Abstract

We consider the non-horizontal distributions of orbital angular momentum biphotons through free-space atmospheric channels, in which case the non-Kolmogorov turbulent effects shall be considered. By considering the case of initial non-perfect resource, i.e., the orbital angular momentum biphotons are initially prepared in an extended Werner-like state, we investigate the non-Kolmogorov effects on the propagations of nonclassical correlations, including quantum entanglement and quantum discord. It is found that universal decay laws of entanglement and discord also exist for non-Kolmogorov turbulence but with their decay curves different from that of entanglement for Kolmogorov turbulence reported by Leonhard et al. [Phys. Rev. A 91, 012345 (2015) [CrossRef]  ]. We show that the universal decay laws are dependent on the power-law exponent of the non-Kolmogorov spectrum and compare the differences of decay properties between entanglement and discord caused by non-Kolmogorov turbulence.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

Orbital angular momenta (OAM) of photons have been playing an important role in quantum information science (QIS) [17], which provide a new degree of freedom for the encoding of nonclassical information, such as entanglement [8]. However, the beam’s wavefront will be distorted during free-space propagation due to the refractive index fluctuation of turbulent atmosphere, leading to random phase aberrations [9] as well as the unavoidable decay of quantum information encoded. Therefore, the understanding of turbulent effects on OAM states is crucial. In recent years, there has been much theoretical [1015] and experimental [1619] research concerning the influences of atmospheric turbulence and the protection of OAM photons from decoherence caused by a turbulent atmosphere.

On the other hand, the quantitative exploration of quantum information encoded in OAM photons needs first to specify a proper measure of nonclassical information. Quantum entanglement is a kind of nonclassical correlation considered as a fundamental resource in QIS [8], and it is usually measured by Wootters’ concurrence [20]. However, it has been found that entanglement is not the unique resource utilized in QIS and there exist other resources such as quantum discord [2123]. A quantifier of discord, termed as local quantum uncertainty (LQU) [24], has been recently put forward by Girolami et al., which meets all the known bona fide criteria. The LQU is exactly computable as Wootters’ concurrence and, more importantly, it has a geometrical and observable interpretation.

Recently, the entanglement decay of OAM states in Kolmogorov atmospheric turbulence has been reported via concurrence [15]. However, the atmospheric turbulence will show obvious non-Kolmogorov characteristics when light beams propagate along the vertical direction [25,26]. The entanglement decay has been investigated with a multiple phase screen approach for non-Kolmogorov turbulence with inner and outer scale [27]. In Ref. [27], the maximally entangled state of two OAM photons has been considered, and concurrence is described by the ratio of the beam waist radius and the spatial coherence radius. Besides, the beams are horizontally propagating, and the power-law exponent of non-Kolmogorov spectrum is only chosen to be 11/3 (Kolmogorov) and 3.6 with weak turbulence. In this paper, we consider the non-horizontal distributions of orbital angular momentum biphotons through free space atmospheric channels, in which case the non-Kolmogorov turbulent effects have to be considered. By considering the case of initial non-perfect resource, i.e., the OAM biphotons are initially prepared in an extended Werner-like state, we investigate the non-Kolmogorov effects on the propagations of nonclassical correlations, including quantum entanglement and quantum discord. We instead explore the nonclassical correlations versus the ratio of phase correlation length and the Fried-like parameter, and find that universal decay laws of entanglement and discord also exist for non-Kolmogorov turbulence, but with their decay curves different from that of entanglement for Kolmogorov turbulence reported by Ref. [15]. The dependence of universal decay laws on the power-law exponent of the non-Kolmogorov spectrum and the differences of decay properties between entanglement and discord caused by non-Kolmogorov turbulence are also discussed.

This paper is organized as follows. In Section 2, we describe the model and derive the photonic OAM state influenced by the non-Kolmogorov turbulent atmosphere. In Section 3, the evolutions of the concurrence and the LQU of the initial extended Werner-like state as well as their decay laws are discussed. Conclusions are presented in Section 4.

2. ENTANGLED OAM STATE THROUGH ATMOSPHERIC TURBULENCE

The model considered is sketched in Fig. 1. Two Laguerre–Gaussian (LG) beams [28], carrying pairs of entangled photons with opposite azimuthal quantum number l and l [6], are emitted non-horizontally from a source at the ground and then propagate through the atmospheric turbulence in two symmetrical channels. We assume the propagating direction is along the z axis and the photons are detected at the altitude h, which form a zenith angle θ of the communication channel. This model can be established for quantum communication including quantum key distribution and quantum teleportation. For instance, Charlie (the sender) in the valley uses the resource and sends the entangled states through turbulent atmosphere to Alice and Bob (the two receivers) on two mountains with the altitude h. The propagation distance is of the order of kilometers (km), and we will consider a distance of approximately 1 km to illustrate the transfer of an entangled resource in weak-to-moderate turbulence. Also, we consider that the mountains are about 0.5 km high and the zenith angle is about π/3. The wavefunction of the LG beam is given by [28]

Up|l|(r,z,ϕ)=2p!π(p+|l|)!1ω(z)[2rω(z)]|l|exp[r2ω2(z)]×Lp|l|[2r2ω2(z)]exp(ilϕ)exp[ikr2z2(z2+zR2)]×exp[i(2p+|l|+1)tan1(zzR)]=Rp|l|(r,z)exp(ilϕ)2π.

 

Fig. 1. Source at the ground produces pairs of OAM-entangled photons propagating through turbulent atmosphere along the z direction and received by two detectors.

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The radial part of the LG beam can be expressed as

Rp|l|(r,z)=2p!(p+|l|)!1ω(z)[2rω(z)]|l|exp[r2ω2(z)]×Lp|l|[2r2ω2(z)]exp[ikr22R(z)]×exp[i(2p+|l|+1)tan1(zzR)],
where l is the azimuthal quantum number, p is the radial quantum number, ω(z)=ω01+(z/zR)2 is the spot size, ω0 is the waist of the input LG modes, R(z)=z[1+(zR/z)2] is the radius of wavefront curvature, zR=12kω02 is the Rayleigh range, k=2π/λ is the wavenumber, and
Lp|l|(x)=m=0p(1)m(p+|l|)!(pm)!(m+|l|)!m!xm,
represents generalized Laguerre polynomials.

