3.1

Quantum channel and protocol parameters

The protocols are essentially influenced by the excess noise on the channel output, ϼ, further fixed to 10^–4 shot-noise units (SNU), which are the vacuum quadrature fluctuations. This complies with the experiment in the 100-km optical fiber with the total attenuation of -20 dB, where the noise on the channel input was estimated as 3% SNU³⁴ and with the recent experiment in the 300 km long low-loss fiber with the total attenuation of -32 dB, where the noise at the channel input was estimated in the worst case as 3.8% SNU.³⁵ Note that the excess noise is mainly concerned with imperfect parameter estimation from the homodyne data on the receiving side of the protocol (even if the channel noise is physically absent, the pessimistic assumption on the level of noise, related to noise estimation error, results in effectively non-zero level of channel noise in order to comply with the required probability of failure of channel estimation procedures²⁵), which substantially depends on the stability of the set-up. It is essential that we fix the noise at the channel output contrary to the standard approach in CV QKD, when noise was fixed as relates to the channel input and then scaled by the channel attenuation, making the assessment of the channel noise in long-distance channels too optimistic. Alternatively and taking into account the fact that the excess noise atop of the calibrated electronic noise of the detector appears most likely due to the imperfect estimation of a noiseless quantum channel, one may assume passive eavesdropping such that no untrusted channel excess noise is present. In the case of satellite-based links, where line of sight between the sender and the receiver suggests the absence of equipment capable of active eavesdropping, this is a particularly sensible assumption and it was applied recently for feasibility study of DV QKD over satellite links.³⁶ We also take this assumption into account in CV QKD by assuming the excess noise to be trusted (i.e., being out of control by an eavesdropper) and including it in the state purification using the scheme similar to the entangling cloner with a strongly unbalanced coupling to the signal prior to detection.³⁰

For the data ensemble size, we rely on the typical passage time of 300 seconds observed for the Micius quantum satellite.¹⁰ Assuming repetition rate of a CV QKD system to be of order of GHz, which is challenging but feasible with the current technology,³⁷ we may expect N = 10¹¹ data points acquired during a satellite passage, half of which will then contribute to the key. Lastly, post-processing efficiency is taken β = 0.95, complying with the currently available algorithms.³⁸

The maximum tolerable channel attenuation gives the idea of the protocols applicability independently of the aperture setting and random atmospheric disturbances, i.e. based only on the link optical budget. The results are summarized in Table 1 for given security assumptions and protocol parameters (signal states and clock rates).

Table 1.

Tolerable levels of channel attenuation (rounded) for various security assumptions and optimized CV QKD protocol settings. Excess noise on the channel output is fixed to ϵ = 10-4 SNU, and optimal squeezing is VS ≥ 0.1 SNU. During a passive attack the excess noise is presumed to be trusted.

Evidently, in the assumption of passive eavesdropping there is no substantial difference between collective and individual attacks. On the other hand, when the channel excess noise is assumed to be untrusted, relaxing the assumptions on the possible attacks from collective to individual ones can substantially extend the tolerable loss. Note that the use of squeezed states typically increases the tolerable channel attenuation by 4-6 dB depending on the attack assumption (the better improvement being observed upon more strict collective attacks). Furthermore, with the high repetition rate the squeezed-state protocol can tolerate from 30 to 44 dB of channel attenuation depending on the attack assumption, making it potentially feasible in the geostationary scenario.

Performance of CV QKD over short-range (terrestrial) free-space links can be essentially limited by quantum channel transmittance fluctuations,^{39, 40} also referred to as fading, which are mainly caused by the atmospheric turbulence effects of beam wander,⁴¹ when a beam spot travels around the receiving aperture. The channel fading then results in increase of the channel noise, detected on the receiver.³⁹ Even though this effect will be partially compensated for in the long-distance realization of CV QKD, where beam spot drastically expands during the propagation, which results in channel stabilization at the cost of increasing the overall loss⁴² as well as by active beam tracking and stabilization systems, it must be taken into account in the realistic CV QKD security analysis. However, as the residual transmittance fluctuations due to atmospheric effects are slow (of the order of KHz⁴³) compared to high achievable repetition rates of CV QKD systems (enabled by GHz rates of homodyne detectors³⁷), the fading can be further compensated for by properly grouping the data according to estimated relatively stable transmittance values,⁴⁴ which we also apply in our study.

