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Satellite-Based Continuous-Variable Quantum Communications: State-of-the-Art and a Predictive Outlook

Nedasadat Hosseinidehaj, Robert Malaney, Soon Xin Ng, Lajos Hanzo

arXiv:1712.09722v2quant-phcs.CRcs.IT

TL;DR

Long-range satellite quantum communication has been demonstrated in the discrete-variable regime, motivating extension to continuous-variable systems. This paper surveys satellite-based CV communication, emphasizing entanglement distribution and CV-QKD, and concludes that experimental deployment is plausible while finite-key effects remain an open issue.

  • Problem

    Satellite quantum communication has advanced through discrete-variable demonstrations, but the capabilities and challenges of extending these systems to continuous-variable communication require systematic characterization.

  • Method

    The paper surveys satellite-based CV quantum communication, focusing on CV entanglement distribution, CV-QKD, security analysis, architectures, and interfaces with terrestrial networks.

  • Results

    Satellite-based CV quantum communication is characterized as experimentally plausible, supported by Micius DV demonstrations and satellite optical-signal homodyne detection demonstrating CV-QKD feasibility.

  • Takeaways & Limitations

    The reported advantages of CV technology warrant experimental deployment toward satellite-based quantum communications.

  • Takeaways & Limitations

    Finite-key effects remain an open issue for space-based QKD because low-Earth-orbit satellites provide short transit times.

Abstract

from arXiv · show

The recent launch of the Micius quantum-enabled satellite heralds a major step forward for long-range quantum communication. Using single-photon discrete-variable quantum states, this exciting new development proves beyond any doubt that all of the quantum protocols previously deployed over limited ranges in terrestrial experiments can, in fact, be translated to global distances via the use of low-orbit satellites. In this work, we survey the imminent extension of space-based quantum communication to the continuous-variable regime - the quantum regime perhaps most closely related to classical wireless communications. The CV regime offers the potential for increased communication performance and represents the next major step forward for quantum communications and the development of the global quantum internet.

I. MOTIVATION AND INTRODUCTION

The paper motivates satellite-based quantum communication as a route beyond terrestrial distance limits and surveys its continuous-variable realization. It frames CV communication around optical quadratures, satellite architectures, and emerging experimental evidence.

  • Quantum communication foundations: Quantum communication uses quantum states prepared from classical information, transmitted over optical or free-space channels, and decoded through quantum measurement.The basic communication sequence is illustrated schematically as preparation, transmission, and detection.
  • Motivation: Quantum communication is motivated by the possibility of more secure information sharing, with entanglement and superposition providing quantum effects absent from classical communication.QKD is identified as a key example whose unconditional security has been theoretically proved.
  • Continuous-variable encoding: CV encoding, the focus of this work, maps information onto optical-field quadratures in an infinite-dimensional Hilbert space.These quadratures are associated with the amplitude and phase of laser light and can be measured using homodyne or heterodyne detectors.
  • Communication channels: Terrestrial fiber and FSO channels have complementary advantages, but terrestrial FSO links remain distance-limited by factors including Earth’s curvature and line-of-sight blockages.Satellite links can avoid the terrestrial horizon limit and place most of the propagation path outside the atmosphere.
  • Experimental progress: Micius demonstrated satellite-based distribution of entangled photon pairs over 1203 km and quantum teleportation of single-photon qubits from ground to satellite.A separate experiment also demonstrated homodyne detection of optical signals transmitted from a geostationary satellite, supporting satellite-based CV-QKD feasibility.
  • Scope of the survey: The work surveys satellite-based CV quantum communication by focusing on CV entanglement distribution and its application to CV-QKD.The survey also considers possible uplink and downlink architectures for satellite implementation.

II. FREE-SPACE CHANNELS TO AND FROM SATELLITES

Free-space satellite channels experience losses from diffraction, atmospheric absorption and scattering, and turbulence, with uplinks and downlinks affected differently. Uplink beam-wandering is typically dominant, whereas downlink losses are more strongly associated with diffraction.