In the case considered here, the non-Kolmogorov effect of the atmospheric turbulence is apparent [25,26]. The non-Kolmogorov spectrum of atmospheric turbulence can be written as [29]

ϕn(α,κ)=A(α)Cn2(z,α)κα,0κ,3<α<4,
where κ is the magnitude of spatial frequency, α is the power-law exponent of the non-Kolmogorov spectrum, Cn2(z,α) is the generalized index-of-refraction structure parameter, and
A(α)=Γ(α1)cos(απ/2)/(4π2),
with Γ() the Gamma function. In non-Kolmogorov atmospheric turbulence, the generalized index-of-refraction structure constant is given by [12]
Cn2(z,α)=0.033(kz)α11/3[0.00594(v/27)2×(zcosθ×105)10×exp(zcosθ/1000)+2.7×1016exp(zcosθ/1500)+C0exp(zcosθ/1000)]/A(α),
with units m3α. Here, we have zcosθ=h as shown in Fig. 1, root-mean-square (rms) wind speed v=21  m/s and C0 denoting a nominal value of Cn2(0) at the ground.

As the increase of the propagation altitude zcosθ, the change of the index-of-refraction structure constant is shown in Fig. 2. The index-of-refraction structure parameter is Cn2, which is important in atmospheric optics for the propagation of laser beam in the atmospheric turbulence [30]. It can be found in Fig. 2 that the index-of-refraction structure constant Cn2 increases with the increase of z at a short distance and then decreases fast, which means that the strength of the atmospheric turbulence decreases sharply when the propagation altitude is high enough [31]. It should be pointed out that, for fixed distance z, the index-of-refraction structure Cn2 increases when α increases, which corresponds to stronger turbulence. In this work, we consider weak-to-moderate turbulence with Cn2<1013  m3α and accordingly choose the proper values of α. The turbulent strength changes as the vertical distance increases during non-horizontal propagation, which is different from the case of horizontal propagation in Refs. [15,27].

 

Fig. 2. Cn2 as a function of z with C0=1014 and θ=π/3.

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Here, we assume that the waist of the input LG modes is ω0=0.1  m, the radial quantum number is p0=0, the azimuthal quantum numbers are l0 and l0, and the wavelength of the LG beams is chosen as 1550 nm, which is in the transmission window of the atmosphere. We study the effects of atmospheric turbulence at a distance near 1 km, and the diffraction of beams can be ignored for ω0=0.1  m [32]. The initial state of the photon pair is prepared in an extended Werner-like state defined as

ρ(0)=1γ4I+γ|Ψ0Ψ0|,
where 0γ1 characterizes the purity of the initial state, and the Bell-like state |Ψ0 reads as
|Ψ0=cos(ϑ2)|l0,l0+eiϕsin(ϑ2)|l0,l0,
with 0ϑπ and 0ϕ2π. Werner-like states, a special class of X-shaped mixed states, include both separable (unentangled) and entangled states, which are thus important for nonclassical experiment tests in QIS [3337]. In realistic experiments, quantum states prepared are usually not determined but mixed with an ensemble of pure states. The purity parameter γ is crucial for whether a Werner-like state is entangled or not, since it is unentangled only for 0γ1/(1+2sinϑ). The parameter ϑ determines the degree of the entanglement for Bell-like states in Eq. (8), which is maximally entangled for ϑ=π/2 as considered in Ref. [15] at γ=1 and ϑ=π/2. The action of the turbulent atmosphere on the entangled photon pair can be treated as a linear operator M, in terms of which the received state at the detectors can be expressed as
ρ=(M1M2)ρ(0),
where M1 and M2 are the individual action of the atmospheric turbulence on each photon. In this model, the photons propagate through two independent turbulence channels, and the turbulence has same influence on the OAM photons with M1=M2=M. The elements Ml˜,l˜l,l=pMpl˜,pl˜p0l,p0l of the linear operator M are given by [15]
Ml˜,l˜l,l=δll,l˜l˜2π0drR0l0(r)R0l0*(r)r×02πdΔφeiΔφ[(l˜+l˜)(l+l)]/2eDS(2r|sin(Δφ/2)|)/2,
where DS(r)=2(r/ϱ0)α2 is the phase structure function of non-Kolmogorov turbulence [31], with the spatial coherence radius of a spherical wave,
ϱ0(α)=[2Γ(3α2)π12k2Γ(2α2)z01Cn2(ξz,α)(1ξ)α2dξ]1/(α2).

The phase structure function of non-Kolmogorov turbulence will reduce to that of Kolmogorov turbulence as Dϕ(r)=6.88(r/r0)5/3 in Ref. [15]. In the following, we will have |l|=|l|=|l˜|=|l˜|=l0.

In the basis {|l0,l0,|l0,l0,|l0,l0,|l0,l0}, the density matrix of the input state from Eq. (7) can be written in an X form,

ρ(0)=(ρ11(0)ρ14(0)ρ22(0)ρ23(0)ρ32(0)ρ33(0)ρ41(0)ρ44(0)),
with
ρ11(0)=1γ4,ρ22(0)=1γ4+γcos2(ϑ2),ρ33(0)=1γ4+γsin2(ϑ2),ρ44(0)=1γ4,ρ14(0)=0,ρ23(0)=γ2eiϕsinϑ,ρ41(0)=ρ14(0)*,ρ32(0)=ρ23(0)*.

According to Ref. [15], the post-selected state at the detectors is given by

ρii,jjll,mmMiillMjjmmρ|lmlm|(0)|ijij|,
with the normalized density matrix elements
ρ11=(a2ρ11(0)+abρ22(0)+abρ33(0)+b2ρ44(0))/(a+b)2,ρ22=(abρ11(0)+a2ρ22(0)+b2ρ33(0)+abρ44(0))/(a+b)2,ρ33=(abρ11(0)+b2ρ22(0)+a2ρ33(0)+abρ44(0))/(a+b)2,ρ44=(b2ρ11(0)+abρ22(0)+abρ33(0)+a2ρ44(0))/(a+b)2,ρ14=a2ρ14(0)/(a+b)2,ρ41=a2ρ41(0)/(a+b)2,ρ23=a2ρ23(0)/(a+b)2,ρ32=a2ρ32(0)/(a+b)2,
and
a=Ml0,l0l0,l0=Ml0,l0l0,l0=Ml0,l0l0,l0=Ml0,l0l0,l0,b=Ml0,l0l0,l0=Ml0,l0l0,l0.

