 5 5 ξ 5 5 π 5 6 r Whereas the limiting values and of b for ξ and ξ respectively can be directly computed from Eq A4) the asymptotics of b for ξ/r or w /r l ) is easier to deduce directly from Eqs A8) A) and A) It follows from A8) and A) that b = Re [J/πa) Using Watson s lemma [4 we obtain the asymptotic series expansion of the integral A): J = e iβ s ) n n= n! q s α ) n Γ + nα) A5) Substituting the values α = 5/ β = π/5 s = il e iπ/5 q = 688 /6 t 5/ l 5/6 e iπ/ and a = π q /5 e iπ/5 Γ+/5) into the ratio b = Re [J/πa) after some simple algebra we obtain )4 ) ) t t 6 t t ) 6 b O[ l ξ = 87 r l )8 ) ξ +78 r l l ) [ 6 ξ ξ O r r ) A6) where the top line reproduces 6) and the bottom line is used in Fig Note an increase of the subsequent expansion coefficients in A6) Starting from some n which value depends on t and l these coefficients dominate the decreasing magnitude of t /l ) 4+5n)/ in the asymptotic expansion of b for t = 5 and l = 5 this occurs at n = 7) This behavior indicates divergence of Eq A5) the common feature of asymptotic series [46 However a finite number of terms in Eq A5) yields an accurate asymptotic description of b We empirically established that the first three terms of the series A6) provide the best overlap with the exact behavior of b at l 5 and t 5 ie for 9 ξ/r 45) see Figs and

8 8 [ L Allen M W Beijersbergen R J C Spreeuw and J P Woerdman Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes Phys Rev A ) [ G Molina-Terriza J P Torres and L Torner Twisted photons Nat Phys 5 7) [ S Franke-Arnold L Allen and M Padgett Advances in optical angular momentum Las & Photon Rev 99 8) [4 G Gibson et al Free-space information transfer using light beams carrying orbital angular momentum Opt Express ) [5 J T Barreiro T-C Wei and P G Kwiat Beating the channel capacity limit for linear photonic superdense coding Nat Phys 4 8 8) [6 H Bechmann-Pasquinucci and A Peres Quantum Cryptography with -State Systems Phys Rev Lett 85 ) [7 M Bourennane A Karlsson and G Björk Quantum key distribution using multilevel encoding Phys Rev A 64 6 ) [8 V I Tatarskii Wave Propagation in a Turbulent Medium McGraw-Hill New York 96) [9 G Vallone et al Free-Space Quantum Key Distribution by Rotation-Invariant Twisted Photons Phys Rev Lett 65 4) [ A Sit et al High-dimensional intracity quantum cryptography with structured photons Optica 4 6 7) [ M Krenn et al Twisted photon entanglement through turbulent air across Vienna Proc Natl Acad Sci USA 497 5) [ M Krenn et al Twisted light transmission over 4 km Proc Natl Acad Sci USA 648 6) [ A K Ekert Quantum cryptography based on Bell s theorem Phys Rev Lett ) [4 F Mintert A R Carvalho M Kuś and A Buchleitner Measures and dynamics of entangled states Phys Rep ) [5 R Horodecki P Horodecki M Horodecki and K Horodecki Quantum entanglement Rev Mod Phys ) [6 A Mair A Vaziri G Weihs and A Zeilinger Entanglement of the orbital angular momentum states of photons Nature 4 ) [7 R Fickler et al Quantum Entanglement of High Angular Momenta Science 8 64 ) [8 M P van Exter E R Eliel and J P Woerdman in The Angular Momentum of Light edited by D L Andrews and M Babiker Cambridge University Press Cambridge UK ) [9 B-J Pors C H Monken E R Eliel and J P Woerdman Transport of orbital-angular-momentum entanglement through a turbulent atmosphere Opt Express ) [ A Hamadou Ibrahim et al Orbital-angular-momentum entanglement in turbulence Phys Rev A 88 ) [ Y Zhang et al Experimentally observed decay of highdimensional entanglement through turbulence Phys Rev A 94 6) [ S K Goyal et al The effect of turbulence on entanglement-based free-space quantum key distribution with photonic orbital angular momentum J Opt ) [ B Ndagano et al Characterizing quantum channels with non-separable states of classical light Nat Phys 97 7) [4 C Gopaul and R Andrews The effect of atmospheric turbulence on entangled orbital angular momentum states New J Phys ) [5 X Sheng et al Effects of low-order atmosphereturbulence aberrations on the entangled orbital angular momentum states Opt Lett 7 67 ) [6 F S Roux Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence Phys Rev A 8 58 ) [7 A H Ibrahim F S Roux and T Konrad Parameter dependence in the atmospheric decoherence of modally entangled photon pairs Phys Rev A ) [8 F S Roux T Wellens and V N Shatokhin Entanglement evolution of twisted photons in strong atmospheric turbulence Phys Rev A 9 6 5) [9 F S Roux Biphoton states in correlated turbulence Phys Rev A ) [ B J Smith and M G Raymer Two-photon wave mechanics Phys Rev A ) [ N D Leonhard V N Shatokhin and A Buchleitner Universal entanglement decay of photonic-orbitalangular-momentum qubit states in atmospheric turbulence Phys Rev A ) [ W K Wootters Entanglement of Formation of an Arbitrary State of Two Qubits Phys Rev Lett ) [ L C Andrews and R L Phillips Laser Beam Propagation through Random Media second ed SPIE Press Bellingham 5) [4 R Buckley Diffraction by a random phase-changing screen: A numerical experiment J Atmos Terr Phys ) [5 C M Kropf C Gneiting and A Buchleitner Effective Dynamics of Disordered Quantum Systems Phys Rev X 6 6) [6 C Paterson Atmospheric turbulence and free-space optical communication using orbital angular momentum of single photons Proc SPIE ) [7 R A Bertlmann K Durstberger and B C Hiesmayr Decoherence of entangled kaons and its connection to entanglement measures Phys Rev A 68 ) [8 D L Fried Statistics of a Geometric Representation of Wavefront Distortion J Opt Soc Am ) [9 D L Fried Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and Very Short Exposures J Opt Soc Am ) [4 J C Leader Atmospheric propagation of partially coherent radiation J Opt Soc Am ) [4 N Chandrasekaran and J H Shapiro Photon Information Efficient Communication Through Atmospheric Turbulence Part I: Channel Model and Propagation Statistics J Lightwave Technol 75 4) [4 G F Carrier M Krook and C E Pearson Functions of a Complex Variable: Theory and Technique SIAM

9 9 New York 5) Vol 49 [4 A P Prudnikov Y A Brychkov and O Marichev Integrals and Series: Direct Laplace Transforms Gordon and Breach Science Publishers New York 99) Vol 4 [44 H Bateman Higher Transcendental Functions McGraw- Hill Book Co Inc New York 95) Vol I [45 Handbook of Mathematical Functions with Formulas edited by M Abramowitz and I A Stegun National Bureau of Standards Washington DC 964) Vol 55 [46 R B Dingle Asymptotic Expansions: Their Derivation and Interpretation Academic Press London 97) [47 A P Prudnikov Y A Brychkov and O Marichev Integrals and Series: Special functions Gordon and Breach Science Publishers New York 99) Vol [48 X Yan et al Decoherence of orbital angular momentum tangled photons in non-kolmogorov turbulence J Opt Soc Am A 8 6) [49 N Leonhard et al Protecting the entanglement of twisted photons by adaptive optics Phys Rev A 97 8) [5 A Erdélyi Asymptotic expansions Dover Publications Inc New York 956)