  • Sources of loss: Diffraction broadens optical beams, while atmospheric absorption and scattering attenuate them in wavelength-dependent ways.The paper assumes absorption and scattering can be largely mitigated through wavelength selection.
  • Sources of loss: Atmospheric turbulence arises from random refractive-index fluctuations and affects optical propagation through turbulent eddies of different scales.Large-scale eddies primarily produce refractive effects, while small-scale eddies mainly cause scintillation.
  • Uplink channels: Uplink beam-wandering is dominant because the beam traverses turbulence while narrower than the large-scale turbulent eddies.Turbulence-induced beam-spreading also occurs, making the time-averaged received beam wider than the diffraction-only beam.
  • Downlink channels: Downlink beam-wandering is less important than in uplinks because the satellite beam enters the turbulent atmosphere wider than the turbulent-eddy scale.Downlink photonic losses are likely dominated by diffraction effects.
  • Atmospheric fading channels: Atmospheric fading is characterized by a probability distribution p(η) for the channel transmission coefficient η = √ηt.The mean fading loss is expressed in decibels using the maximum transmission coefficient η0.

D. Beam-wandering model

The beam-wandering model describes random beam-center displacement and its resulting transmission fluctuations, initially for fixed beam width and then with turbulence-induced beam-spreading. The resulting fading is modeled with a log-negative Weibull distribution whose mean loss increases with beam-spot radius and beam-wandering variance, while downlink losses are generally much lower than uplink losses.

  • Beam-wandering model: Beam-wandering displaces the beam center randomly around a fixed point in the receiver aperture plane.The beam-center coordinates follow a two-dimensional Gaussian distribution, and the deflection distance follows a Ricean distribution.
  • Beam-wandering model: The channel transmission coefficient η depends on beam-deflection distance and is parameterized using γ, S, and η0.The receiver aperture radius β and beam-spot radius W determine the parameter h = (β/W)^2.
  • Beam-wandering model: With fixed beam width, the transmission-coefficient distribution is described by a log-negative Weibull distribution over η ∈ [0, η0].The model is reported to describe the operationally important distribution tail more accurately than the previously used log-normal distribution.
  • Beam-spreading effects: Including turbulence-induced beam-spreading makes beam-spot radius W and maximum transmission η0 fluctuate rather than remain fixed.The analysis assumes the beam remains circular and averages over the fluctuating beam width.
  • Beam-spreading effects: Increasing beam-spot radius W raises mean fading loss, with W = 0.8 producing 2.7 dB and W = 2 producing 5.5 dB under the stated parameters.The comparison fixes σb = 0.7, β = 1, and d = 0.
  • Beam-spreading effects: Increasing beam-wandering standard deviation σb raises mean fading loss, with σb = 1.5 producing 7.4 dB and σb = 5.5 producing 17.8 dB under the stated parameters.The comparison fixes W = 1.1, β = 1, and d = 0.
  • Downlink implications: Downlink optical losses are usually orders of magnitude lower than uplink losses because downlink diffraction dominates rather than uplink beam-wandering.The paper links this asymmetry to the beam encountering the main turbulence-inducing atmospheric layers close to its target.

E. Estimation of a FSO channel

The section develops the continuous-variable field formalism underlying free-space optical channels, representing each mode as a quantum harmonic oscillator and describing its quadratures, operators, and multimode structure.

  • FSO channel estimation: Atmospheric transmission can be estimated in real time using auxiliary classical pulses or a strong local oscillator because atmospheric fluctuations are much slower than typical transmission rates.The considered fluctuations occur at rates of a few kHz, at least a thousand times slower than typical transmission rates.
  • CV mode representation: A single electromagnetic mode is modeled as a quantum harmonic oscillator, with electric and magnetic fields playing roles analogous to position and momentum.The single-mode Hamiltonian is formally equivalent to that of a harmonic oscillator of unity mass.
  • Field operators: Bosonic annihilation and creation operators obey [a, a†] = 1 and provide a non-Hermitian operator description of the field.These operators have time-dependent free-evolution forms and are used to define the quadrature operators.
  • Quadratures: The quadrature operators represent the amplitudes of the electric field’s cosine and sine components and satisfy a noncommuting relation.The quadratures are 90° out of phase, and their commutation relation imposes quantum uncertainty.
  • Multimode systems: An N-mode CV system uses a tensor-product Hilbert space, with mode-specific bosonic operators and Fock states labeled by photon number.Each mode has a vacuum state containing no field quanta.
  • Conventions and assumptions: The quadrature convention is not uniform across the literature because authors use different normalizations for ℏ, ω, and the quadrature operators.The section notes that dimensionless conventions and alternative prefactors are common.