It is worth noting that ρ14=ρ41=0 since ρ14(0) and ρ41(0) vanish. Other non-vanishing matrix elements can be numerically evaluated by the integrals in Eq. (10).

3. NONCLASSICAL CORRELATION OF OAM STATE IN ATMOSPHERIC TURBULENCE

We proceed to study the nonclassical information contained in the output state. The first kind of nonclassical information we investigate is entanglement. To quantify the degree of entanglement, a measure of entanglement defined as concurrence C [20] will be used, which varies from C=0 for a separable state to C=1 for a maximally entangled state. For a two-qubit state ρ, the concurrence of this state is given by [20]

C(ρ)=max{λ1λ2λ3λ4,0},
where λi(i=1,2,3,4) are the eigenvalues of the matrix
R=ρ(σyσy)ρ*(σyσy),
with ρ* denoting the complex conjugation of ρ.

Then, we plot the concurrence C(ρ) as a function of the propagation distance z and the initial state parameter ϑ in Fig. 3(a). It is seen that the concurrence decays with the increase of the propagation distance z to vanishing in a non-asymptotical manner, a phenomenon that is termed as entangled sudden death (ESD) [38,39]. As clearly shown for ϑ near π/2, the decay rate at a short distance is much faster than that at long distance. The reason why concurrence decays fast at the beginning and then slows down is that the strength of the atmospheric turbulence is strong when the OAM beams propagate near the ground, and then the strength decreases fast with the altitude increasing, which is shown in Fig. 2. Besides, we plot the concurrence C(ρ) as a function of the propagation distance z and the initial state purity γ in Fig. 3(b). The concurrence goes to vanishing more quickly as the purity decreases. For 0γ1/3 with ϑ=π/2, the concurrence is always vanishing, since the initial state is unentangled, and no entanglement can be created by independent turbulent channels.

 

Fig. 3. Concurrence as a function of (a) z and ϑ and of (b) z and γ. The parameters are chosen as l0=1, C0=1014, α=3.8, θ=π/3 for (a) γ=1 and (b) ϑ=π/2.

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To explore the ESD phenomenon more precisely, we display the vanishing distance zc as a function of ϑ and of γ in Figs. 4(a) and 4(b), respectively. In Fig. 4(a), the vanishing distance zc increases rapidly as ϑ increases to π/2, and then it decreases fast as ϑ increases from π/2 to π, exhibiting a symmetric distribution as well as a peak at ϑ=π/2. It is demonstrated in Fig. 4(b) that the vanishing distance zc also decreases non-asymptotically to vanishing as the purity γ decreases from 1 to 1/3 for ϑ=π/2.

 

Fig. 4. Vanishing distance as functions of (a) ϑ and of (b) γ. The parameters are chosen the same as Fig. 3.

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Another important type of nonclassical information in QIS is quantum discord [2123]. In order to quantify the degree of discord, we use a measure termed as local quantum uncertainty (LQU) [24], which is defined by the minimal Wigner–Yanase skew information [40],

UA(ρAB)=min{KA}{I(ρAB,KAIB)}=12min{KA}{Tr([ρAB,KAIB]2)},
with {KA} a set of local observables measured on subsystem A, and IB the identity operator of subsystem B. For a 2×d quantum system, the LQU can be simply calculated as
UA(ρAB)=1λmax{WAB},
where λmax is the maximal eigenvalue of matrix WAB defined by
(WAB)ij=Tr[ρAB(σiAIB)ρAB(σjAIB)],
with σiA(i=x,y,z) the Pauli matrixes of subsystem A.

In Fig. 5(a), we plot the LQU as a function of the propagation distance z and the initial state parameter ϑ. Similar to the concurrence, the LQU also decays fast at the beginning and the decay gradually becomes slow. Different from the concurrence, the LQU does not vanish even when the OAM photons in the atmospheric turbulence have propagated at a long distance. When the propagation distance z is long enough, the LQU becomes insensitive to the increase of z. The LQU as a function of z and γ is also plotted in Fig. 5(b). The LQU decreases only in an asymptotical manner when the purity γ decreases from 1 to 0, which is also different from that of the concurrence shown in Fig. 3(b).

 

Fig. 5. LQU as a function of (a) z and ϑ and of (b) z and γ. The parameters are chosen the same as Fig. 3.

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Moreover, we also explore the effect of azimuthal quantum number l0 on the decay of concurrence and LQU. The decays of the concurrence and LQU with the increase of z with different azimuthal quantum numbers are shown in Figs. 6(a) and 6(b). Both concurrence and LQU decay more slowly with the increase of the azimuthal quantum number l0. It has been reported that the decay of concurrence may finally collapse onto a universal curve with the increase of l0 for the case of Kolmogorov atmospheric turbulence [15]. Here, we have shown that the larger azimuthal quantum number OAM photons carry, the more quantum-ness can be preserved through non-Kolmogorov atmospheric turbulence, which is rather different from that of Kolmogorov atmospheric turbulence.

 

Fig. 6. (a) Concurrence and (b) LQU as functions of z for different l0. The parameters are chosen as C0=1014, α=3.8, θ=π/3, γ=1, and ϑ=π/2.

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The effects of the power-law exponent α of the non-Kolmogorov spectrum on the decay of concurrence and LQU are also shown in Figs. 7(a) and 7(b), respectively. In Fig. 7(a), we see that the decay rates of concurrence and LQU become faster with the value of α increasing, which indicates the OAM photons are more deeply influenced by the atmospheric turbulence with the higher power-law exponent α. The ESD also becomes clearer as the power-law exponent increases (see purple dashed curve for α=3.9). In Fig. 7(b), the LQU decays faster at first, with α increasing while the decay rates become almost the same, and the curves for different α become almost parallel at long propagation distance z, which is also different from that of the concurrence in Fig. 7(a).