3.2

Satellite-to-ground channel model

In our study we consider downlink satellite channels, as the turbulence effects, being destructive for CV QKD,³⁹ are less pronounced in this regime compared to the uplink scenario, where the signal is strongly affected by the atmosphere already in the beginning of the channel.⁴⁵ Optical Gaussian beam in the satellite-to-ground link is influenced by analytically predictable systematic and statistical effects occurring during the communication window. Aside from losses within the receiver optical system η_det, due to coupling and detection inefficiencies, systematic effects also include diffraction and refraction induced losses. The main source of loss is beam-spot broadening caused by diffraction. The broadening limits the maximal transmission efficiency η₀ as finite receiving aperture of size a truncates and measures only a part of the incoming collimated beam with spot-size W. The latter is largely determined as W = θ_dL(ζ) by divergence of the beam θ_d and the channel length L(ζ). Line-of-sight distance, also referred to as slant range, between the observer and the satellite depends on the exact position of the latter. In following we assume a perfectly circular Low-Earth orbit within observers meridian plane, characterized by the altitude above the ground H, ranging from 200 km to 2000 km. The slant range is obtained as

where R_⊕ is Earth radius and ζ is zenith angle i.e. between the line pointing in opposite direction to the gravity from the observer and the slant range, see also Figure 2. Additionally, air density and optical refractive index change with the altitude causing the bending of the beam and making the overall traveling distance longer than the straight geometrical slant range. The refraction induced elongation also depends on the signal wavelength, geographical position and altitude of the receiver, as well as atmospheric conditions (relative humidity, temperature, wind, pressure). We assume the communication window is established up to the zenith angle ζ_max = 70°, when the difference between actual and perceived slant ranges is small.⁴⁶ While such limitation reduces the communication window and the size of data block, it also avoids contributions from the longest propagation distance overall and through the thickest air mass.^{47, 48} Duration of communication window, i.e. total time in view of the satellite t, determines the amount of accumulated data points N for given source repetition rate. The time in view is calculated from geometrical considerations and satellite orbital velocity, which for circular orbit in the observers meridian plane can be simplified as:^?

Figure 2.

Geometrical representation of satellite-to-ground communication scheme. H is the circular orbit altitude; R_⊕ is Earth radius; ζ ∈ [−Pi/2,Pi/2] is the angle between the ray in the meridian plane, pointing above the observer (Bob), and the direct line connecting Bob and satellite (Alice), in practice limited to ±ζ_max; and L(ζ) is the slant range.

where G is the gravitational constant, and M_⊕ is Earth mass.

The volume of air mass is related to the atmospheric extinction ratio η_ext, which is a measure of Rayleigh scattering, scattering due to aerosols, and molecular absorption experienced by the signal beam in terrestrial atmosphere. The value of extinction ratio η_ext,ζ=0 depends on the the signal wavelength, air constituents, temperature and weather conditions, and its change with increase of the path length through the atmosphere (in terms of zenith angle) can be approximated (for angles up to ζ_max where refraction effects are small) as:⁴⁹

The ratio at zenith can be obtained based on MODTRAN atmospheric transmittance and radiance model,⁵⁰ which for rural or urban sea-level Mid-latitude location with clear sky visibility yields η_ext = 0.908.

Aside from aforementioned effects, refraction and diffraction are also caused by wind shear and temperature fluctuations and consequently by spatial and temporal variations of refractive index in the channel. Such perturbations result in scintillation, deviation of beam-spot from the center of the receiving aperture, and deformation of the beam-spot. For satellite links, the beam-spot radius is always significantly larger than the aperture, i.e., W > a, which allows us to ignore the deformation of the Gaussian beam profile, hence making beam wandering the dominant effect governing the fluctuation statistic of the channel transmittance.