A. Gaussian quantum states

Gaussian quantum states are characterized by first moments and covariance matrices, providing a compact description of common CV states, squeezing, and Gaussian entanglement.

  • State characterization: Gaussian states are completely characterized by the first moments of their quadratures and a covariance matrix of second moments.Their covariance matrix must satisfy the uncertainty condition M + iΩ ≥ 0.
  • Examples: Vacuum, coherent, thermal, and squeezed states are identified as important Gaussian-state examples, with thermal noise representing generic quantum-channel noise.The Wigner function of a Gaussian state is a Gaussian distribution of quadrature variables.
  • Single-mode squeezing: Single-mode squeezing reduces one quadrature variance below vacuum noise while increasing the variance of the conjugate quadrature.For positive squeezing parameter r_s, V(q) < 1 and V(p) > 1 in the stated convention.
  • Two-mode squeezing: Two-mode squeezing acts on superpositions of modes rather than independently squeezing each mode, producing complementary squeezed and anti-squeezed variances.The stated relations are V(q−) = V(p+) = exp(−2r) and V(q+) = V(p−) = exp(2r).
  • Gaussian entanglement: For r > 0, EPR quadrature correlations indicate bipartite entanglement, which increases with the two-mode squeezing parameter.The two-mode squeezed vacuum is identified as the most commonly used Gaussian entangled state.
  • State generation: Gaussian entangled squeezed states can be generated by parametric down conversion in a non-degenerate optical parametric amplifier.A pumped nonlinear crystal converts an incoming photon into lower-energy signal and idler photons.

B. Homodyne detection

Homodyne detection measures a selected field quadrature by interfering a target mode with a bright local oscillator and taking the difference of two photodetector currents.

  • Detection setup: Homodyne detection combines the target mode and a bright coherent local oscillator at a balanced beam splitter.The target mode is described by annihilation operator a, while the local oscillator has amplitude α_LO.
  • Photodetection: The two output modes are detected by photodetectors whose photocurrents are proportional to the corresponding photon numbers.The detectors convert electromagnetic-mode photons into electrons and then electrical currents.
  • Difference measurement: The difference current is proportional to α_LO* a + α_LO a† and provides the homodyne measurement signal.This difference removes the common contribution and couples the measurement to the field quadrature.
  • Phase selection: The measured quadrature is selected through the local oscillator phase: Θ = 0 measures q, whereas Θ = π/2 measures p.The local oscillator is written as α_LO = |α_LO| exp(iΘ).
  • Heterodyne contrast: Heterodyne detection measures q and p simultaneously by mixing the target with a vacuum ancillary mode and applying homodyne detection to conjugate output quadratures.This simultaneous measurement introduces an additional noise term from the injected vacuum state.

C. CV entanglement

The section introduces measures of bipartite CV entanglement, including entanglement of formation, entropy of entanglement, and logarithmic negativity, with covariance-matrix formulas for Gaussian states.

  • CV entanglement: Bipartite CV entanglement concerns correlations between two continuous-variable quantum systems and is treated as a resource for quantum information and communications.The systems are associated with Hilbert spaces H_A and H_B.
  • Entropy of entanglement: For pure bipartite states, entropy of entanglement quantifies the entanglement through the von Neumann entropy of reduced density operators.It represents the number of entangled qubits that can be extracted or the entanglement required to generate the state.
  • Gaussian-state entropy: For Gaussian states, the von Neumann entropy is computed from the symplectic eigenvalues of the covariance matrix.The entropy is expressed as a sum of g(ν_k) over the symplectic eigenvalues.
  • Entanglement of formation: Entanglement of formation is the minimum entanglement required to prepare a mixed state from ensembles of pure entangled states.It is generally difficult to calculate because it involves optimization over pure-state decompositions.
  • Logarithmic negativity: Logarithmic negativity is an easily computable upper bound on distillable entanglement and does not increase on average under local operations and classical communication.It is zero for separable states and is based on the negativity of the partially transposed state.
  • Gaussian logarithmic negativity: For two-mode Gaussian states, logarithmic negativity can be obtained from the smallest symplectic eigenvalue of the partially transposed covariance matrix.The covariance matrix is represented in block form using 2 × 2 real matrices A, B, and C.