 

Fig. 7. (a) Concurrence and (b) LQU as functions of z for different α. The parameters are chosen as l0=1, C0=1014, θ=π/3, γ=1, and ϑ=π/2.

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In order to compare our results with those of Ref. [15], we introduce the phase correlation length of the LG beam [15],

ξ(l0)=sin(π2|l0|)ω02Γ(|l0|+3/2)Γ(|l0|+1),
and the Fried-like parameter [12],
r0(α)=[2Γ(3α2)(8α2Γ(2α2))(α2)/2π12k2Γ(2α2)z01Cn2(ξz,α)(1ξ)α2dξ]1/(α2).

We then plot the concurrence as a function of the ratio ξ(l0)/r0 for weak (α=3.1) and moderate (α=3.8) turbulence in Figs. 8(a) and 8(b). We gain two universal decay laws for α=3.1 and α=3.8 in the case of a large l0. The function of the universal curve in Fig. 8(a) is f(x)=exp(2.89x2.10), and in Fig. 8(b) is g(x)=exp(4.71x3.51), which are different from that for Kolmogorov turbulence reported in Ref. [15]. Besides, we also would like to explore the existence of a decay law for discord in Fig. 9. Two universal decay curves u(x)=0.93exp(2.86x1.22)+0.069 and w(x)=0.93exp(4.00x2.22)+0.067 are also found for weak [α=3.1 in Fig. 9(a)] and moderate [α=3.8 in Fig. 9(b)] turbulence, respectively. To discuss the power-law exponent α of the non-Kolmogorov spectrum on the decay law of entanglement and discord, in Fig. 10, we plot concurrence and LQU as functions of ξ(l0)/r0 and α. It can be found that the universal decay to vanishing becomes more fast when α increases with stronger turbulence.

 

Fig. 8. Concurrence as a function of the ratio ξ(l0)/r0 for different l0. The parameters are chosen as C0=1014, θ=π/3, γ=1ϑ=π/2, (a) α=3.1 and (b) α=3.8.

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Fig. 9. LQU as a function of the ratio ξ(l0)/r0 for different l0. The parameters are chosen the same as Fig. 8.

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Fig. 10. (a) Concurrence and (b) LQU as functions of ξ(l0)/r0 and α. The parameters are chosen as l0=100, C0=1014, θ=π/3, γ=1, and ϑ=π/2.

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4. CONCLUSIONS

In conclusion, we have investigated the decay properties of the nonclassical correlations (entanglement and quantum discord) of two photonic qubits propagating through the non-Kolmogorov atmospheric turbulence via concurrence and LQU. By considering that the photonic OAM beams are initially prepared in an extended Werner-like state and non-horizontally propagate through turbulence, we show that entanglement and quantum discord have different decay properties in non-Kolmogorov atmospheric turbulence for different cases, including the change of the propagation distance, the azimuthal quantum number, and the power-law exponent of the non-Kolmogorov spectrum. Comparing the decay properties of the concurrence and LQU with the increase of the propagation distance, we find that there exists ESD in the propagation of the entanglement while the LQU decays merely in an asymptotical manner. The effects of value of the azimuthal quantum number are also explored. The nonclassical correlations with a larger azimuthal quantum number can be more robust against the influence of atmospheric turbulence.

In addition, we also investigate the effects of the power-law exponent of the non-Kolmogorov spectrum. With the increase of the power-law exponent, the decay rates of the concurrence and LQU both become faster at the beginning, due to a larger α indicating stronger turbulence. The difference is that the entanglement suffers from ESD in shorter distances while quantum discord keeps almost constant at long distances. Moreover, we explore the decay properties of concurrence and LQU versus the ratio of the phase correlation length to the Fried-like parameter and find that universal decay laws of entanglement and discord also exist for non-Kolmogorov turbulence but with their decay curves different from that of entanglement for Kolmogorov turbulence reported by [15]. The universal decay laws are dependent on the power-law exponent of the non-Kolmogorov spectrum, and the universal decay to vanishing becomes faster with stronger turbulence.

Before ending, we shall remark that the results here, such as an increased stability for higher-order OAM modes, may be valid only for weak-to-moderate turbulence. For strong turbulence, different results have been shown by Roux et al. [41]. It would also be interesting to explore the turbulence in a strong regime, which is beyond the scope of this work. We believe our results are the first theoretical attempt to explore quantum discord in non-Kolmogorov atmospheric turbulence and may contribute to the understanding of OAM quantum communication in free-space communication with new quantum resources.

Funding

National Natural Science Foundation of China (NSFC) (11504140; 11504139); Natural Science Foundation of Jiangsu Province (BK20140128, BK20140167); Fundamental Research Funds for the Central Universities (JUSRP51517).

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24. D. Girolami, T. Tufarelli, and G. Adesso, “Characterizing nonclassical correlations via local quantum uncertainty,” Phys. Rev. Lett. 110, 240402 (2013). [CrossRef]  

25. A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008). [CrossRef]  

26. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007). [CrossRef]  

27. X. Yan, P. F. Zhang, J. H. Zhang, H. Q. Chun, and C. Y. Fan, “Decoherence of orbital angular momentum tangled photons in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 33, 1831–1835 (2016). [CrossRef]  

28. 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 45, 8185–8189 (1992). [CrossRef]  

29. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]  

30. J. I. Davis, “Consideration of atmospheric turbulence in laser system design,” Appl. Opt. 5, 139–147 (1966). [CrossRef]  

31. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

32. C. Paterson, “Atmospheric turbulence and free-space optical communication using orbital angular momentum of single photons,” Proc. SPIE 5572, 187–198 (2004). [CrossRef]  

33. R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989). [CrossRef]  

34. B. Bellomo, R. L. Franco, and G. Compagno, “Entanglement dynamics of two independent qubits in environments with and without memory,” Phys. Rev. A 77, 032342 (2008). [CrossRef]  

35. J. Lee and M. S. Kim, “Entanglement teleportation via Werner states,” Phys. Rev. Lett. 84, 4236–4239 (2000). [CrossRef]  