The maximal transmission efficiency is reached when incoming signal is perfectly aligned with the receiving aperture (r = 0), and is defined by the ratio a/W of the aperture and beam spot sizes as follows

Efficiency η ∈ [0, η₀] decreases with the increase of deflection distance r, with approximate analytical solution⁴¹ being

where λ and R are shape and scale parameters respectively:

where I_n is n-th order Bessel function. The position of deflected beam center is assumed to be normally fluctuating around the aperture center,⁵¹ and described by random transverse vector r₀:

where beam-wandering variance is limited by tracking accuracy and beam stabilization σ_BW = θ_pL(ζ), and r = |r₀|². The analytical form of the probability distribution of atmospheric transmittance is the log-negative Weibull distribution:⁵¹

We simulate the values of η for each zenith angle ζ ∈ [−ζ_max, ζ_max] within the communication window with respect to the clock rate of the CV QKD protocol, and combine it with systematic receiver loss η_det, and respective extinction ratio η_ext (ζ) to obtain an overall mean transmittance 〈η_tot〉 (as well as of the satellite pass. Hence the transmittance statistics during an overall communication window consists of contributions from individual simulated free-space channels at every permitted zenith angle ζ. Both 〈η_tot〉 and govern the evolution of a covariance matrix of the state shared between Alice and Bob over the fluctuating channel.⁵² Note that we evaluate the mutual information between the trusted parties, I_AB, from the overall covariance matrix, averaged over the whole transmittance distribution (similarly to evaluating the Holevo bound), which we observe to be lower than the average mutual information, hence being the pessimistic estimate. Following is the list of parameters used in the simulation: the receiver aperture radius a = 0.5 (m) (respective plots shown in green color) 0.75 (m) (shown in blue), and 1 (m) (red), wavelength λ = 1550(nm), detection efficiency η_det = −3 dB,⁴⁷ tracking and pointing accuracy θ_p = 1.2(μrad),²³ and beam divergence θ_d = 10(μrad).¹⁰ The resulting mean transmittance for a single pass at altitude H is shown in Figure 3.

Figure 3.

Mean signal loss in satellite-to-ground channel dependence on orbit altitude H with receiver aperture radius a = 0.5, 0.75, 1 m (from top to bottom respectively). Single pass transmittance distribution is assembled from simulated individual links at every zenith angle within -ζ_max < ζ < ζ_max.

3.3

Security evaluation

We first assess the security by considering the individual attacks in asymptotic regime (see Figure 4 left). In this regime, security can be established for every LEO satellite altitude and feasible squeezing V_S ≥ 0.1 (SNU) provides a noticeable rate gain. Note that under the individual attacks, the higher levels of squeezing always translate into a higher secure key rate, which is not the case for collective attacks where the transmittance fluctuations limit the applicable values of squeezing⁵³ and require active squeezing optimization based on the estimated atmospheric transmittance distribution. The optimization of the latter is a crucial step required for the establishing of secure key rate based on the data generated from a single pass of the satellite.

Figure 4.

(Left) Asymptotic lower bound on the key rate (in bits per channel use) of the optimized coherent-state (dashed) and squeezed-state CV QKD protocols (solid) secure against active individual attacks for a single pass of a LEO satellite with orbit altitude H and receiver aperture radius a = 0.5, 0.75m (green and blue lines respectively). Channel excess noise at the output ϵ = 10^-4 SNU. (Right) The key rate (in bits per channel use) of optimized coherent-state (dashed) and squeezed-state protocol (solid) under passive collective attacks for a single pass of a LEO satellite with orbit altitude H, and receiver aperture radius a = 0.5, 0.75m (green and blue lines respectively). Note that the coherent-state protocol can only be securely established with larger aperture a = 0.75m. The overall block size depends on the length of communication window (given by the orbit altitude h) and the clock rate (from bottom to top) 100 MHz, 1 GHz, and 10 GHz. Trusted channel noise at the output is 10^-4 SNU. Both squeezing and modulation variance are optimized (with optimal squeezing limited by a feasible value as V_S ≥ 0.1) in order to establish a secure key regardless of the altitude.

The performance of the optimized CV QKD protocol under passive collective attacks with trusted noise ϵ = 10^-4 is depicted in Figure 4 (right). The impact of finite-size effects reduces with an increase of the altitude H as the communication window gets longer and consequently the overall block size N gets larger. Low altitude satellite downlinks exhibit less mean attenuation, but such advantage is partially offset by larger confidence intervals of estimated channel parameters and shorter raw key. While optimization of estimation block-size N − n can lengthen the key to some extent, increasing the repetition rate of the system is necessary to greatly extend the raw key. Higher repetition rate is especially crucial for the coherent-state protocol that can be implemented at the altitudes above 500 km only with 10 GHz clock rates. Increasing the size of receiving aperture also leads to significant improvement of the secure key rate.