D. Gaussian lossy quantum channel

A fixed-attenuation channel is modeled as a Gaussian channel that preserves Gaussian states while applying transmissivity and thermal noise. Beam-splitter transformations describe the resulting quadrature variances for one-mode and two-mode transfers.

  • Channel model: A fixed-attenuation channel is characterized by transmissivity τ and thermal noise variance Vn.In the optical frequency domain, thermal noise is effectively vacuum noise, whereas millimeter-wave operation can require cryogenic temperatures to suppress it.
  • Channel model: A beam splitter models how channel loss and thermal noise transform a single-mode Gaussian input state.The channel maps Gaussian states to Gaussian output states despite attenuation.
  • Single-mode transfer: The received quadrature variance follows V(q′1) = τV(p1) + (1 −τ)Vn for the stated channel representation.The same channel representation is then used to analyze two-mode Gaussian states.
  • Single-mode transfer: In single-mode transfer, one mode of a two-mode squeezed vacuum state traverses the channel while the other remains unaffected.The resulting Gaussian state is represented by a zero-mean covariance matrix.
  • Two-mode transfer: In two-mode transfer, each mode traverses an independent fixed-attenuation channel with its own transmissivity and thermal-noise variance.The model assumes the two thermal noises are uncorrelated.

IV. CV-QKD

CV-QKD encodes information in optical-field quadratures and detects them with homodyne or heterodyne measurements. The section presents Gaussian prepare-and-measure protocols using squeezed or coherent states, followed by classical parameter estimation and reconciliation.

  • Protocol foundations: CV-QKD maps key information to optical-field quadratures and uses homodyne or heterodyne detectors instead of single-photon detection.The review focuses on CV technology as an alternative to discrete-variable QKD.
  • Protocol foundations: In Gaussian prepare-and-measure schemes, Alice sends modulated squeezed or coherent states through an insecure channel and Bob measures the received states.Gaussian states are modulated by Gaussian distributions in the standard scheme.
  • Squeezed-state protocols: Squeezed-state homodyne protocols randomly select one quadrature for preparation and measurement, then sift mismatched choices.Alice and Bob retain data when their selected quadratures agree.
  • Squeezed-state protocols: Heterodyne detection measures both quadratures and introduces vacuum noise, while offering better robustness against channel noise when Bob’s data reference error correction.This comparison concerns squeezed-state protocols.
  • Coherent-state protocols: Coherent-state heterodyne protocols use both real variables for key generation, eliminating sifting and potentially increasing secret key rates.The benefit comes at the cost of added vacuum noise from heterodyne detection.
  • Classical post-processing: Parameter estimation reveals a random subset of data to estimate channel transmissivity and noise before reconciliation and error correction.The estimates constrain the maximum information available to Eve.
  • Protocol variants: Gaussian CV-QKD has eight prepare-and-measure protocol choices combining squeezed or coherent states, homodyne or heterodyne detection, and direct or reverse reconciliation.Each prepare-and-measure scheme has an equivalent entanglement-based scheme.

A. CV-QKD security analysis

CV-QKD security analysis models Eve’s attacks and evaluates asymptotic key rates using equivalent entanglement-based representations. Gaussian collective attacks can be implemented with an entangling cloner, but their equivalence to coherent attacks is not established for finite keys.

  • Attack models: Coherent attacks allow Eve to prepare a globally entangled ancilla, interact it with signals, and store the output for later measurement.The stored quantum memory is measured after public classical post-processing.
  • Key-rate analysis: The asymptotic key rate against collective attacks is K = ξIAB − IE, combining reconciliation efficiency, Alice–Bob mutual information, and Eve’s Holevo bound.Perfect reconciliation corresponds to ξ = 1.
  • Gaussian collective attacks: In the entangling-cloner model, Eve reproduces channel attenuation and noise with a TMSV ancilla and a beam splitter of transmissivity τ.Bob receives one beam-splitter output, while Eve stores the other output and the ancilla mode.
  • Gaussian analysis: Gaussian operations preserve the Gaussian character of Gaussian states because they are linear in the quadrature amplitudes.This property supports Gaussian-state security analysis.
  • Scope limitation: Gaussian collective attacks are as strong as coherent attacks asymptotically, but this equivalence is not known for realistic finite-length keys.The limitation concerns finite-size security rather than the infinite-sample setting.