36. M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation,” Phys. Rev. A 60, 1888–1898 (1999). [CrossRef]  

37. A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006). [CrossRef]  

38. T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97, 140403 (2006). [CrossRef]  

39. T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323, 598–601 (2009). [CrossRef]  

40. E. P. Wigner and M. M. Yanase, “Information contents of distributions,” Proc. Natl. Acad. Sci. USA 49, 910–918 (1963). [CrossRef]  

41. F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015). [CrossRef]  

References

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  1. J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
    [Crossref]
  2. A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
    [Crossref]
  3. G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
    [Crossref]
  4. E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
    [Crossref]
  5. B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
    [Crossref]
  6. R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
    [Crossref]
  7. S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
    [Crossref]
  8. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
    [Crossref]
  9. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
    [Crossref]
  10. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94–101 (2007).
    [Crossref]
  11. F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
    [Crossref]
  12. X. Sheng, Y. Zhang, F. Zhao, L. Zhang, and Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. 37, 2607–2609 (2012).
    [Crossref]
  13. T. Brünner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
    [Crossref]
  14. J. R. Gonzalez Alonso and T. A. Brun, “Protecting orbitalangular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
    [Crossref]
  15. N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
    [Crossref]
  16. 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 19, 6671–6683 (2011).
    [Crossref]
  17. M. Malik, M. O’sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
    [Crossref]
  18. M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled twophoton beams,” Phys. Rev. A 88, 053836 (2013).
    [Crossref]
  19. B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
    [Crossref]
  20. W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
    [Crossref]
  21. L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899–6905 (2001).
    [Crossref]
  22. H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phys. Rev. Lett. 88, 017901 (2001).
    [Crossref]
  23. K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
    [Crossref]
  24. D. Girolami, T. Tufarelli, and G. Adesso, “Characterizing nonclassical correlations via local quantum uncertainty,” Phys. Rev. Lett. 110, 240402 (2013).
    [Crossref]
  25. A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
    [Crossref]
  26. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
    [Crossref]
  27. X. Yan, P. F. Zhang, J. H. Zhang, H. Q. Chun, and C. Y. Fan, “Decoherence of orbital angular momentum tangled photons in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A 33, 1831–1835 (2016).
    [Crossref]
  28. 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 45, 8185–8189 (1992).
    [Crossref]
  29. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
    [Crossref]
  30. J. I. Davis, “Consideration of atmospheric turbulence in laser system design,” Appl. Opt. 5, 139–147 (1966).
    [Crossref]
  31. L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).
  32. C. Paterson, “Atmospheric turbulence and free-space optical communication using orbital angular momentum of single photons,” Proc. SPIE 5572, 187–198 (2004).
    [Crossref]
  33. R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
    [Crossref]
  34. B. Bellomo, R. L. Franco, and G. Compagno, “Entanglement dynamics of two independent qubits in environments with and without memory,” Phys. Rev. A 77, 032342 (2008).
    [Crossref]
  35. J. Lee and M. S. Kim, “Entanglement teleportation via Werner states,” Phys. Rev. Lett. 84, 4236–4239 (2000).
    [Crossref]
  36. M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation,” Phys. Rev. A 60, 1888–1898 (1999).
    [Crossref]
  37. A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
    [Crossref]
  38. T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97, 140403 (2006).
    [Crossref]
  39. T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323, 598–601 (2009).
    [Crossref]
  40. E. P. Wigner and M. M. Yanase, “Information contents of distributions,” Proc. Natl. Acad. Sci. USA 49, 910–918 (1963).
    [Crossref]
  41. F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
    [Crossref]

2017 (1)

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

2016 (1)

2015 (2)

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

2014 (1)

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

2013 (4)

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled twophoton beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

D. Girolami, T. Tufarelli, and G. Adesso, “Characterizing nonclassical correlations via local quantum uncertainty,” Phys. Rev. Lett. 110, 240402 (2013).
[Crossref]

T. Brünner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
[Crossref]

J. R. Gonzalez Alonso and T. A. Brun, “Protecting orbitalangular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
[Crossref]

2012 (4)

X. Sheng, Y. Zhang, F. Zhao, L. Zhang, and Y. Zhu, “Effects of low-order atmosphere-turbulence aberrations on the entangled orbital angular momentum states,” Opt. Lett. 37, 2607–2609 (2012).
[Crossref]

M. Malik, M. O’sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
[Crossref]

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref]

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

2011 (3)

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

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 19, 6671–6683 (2011).
[Crossref]

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

2009 (3)

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
[Crossref]

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
[Crossref]

T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323, 598–601 (2009).
[Crossref]

2008 (2)

A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
[Crossref]

B. Bellomo, R. L. Franco, and G. Compagno, “Entanglement dynamics of two independent qubits in environments with and without memory,” Phys. Rev. A 77, 032342 (2008).
[Crossref]

2007 (3)

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[Crossref]

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94–101 (2007).
[Crossref]

2006 (2)

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97, 140403 (2006).
[Crossref]

2005 (1)

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[Crossref]

2004 (1)

C. Paterson, “Atmospheric turbulence and free-space optical communication using orbital angular momentum of single photons,” Proc. SPIE 5572, 187–198 (2004).
[Crossref]

2003 (1)

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref]

2002 (1)

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

2001 (2)

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899–6905 (2001).
[Crossref]

H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phys. Rev. Lett. 88, 017901 (2001).
[Crossref]

2000 (1)

J. Lee and M. S. Kim, “Entanglement teleportation via Werner states,” Phys. Rev. Lett. 84, 4236–4239 (2000).
[Crossref]

1999 (1)

M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation,” Phys. Rev. A 60, 1888–1898 (1999).
[Crossref]

1998 (1)

W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
[Crossref]

1995 (1)

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

1992 (1)

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 45, 8185–8189 (1992).
[Crossref]

1989 (1)

R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
[Crossref]

1966 (1)

1963 (1)

E. P. Wigner and M. M. Yanase, “Information contents of distributions,” Proc. Natl. Acad. Sci. USA 49, 910–918 (1963).
[Crossref]

’t Hooft, G. W.

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

Acín, A.

A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
[Crossref]

Adesso, G.

D. Girolami, T. Tufarelli, and G. Adesso, “Characterizing nonclassical correlations via local quantum uncertainty,” Phys. Rev. Lett. 110, 240402 (2013).
[Crossref]

Allen, L.

