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Continuous-Variable Quantum Key Distribution with Gaussian Modulation -- The Theory of Practical Implementations
Fabian Laudenbach, Christoph Pacher, Chi-Hang Fred Fung, Andreas Poppe, Momtchil Peev, Bernhard Schrenk, Michael Hentschel, Philip Walther, Hannes Hübel
TL;DR
Practical CV-QKD with Gaussian-modulated coherent states is promising but requires a comparatively involved security analysis. The paper rederives its theoretical foundations and develops hardware-linked noise models, while focusing on asymptotic security and practical implementation effects. Its models support estimates of how transmittance, excess noise, reconciliation efficiency, and hardware choices affect secure-key performance.
Problem
CV-QKD security analysis is more involved than the protocol itself, while practical implementations must account for hardware imperfections and security assumptions.
Source: §1 · p. 5Method
The paper pedagogically rederives CV-QKD relations and develops analytical noise models linked to hardware specifications, including covariance-based Eve-information calculations.
Source: §1 · p. 5, §7 · p. 20, §7.2 · p. 24Results
The models characterize secure-key performance through transmittance, excess noise, reconciliation efficiency, modulation variance, and hardware-dependent noise effects.
Source: §10 · pp. 46–47Takeaways & Limitations
The framework provides practical guidance for estimating implementation performance and identifying hardware choices that influence expected secure-key rates.
Source: §10 · p. 47The main security analysis is restricted to asymptotically long keys and collective attacks, although extensions are needed for finite-key effects.
Source: §1 · p. 5, §7.3 · p. 26
Abstract
from arXiv · showhide
Quantum key distribution using weak coherent states and homodyne detection is a promising candidate for practical quantum-cryptographic implementations due to its compatibility with existing telecom equipment and high detection efficiencies. However, despite the actual simplicity of the protocol, the security analysis of this method is rather involved compared to discrete-variable QKD. In this article we review the theoretical foundations of continuous-variable quantum key distribution (CV-QKD) with Gaussian modulation and rederive the essential relations from scratch in a pedagogical way. The aim of this paper is to be as comprehensive and self-contained as possible in order to be well intelligible even for readers with little pre-knowledge on the subject. Although the present article is a theoretical discussion of CV-QKD, its focus lies on practical implementations, taking into account various kinds of hardware imperfections and suggesting practical methods to perform the security analysis subsequent to the key exchange. Apart from a review of well known results, this manuscript presents a set of new original noise models which are helpful to get an estimate of how well a given set of hardware will perform in practice.
1. Introduction
The paper presents Gaussian-modulated coherent-state CV-QKD as a practical alternative whose security analysis remains more involved than the protocol itself. It combines a pedagogical review with hardware-oriented noise models and practical security considerations.
1. Introduction
- The tutorial derives CV-QKD security relations from scratch and develops analytical noise models linking quantum-information formalism to hardware datasheets.The models target practical implementation choices and are intended for beginners and experimentalists.Source: §1 · pp. 5–6
- CV-QKD with coherent states offers efficient, high-rate, cost-effective homodyne detection while avoiding squeezed-light generation.The paper contrasts homodyne receivers with single-photon counters and coherent-state protocols with squeezed-state protocols.Source: §1 · p. 5
- Experimental secure-key-rate comparisons are difficult because demonstrations use different assumptions about attacks, reconciliation, finite-key effects, trusted noise, modulation security, and side channels.Long-distance demonstrations transmitted the local oscillator through the quantum channel, creating security loopholes; newer work uses locally generated oscillators.Source: §1 · p. 5
- The article restricts its security analysis to infinitely long keys, although the derived formalism remains applicable to finite-key effects with suitable extensions.Finite-key security for coherent-state Gaussian-modulated CV-QKD is described as less advanced and still under active development.Source: §1 · p. 5
- Implementation imperfections are central because reported QKD attacks have targeted deviations from the theoretical model.This motivates the paper’s emphasis on realistic hardware and noise modeling.Source: §1 · pp. 5–6
2. Notions of Security and Secure-Key Rate
This section defines the adversary models and asymptotic security setting used for CV-QKD. It then formulates the practical reverse-reconciliation secret fraction from Alice–Bob information, Eve’s bound, and post-processing imperfections.