A. Entanglement distribution and standard QKD protocols

Satellite architectures distribute continuous-variable entanglement between ground stations and then apply homodyne or heterodyne CV-QKD. The reviewed protocols include trusted-source, untrusted-source, and measurement-device-independent configurations, with practical limits from atmospheric loss and squeezing.

  • Entanglement distribution: Satellite architectures distribute a two-mode entangled state between ground stations by transmitting modes through satellite links.Entanglement swapping at the satellite provides an explicit relay-based implementation.
  • Standard QKD protocols: After entanglement distribution, Alice and Bob can apply homodyne or heterodyne detection to their modes and invoke entanglement-based CV-QKD.The resulting shared entanglement supports key generation across the considered architectures.
  • Untrusted sources: Secure keys can still be generated when the satellite controls the entanglement source.This result applies to entanglement-based CV-QKD protocols.
  • Measurement-device-independent QKD: In measurement-device-independent CV-QKD, Alice and Bob can generate a secret key even when the satellite-controlled Bell measurement is untrusted.The relay acts as the intermediate measurement device rather than a trusted endpoint.
  • Measurement-device-independent QKD: The equivalent entanglement-based MDI scheme starts with one TMSV state owned by each party and performs a continuous-variable Bell measurement at the relay.The relay receives one mode from each party and announces the measurement results.
  • Measurement-device-independent QKD: Homodyne and heterodyne detection at the trusted parties correspond respectively to squeezed-state and coherent-state prepare-and-measure MDI protocols.The satellite announces Bell-measurement results, after which Alice and Bob modify their data and perform classical post-processing.
  • Performance and assumptions: CV-MDI-QKD is feasible for low-loss fixed-attenuation channels and can provide beneficial secure key rates over high-loss atmospheric channels.The protocol assumes trusted devices at Alice and Bob, while the satellite measurement device may be adversarial.
  • Performance and assumptions: MDI protocols reduce assumptions about the measurement device while retaining unconditional security, although an adversary can reduce the key rate to zero.Full device-independent CV-QKD remains practically limited by very low expected key rates.

C. Entanglement determination and quantum key rate computation

The analysis distinguishes unknown and independently varying fading coefficients, models the resulting ensemble-averaged states, and notes that covariance-matrix calculations capture only Gaussian entanglement for non-Gaussian mixtures.

  • The channel is treated in two operational settings: single-mode transfer and two-mode transfer through atmospheric fading channels.
  • Single-mode transfer: For single-mode transfer, one mode remains at the ground station or satellite while the other crosses a fading uplink or downlink.The transmitted mode is described by a random coefficient η with distribution p(η) and maximum transmission coefficient η0; the output state is averaged over η.
  • Ensemble-averaged states: When the initial state is Gaussian, fading produces a non-Gaussian mixture of Gaussian states rather than a state fully characterized by first and second moments.Consequently, covariance-matrix entanglement quantifies only the Gaussian component, not the total distributed entanglement.
  • Quantum key rate: The same ensemble-averaged-state concept is used when calculating quantum key rates for entanglement-based continuous-variable QKD over atmospheric fading channels.The supplied text notes that covariance-matrix-based key-rate calculations concern the Gaussian component.
  • Two-mode transfer: For two-mode transfer, each mode crosses an independent fading downlink characterized by its own distribution and maximum transmission coefficient.The resulting density operator is formed by averaging over both transmission coefficients η1 and η2.

2) Scenario 2. The transmission coefficient of the fading channel can be measured:

Measuring the fading coefficient enables calculations within individual fading bins, where Gaussian input states remain Gaussian and entanglement or key rates can be accumulated across channel realizations. High satellite losses still motivate post-selection and distillation strategies.