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 45, 8185–8189 (1992).
[Crossref]

Andrews, L. C.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[Crossref]

L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media, 2nd ed. (SPIE, 2005).

Andrews, R.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94–101 (2007).
[Crossref]

Babiker, M.

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

Barnett, S. M.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

Beijersbergen, M. W.

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 45, 8185–8189 (1992).
[Crossref]

Bellomo, B.

B. Bellomo, R. L. Franco, and G. Compagno, “Entanglement dynamics of two independent qubits in environments with and without memory,” Phys. Rev. A 77, 032342 (2008).
[Crossref]

Boyd, R. W.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

M. Malik, M. O’sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
[Crossref]

Brodutch, A.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

Brun, T. A.

J. R. Gonzalez Alonso and T. A. Brun, “Protecting orbitalangular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
[Crossref]

Brünner, T.

T. Brünner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
[Crossref]

Buchleitner, A.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

Cable, H.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

Chun, H. Q.

Compagno, G.

B. Bellomo, R. L. Franco, and G. Compagno, “Entanglement dynamics of two independent qubits in environments with and without memory,” Phys. Rev. A 77, 032342 (2008).
[Crossref]

Courtial, J.

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[Crossref]

da Cunha Pereira, M. V.

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled twophoton beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

Davis, J. I.

De Martini, F.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
[Crossref]

Eberly, J. H.

T. Yu and J. H. Eberly, “Sudden death of entanglement,” Science 323, 598–601 (2009).
[Crossref]

T. Yu and J. H. Eberly, “Quantum open system theory: bipartite aspects,” Phys. Rev. Lett. 97, 140403 (2006).
[Crossref]

Eliel, E. R.

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

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 19, 6671–6683 (2011).
[Crossref]

Fan, C. Y.

Ferrero, V.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[Crossref]

Fickler, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref]

Filpi, L. A. P.

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled twophoton beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

Franco, R. L.

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Girolami, D.

D. Girolami, T. Tufarelli, and G. Adesso, “Characterizing nonclassical correlations via local quantum uncertainty,” Phys. Rev. Lett. 110, 240402 (2013).
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A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
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Golbraikh, E.

A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
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Gonzalez Alonso, J. R.

J. R. Gonzalez Alonso and T. A. Brun, “Protecting orbitalangular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
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Gopaul, C.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94–101 (2007).
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L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899–6905 (2001).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation,” Phys. Rev. A 60, 1888–1898 (1999).
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Horodecki, P.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation,” Phys. Rev. A 60, 1888–1898 (1999).
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Horodecki, R.

R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81, 865–942 (2009).
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M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation,” Phys. Rev. A 60, 1888–1898 (1999).
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Jennewein, T.

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
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Karimi, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
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A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
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Krenn, M.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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Kupershmidt, I.

A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
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Lapkiewicz, R.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
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Leach, J.

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J. Lee and M. S. Kim, “Entanglement teleportation via Werner states,” Phys. Rev. Lett. 84, 4236–4239 (2000).
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N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
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S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
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Magaña-Loaiza, O. S.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
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Maher, L.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
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Malik, M.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
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M. Malik, M. O’sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
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Marrucci, L.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
[Crossref]

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A. Acín, N. Gisin, and L. Masanes, “From Bell’s theorem to secure quantum key distribution,” Phys. Rev. Lett. 97, 120405 (2006).
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B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
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B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

M. Malik, M. O’sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
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G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
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M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled twophoton beams,” Phys. Rev. A 88, 053836 (2013).
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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 19, 6671–6683 (2011).
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E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
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Ollivier, H.

H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phys. Rev. Lett. 88, 017901 (2001).
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Padgett, M. J.

Pan, J.-W.

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
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Paterek, T.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
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Piccirillo, B.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
[Crossref]

Plick, W. N.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref]

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B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

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 19, 6671–6683 (2011).
[Crossref]

Ramelow, S.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref]

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B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

M. Malik, M. O’sullivan, B. Rodenburg, M. Mirhosseini, J. Leach, M. P. J. Lavery, M. J. Padgett, and R. W. Boyd, “Influence of atmospheric turbulence on optical communications using orbital angular momentum for encoding,” Opt. Express 20, 13195–13200 (2012).
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B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
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F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
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T. Brünner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
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F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

Sansoni, L.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
[Crossref]

Santamato, E.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
[Crossref]

Schaeff, C.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref]

Sciarrino, F.

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
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Shatokhin, V. N.

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
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F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

Sheng, X.

Shtemler, Y.

A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
[Crossref]

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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 45, 8185–8189 (1992).
[Crossref]

Steinhoff, N. K.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Stribling, B. E.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
[Crossref]

Thirunavukkarasu, G.

S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
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Torner, L.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
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Torres, J. P.

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

Toselli, I.

I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free space optical system performance for laser beam propagation through non-Kolmogorov turbulence for uplink and downlink paths,” Proc. SPIE 6708, 670803 (2007).
[Crossref]

Tufarelli, T.

D. Girolami, T. Tufarelli, and G. Adesso, “Characterizing nonclassical correlations via local quantum uncertainty,” Phys. Rev. Lett. 110, 240402 (2013).
[Crossref]

Tyler, G. A.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Vaziri, A.

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref]

Vedral, V.

K. Modi, A. Brodutch, H. Cable, T. Paterek, and V. Vedral, “The classical-quantum boundary for correlations: discord and related measures,” Rev. Mod. Phys. 84, 1655–1707 (2012).
[Crossref]

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899–6905 (2001).
[Crossref]

Virtser, A.

A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
[Crossref]

Weihs, G.

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref]

Wellens, T.

F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92, 012326 (2015).
[Crossref]

Welsh, B. M.

B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-Kolmogorov atmospheric turbulence,” Proc. SPIE 2471, 181–196 (1995).
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B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

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 19, 6671–6683 (2011).
[Crossref]

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 45, 8185–8189 (1992).
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W. K. Wootters, “Entanglement of formation of an arbitrary state of two qubits,” Phys. Rev. Lett. 80, 2245–2248 (1998).
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Yanakas, M.