Mean channel attenuation and fading noise, originating from transmittance fluctuations, diminish the tolerance to the channel excess noise, as shown in Figure 5. Evidently, both protocols are extremely sensitive to excess noise and no secure key can be generated at any orbit altitude if the noise at the output of the squeezed-state protocol is ϵ ≥ 2 · 10^-3 SNU for a = 0.5, or if the noise ϵ ≥ 5 · 10^-3 for a = 0.75, with orders of magnitude lower values needed for security of the coherent-state protocol (ϵ > 6 · 10^-5 at a = 0.5, or ϵ ≥ 4 · 10^-4 at a = 0.75). Note that with an increase of satellite orbit altitude the rate at which the protocol looses noise tolerance decreases. As the optical channel becomes longer it also becomes more stable,^{39, 42} i.e. both mean attenuation and fading (viewed as the variance of transmittance fluctuations are simultaneously reduced.

Figure 5.

Maximal tolerable excess noise in the case of active collective attacks on squeezed- or coherent-state protocols (solid and dashed lines respectively) with receiver aperture size a = 0.5 (left) and 0.75 (right). From top to bottom: asymptotic regime, finite-size regime with repetition rate (from bottom to top) 100 MHz, 1 GHz, and 10 GHz. Squeezing V_S ≥ 0.1 and modulation variance V_M are optimized.

While the effect at LEO altitudes is more apparent for the coherent-state protocol, the same dependency can be expected for the squeezed-state protocol at higher altitudes (MEO or GEO). Furthermore, channel stabilization will be more pronounced with less accurate beam-tracking θ_p which directly limits the beam-wandering and consequently channel fading.

Clearly, in order to operate under collective attacks with untrusted channel noise, the noise has to be limited to very low values. This can be achieved by proper control of the set-up or precise parameter estimation, however one can as well reduce the amount of fading noise by dividing the overall single-pass data block into a subset of smaller blocks.³⁹ Data clusterization with respect to channel attenuation allows to compensate the effect of channel transmittance fluctuations,⁴⁴ however such post-processing can be demanding for satellite-based QKD and is not needed for slow systematic changes of transmittance during the satellite pass. In the current work we therefore adopt simpler albeit similar method by splitting the satellite tracked pass into a set of segments and generating the key for each one. This allows to achieve the following lower bound on the overall secure key rate as a weighed sum of the key rates from individual segments:

where n_i is the raw key length for a given segment i with number of data points N_i, so that ∑_iN_i = N, N_i − n_i of data points is used for segment channel estimation, and the weight is determined by the relative size of the segment n_i/N, with N being the overall block size for a given satellite pass. The segments are chosen in accordance with the zenith angle ζ, so that for i = 1, 2, 3 we obtain 3 segments each containing measurement results at respectively [−ζ_max, −2/3ζ_max) ∪ (2/3ζ_max, ζ_max], [−2/3ζ_max, −1/3ζ_max) ∪ (1/3ζ_max, 2/3ζ_max], and [−1/3ζ_max, 1/3ζ_max]. The finite-size effects are stronger within each segment and transmittance fluctuations are substantial, yet reduced. Three segments are already sufficient to attain an enhanced positive secure key rate and extend the range of secure altitudes, as shown in Figure 6. For systems with smaller apertures this implies an effective increase of no tracking zone ζ_max, as the segment characterized by the longer slant ranges L(ζ) might not contribute to the overall key.

Figure 6.

Secure key rate without channel subdivision (solid lines) and with channel subdivision in 3 segments (dashed lines) versus orbit altitude H (km), secure against collective attacks in the finite-size regime with number of data points determined by (from darker to lighter shade) repetition rate 100MHz, 1GHz, 10GHz for the squeezed-state protocol with optimized squeezing V_S≥ 0.1 in the presence of untrusted excess noise ϵ = 10^-4 related to the channel output. Aperture size a = 0.5m (left) and a = 0.75m (right).