  • Measuring η with a separate coherent signal increases system complexity but enables entanglement and key-rate calculations conditioned on the channel coefficient.A polarized local oscillator can be sent through the channel to estimate the transmission coefficient at the receiver.
  • Within each sufficiently small fading bin, a Gaussian input produces a Gaussian received state, so covariance matrices determine the total entanglement for that bin.The transmission coefficient is treated as constant during each bin, preserving Gaussianity of the collected state.
  • For EB CV-QKD, relatively long atmospheric coherence times may allow key rates to be derived for each fading realization and summed.The same bin-wise treatment is described for entanglement and quantum key-rate calculations.
  • 25-30 dB uplink losses and 5-10 dB downlink losses are cited for LEO satellite channels, motivating post-selection and entanglement distillation.The passage states that high losses make satellite entanglement distribution and QKD impractical without such interventions.
  • Post-selection retains states from high-transmission windows, while distillation uses quantum measurements to extract states with higher entanglement.Post-selection estimates η using coherent pulses and applies a threshold; the described distillation uses a tapped beam, homodyne detection, and a threshold qth.
  • Both strategies act as Gaussification methods, but the resulting entanglement increase is probabilistic.Conditioned states approach a Gaussian form because low-loss states become more concentrated in the final ensemble.

A. Non-Gaussian entangled states

The section introduces probabilistically generated non-Gaussian entangled states and their channel analysis, emphasizing that non-Gaussian states require density-operator methods beyond covariance matrices. Their benefit depends on creation probability, transmission conditions, and rate accounting.

  • Non-Gaussian entangled states are generated by applying photon subtraction, photon addition, or photon replacement to Gaussian TMSV states.The operations are applied probabilistically, typically at the receiver, to create photon-subtracted, photon-added, or photon-replaced squeezed states.
  • Photon subtraction: Photon subtraction heralds a pure state when both photon-number-resolving detectors register k photons, with creation probability 0 < Psb < 1.For the illustrated operation, k = 1.
  • Photon addition and replacement: Photon addition and photon replacement likewise herald pure non-Gaussian states through simultaneous detector outcomes, each with probability strictly between zero and one.Photon addition uses on/off detectors registering vacuum, while photon replacement uses photon-number-resolving detectors registering one photon; both require creation of additional photons.
  • Channel analysis: Unlike Gaussian states, non-Gaussian-state evolution cannot be analyzed solely through a covariance matrix, so the section uses a Kraus representation to analyze channel output states.The output is a non-Gaussian mixed state, and numerical truncation is required to approximate its infinite-dimensional density operator while keeping the trace near one.
  • Performance indicators: Non-Gaussian entanglement distribution is evaluated using both entanglement E and entanglement-generation rate RE = Pc E.Pc is the creation probability of the initial non-Gaussian state.
  • Performance comparison: With just-in-time state creation, direct Gaussian transmission maximizes entanglement-generation rate, but some non-Gaussian states enhance entanglement transfer when transmission rates are equalized.The equalization example is associated with quantum memory.
  • CV-QKD: For PSS-based CV-QKD, the reported key-rate expression is a lower bound, while actual rates require evaluating K(η) from the mixed non-Gaussian output density operator.

VII. COMPARISON WITH DISCRETE-VARIABLE TECHNOLOGIES

Space-based CV quantum communication is compared with DV systems across implementation costs, channel losses, detector robustness, and unresolved deployment choices. The paper also identifies finite-key security, adaptive reconciliation, error correction, and network integration as priorities for satellite CV systems.

  • DV versus CV: DV systems are less severely affected by photon loss, whereas CV losses introduce vacuum noise that reduces Alice–Bob correlations.
  • DV versus CV: CV-QKD uses lower-cost implementations and may support higher key rates than DV-QKD, whose performance is constrained by single-photon sources and detectors.
  • DV versus CV: Homodyne detectors in CV systems are comparatively robust to background light and can operate in daylight with reduced additional filtering.
  • Deployment choices: Whether DV or CV should carry quantum information in space remains open, with hybrid architectures proposed for time-varying atmospheric conditions.
  • Open research issues: Finite-key effects are especially important for satellite QKD because LEO satellites provide short transit times, and the analysis depends on the selected CV-QKD protocol.
  • Open research issues: Satellite CV systems require adaptive-rate reconciliation that maintains high efficiency across rapidly varying SNRs, alongside further work on non-Gaussian error correction.
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