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
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E. P. Wigner and M. M. Yanase, “Information contents of distributions,” Proc. Natl. Acad. Sci. USA 49, 910–918 (1963).
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S. M. Lloyd, M. Babiker, G. Thirunavukkarasu, and J. Yuan, “Electron vortices: beams with orbital angular momentum,” Rev. Mod. Phys. 89, 035004 (2017).
[Crossref]

Zeilinger, A.

R. Fickler, R. Lapkiewicz, W. N. Plick, M. Krenn, C. Schaeff, S. Ramelow, and A. Zeilinger, “Quantum entanglement of high angular momenta,” Science 338, 640–643 (2012).
[Crossref]

A. Vaziri, J.-W. Pan, T. Jennewein, G. Weihs, and A. Zeilinger, “Concentration of higher dimensional entanglement: qutrits of photon orbital angular momentum,” Phys. Rev. Lett. 91, 227902 (2003).
[Crossref]

Zhang, J. H.

Zhang, L.

Zhang, P. F.

Zhang, Y.

Zhao, F.

Zhu, Y.

Ziberman, A.

A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
[Crossref]

Zurek, W. H.

H. Ollivier and W. H. Zurek, “Quantum discord: a measure of the quantumness of correlations,” Phys. Rev. Lett. 88, 017901 (2001).
[Crossref]

Appl. Opt. (1)

Atmos. Res. (1)

A. Ziberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov turbulence,” Atmos. Res. 88, 66–77 (2008).
[Crossref]

J. Opt. (1)

B.-J. Pors, F. Miatto, G. W. ’t Hooft, E. R. Eliel, and J. P. Woerdman, “High-dimensional entanglement with orbital-angular-momentum states of light,” J. Opt. 13, 064008 (2011).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. A (1)

L. Henderson and V. Vedral, “Classical, quantum and total correlations,” J. Phys. A 34, 6899–6905 (2001).
[Crossref]

Nat. Photonics (1)

E. Nagali, L. Sansoni, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, “Optimal quantum cloning of orbital angular momentum photon qubits through Hong-Ou-Mandel coalescence,” Nat. Photonics 3, 720–723 (2009).
[Crossref]

Nat. Phys. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nat. Phys. 3, 305–310 (2007).
[Crossref]

New J. Phys. (3)

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94–101 (2007).
[Crossref]

T. Brünner and F. S. Roux, “Robust entangled qutrit states in atmospheric turbulence,” New J. Phys. 15, 063005 (2013).
[Crossref]

B. Rodenburg, M. Mirhosseini, M. Malik, O. S. Magaña-Loaiza, M. Yanakas, L. Maher, N. K. Steinhoff, G. A. Tyler, and R. W. Boyd, “Simulating thick atmospheric turbulence in the laboratory with application to orbital angular momentum communication,” New J. Phys. 16, 033020 (2014).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. A (9)

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[Crossref]

M. V. da Cunha Pereira, L. A. P. Filpi, and C. H. Monken, “Cancellation of atmospheric turbulence effects in entangled twophoton beams,” Phys. Rev. A 88, 053836 (2013).
[Crossref]

J. R. Gonzalez Alonso and T. A. Brun, “Protecting orbitalangular-momentum photons from decoherence in a turbulent atmosphere,” Phys. Rev. A 88, 022326 (2013).
[Crossref]

N. D. Leonhard, V. N. Shatokhin, and A. Buchleitner, “Universal entanglement decay of photonic-orbital-angular-momentum qubit states in atmospheric turbulence,” Phys. Rev. A 91, 012345 (2015).
[Crossref]

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 45, 8185–8189 (1992).
[Crossref]

M. Horodecki, P. Horodecki, and R. Horodecki, “General teleportation channel, singlet fraction, and quasidistillation,” Phys. Rev. A 60, 1888–1898 (1999).
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Figures (10)

Fig. 1.
Fig. 1. Source at the ground produces pairs of OAM-entangled photons propagating through turbulent atmosphere along the z direction and received by two detectors.
Fig. 2.
Fig. 2. C n 2 as a function of z with C 0 = 10 14 and θ = π / 3 .
Fig. 3.
Fig. 3. Concurrence as a function of (a)  z and ϑ and of (b)  z and γ . The parameters are chosen as l 0 = 1 , C 0 = 10 14 , α = 3.8 , θ = π / 3 for (a)  γ = 1 and (b)  ϑ = π / 2 .
Fig. 4.
Fig. 4. Vanishing distance as functions of (a)  ϑ and of (b)  γ . The parameters are chosen the same as Fig. 3.
Fig. 5.
Fig. 5. LQU as a function of (a)  z and ϑ and of (b)  z and γ . The parameters are chosen the same as Fig. 3.
Fig. 6.
Fig. 6. (a) Concurrence and (b) LQU as functions of z for different l 0 . The parameters are chosen as C 0 = 10 14 , α = 3.8 , θ = π / 3 , γ = 1 , and ϑ = π / 2 .
Fig. 7.
Fig. 7. (a) Concurrence and (b) LQU as functions of z for different α . The parameters are chosen as l 0 = 1 , C 0 = 10 14 , θ = π / 3 , γ = 1 , and ϑ = π / 2 .
Fig. 8.
Fig. 8. Concurrence as a function of the ratio ξ ( l 0 ) / r 0 for different l 0 . The parameters are chosen as C 0 = 10 14 , θ = π / 3 , γ = 1 ϑ = π / 2 , (a)  α = 3.1 and (b)  α = 3.8 .
Fig. 9.
Fig. 9. LQU as a function of the ratio ξ ( l 0 ) / r 0 for different l 0 . The parameters are chosen the same as Fig. 8.
Fig. 10.
Fig. 10. (a) Concurrence and (b) LQU as functions of ξ ( l 0 ) / r 0 and α . The parameters are chosen as l 0 = 100 , C 0 = 10 14 , θ = π / 3 , γ = 1 , and ϑ = π / 2 .

Equations (23)

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U p | l | ( r , z , ϕ ) = 2 p ! π ( p + | l | ) ! 1 ω ( z ) [ 2 r ω ( z ) ] | l | exp [ r 2 ω 2 ( z ) ] × L p | l | [ 2 r 2 ω 2 ( z ) ] exp ( i l ϕ ) exp [ i k r 2 z 2 ( z 2 + z R 2 ) ] × exp [ i ( 2 p + | l | + 1 ) tan 1 ( z z R ) ] = R p | l | ( r , z ) exp ( i l ϕ ) 2 π .
R p | l | ( r , z ) = 2 p ! ( p + | l | ) ! 1 ω ( z ) [ 2 r ω ( z ) ] | l | exp [ r 2 ω 2 ( z ) ] × L p | l | [ 2 r 2 ω 2 ( z ) ] exp [ i k r 2 2 R ( z ) ] × exp [ i ( 2 p + | l | + 1 ) tan 1 ( z z R ) ] ,
L p | l | ( x ) = m = 0 p ( 1 ) m ( p + | l | ) ! ( p m ) ! ( m + | l | ) ! m ! x m ,
ϕ n ( α , κ ) = A ( α ) C n 2 ( z , α ) κ α , 0 κ , 3 < α < 4 ,
A ( α ) = Γ ( α 1 ) cos ( α π / 2 ) / ( 4 π 2 ) ,
C n 2 ( z , α ) = 0.033 ( k z ) α 11 / 3 [ 0.00594 ( v / 27 ) 2 × ( z cos θ × 10 5 ) 10 × exp ( z cos θ / 1000 ) + 2.7 × 10 16 exp ( z cos θ / 1500 ) + C 0 exp ( z cos θ / 1000 ) ] / A ( α ) ,
ρ ( 0 ) = 1 γ 4 I + γ | Ψ 0 Ψ 0 | ,
| Ψ 0 = cos ( ϑ 2 ) | l 0 , l 0 + e i ϕ sin ( ϑ 2 ) | l 0 , l 0 ,
ρ = ( M 1 M 2 ) ρ ( 0 ) ,
M l ˜ , l ˜ l , l = δ l l , l ˜ l ˜ 2 π 0 d r R 0 l 0 ( r ) R 0 l 0 * ( r ) r × 0 2 π d Δ φ e i Δ φ [ ( l ˜ + l ˜ ) ( l + l ) ] / 2 e D S ( 2 r | sin ( Δ φ / 2 ) | ) / 2 ,
ϱ 0 ( α ) = [ 2 Γ ( 3 α 2 ) π 1 2 k 2 Γ ( 2 α 2 ) z 0 1 C n 2 ( ξ z , α ) ( 1 ξ ) α 2 d ξ ] 1 / ( α 2 ) .
ρ ( 0 ) = ( ρ 11 ( 0 ) ρ 14 ( 0 ) ρ 22 ( 0 ) ρ 23 ( 0 ) ρ 32 ( 0 ) ρ 33 ( 0 ) ρ 41 ( 0 ) ρ 44 ( 0 ) ) ,
ρ 11 ( 0 ) = 1 γ 4 , ρ 22 ( 0 ) = 1 γ 4 + γ cos 2 ( ϑ 2 ) , ρ 33 ( 0 ) = 1 γ 4 + γ sin 2 ( ϑ 2 ) , ρ 44 ( 0 ) = 1 γ 4 , ρ 14 ( 0 ) = 0 , ρ 23 ( 0 ) = γ 2 e i ϕ sin ϑ , ρ 41 ( 0 ) = ρ 14 ( 0 ) * , ρ 32 ( 0 ) = ρ 23 ( 0 ) * .
ρ i i , j j l l , m m M i i l l M j j m m ρ | l m l m | ( 0 ) | i j i j | ,
ρ 11 = ( a 2 ρ 11 ( 0 ) + a b ρ 22 ( 0 ) + a b ρ 33 ( 0 ) + b 2 ρ 44 ( 0 ) ) / ( a + b ) 2 , ρ 22 = ( a b ρ 11 ( 0 ) + a 2 ρ 22 ( 0 ) + b 2 ρ 33 ( 0 ) + a b ρ 44 ( 0 ) ) / ( a + b ) 2 , ρ 33 = ( a b ρ 11 ( 0 ) + b 2 ρ 22 ( 0 ) + a 2 ρ 33 ( 0 ) + a b ρ 44 ( 0 ) ) / ( a + b ) 2 , ρ 44 = ( b 2 ρ 11 ( 0 ) + a b ρ 22 ( 0 ) + a b ρ 33 ( 0 ) + a 2 ρ 44 ( 0 ) ) / ( a + b ) 2 , ρ 14 = a 2 ρ 14 ( 0 ) / ( a + b ) 2 , ρ 41 = a 2 ρ 41 ( 0 ) / ( a + b ) 2 , ρ 23 = a 2 ρ 23 ( 0 ) / ( a + b ) 2 , ρ 32 = a 2 ρ 32 ( 0 ) / ( a + b ) 2 ,
a = M l 0 , l 0 l 0 , l 0 = M l 0 , l 0 l 0 , l 0 = M l 0 , l 0 l 0 , l 0 = M l 0 , l 0 l 0 , l 0 , b = M l 0 , l 0 l 0 , l 0 = M l 0 , l 0 l 0 , l 0 .
C ( ρ ) = max { λ 1 λ 2 λ 3 λ 4 , 0 } ,
R = ρ ( σ y σ y ) ρ * ( σ y σ y ) ,
U A ( ρ A B ) = min { K A } { I ( ρ A B , K A I B ) } = 1 2 min { K A } { Tr ( [ ρ A B , K A I B ] 2 ) } ,
U A ( ρ A B ) = 1 λ max { W A B } ,
( W A B ) i j = Tr [ ρ A B ( σ i A I B ) ρ A B ( σ j A I B ) ] ,
ξ ( l 0 ) = sin ( π 2 | l 0 | ) ω 0 2 Γ ( | l 0 | + 3 / 2 ) Γ ( | l 0 | + 1 ) ,
r 0 ( α ) = [ 2 Γ ( 3 α 2 ) ( 8 α 2 Γ ( 2 α 2 ) ) ( α 2 ) / 2 π 1 2 k 2 Γ ( 2 α 2 ) z 0 1 C n 2 ( ξ z , α ) ( 1 ξ ) α 2 d ξ ] 1 / ( α 2 ) .

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