2. Notions of Security and Secure-Key Rate
- Eve’s attack models range from i.i.d. individual attacks to collective attacks with delayed joint measurement and fully general coherent attacks without an i.i.d. assumption.The coherent model permits globally correlated ancillas and collective measurement after classical post-processing.Source: §2 · p. 7
- Security against coherent attacks was reduced to covariance-matrix analyses using de Finetti methods and Gaussian-attack optimality, but finite-block security remains difficult [40, 41, 42, 43, 44, 15, 45, 46, 47, 48, 49].A later Gaussian de Finetti result was reported as addressing limitations of earlier finite-size approaches.Source: §2 · p. 7
2.1. Asymptotic Security against Collective Attacks
- The tutorial assumes collective attacks and asymptotically long keys for its main secure-key-rate analysis.This choice simplifies the derivation, while asymptotic rates provide an upper bound on corresponding non-asymptotic results.Source: §2.1 · p. 7, §2 · p. 7
- The asymptotic secret fraction follows the Devetak–Winter framework as mutual information IAB minus Eve’s Holevo information χ, with reverse reconciliation using χEB.The paper derives IAB and χEB for Gaussian-modulated coherent-state CV-QKD under reverse reconciliation.Source: §2.1 · p. 8
- Practical reconciliation modifies the secret fraction through reconciliation efficiency β, frame-error rate FER, and disclosed fraction ν used for covariance-matrix estimation.The resulting expression is a lower bound because χEB is an upper bound on Eve’s information.Source: §2.1 · p. 8
3. Basics of a Coherent-State Protocol
The protocol prepares Gaussian-modulated coherent states, transmits them through a Gaussian channel, and measures quadratures before classical post-processing. Parameter estimation, reconciliation, confirmation, privacy amplification, and authentication convert these data into a secure key.
3.1. Gaussian Modulation
- Alice samples independent Gaussian quadratures q and p to prepare displaced coherent states, whose ensemble variance equals modulation variance plus one shot-noise unit.The relation is V(q)=V(p)=Vmod+1, with Vmod=4˜Vmod and inherent variance V0=1.Source: §3.1 · p. 9, Figure 3.1 · p. 10
3. Basics of a Coherent-State Protocol
- Bob receives each coherent state through a Gaussian quantum channel and measures one or both quadratures using homodyne or heterodyne detection.The protocol’s quantum transmission is followed by classical processing.Source: §3 · p. 9, §3.1 · p. 10
3.2. Classical Post-Processing
- Alice and Bob reveal a random data subset to estimate transmittance and excess noise, then compute IAB and bound Eve’s information before deciding whether to abort.The protocol proceeds only when βIAB exceeds the bound χ.Source: §3.2 · pp. 10–11
- Reverse reconciliation overcomes the 3 dB loss limit of forward reconciliation by having Alice correct her data to Bob’s, although low transmission makes excess noise increasingly critical.The paper notes that reverse reconciliation can retain IAB greater than χ for any total transmission T under the stated assumptions.Source: §3.2 · p. 11
- Information reconciliation uses error-correction schemes such as slice or multidimensional reconciliation, with LDPC codes including MET LDPC codes for high-loss regimes.Confirmation uses universal hashing, privacy amplification uses seeded extraction, and authenticated classical communication prevents man-in-the-middle attacks.Source: §3.2 · p. 11
4. Transmittance and Noise
The paper separates transmittance from noise and defines excess noise as the aggregate of independent hardware and channel contributions. Its measurement-location convention supports analytical models for estimating practical implementation performance.
4. Transmittance and Noise
- CV-QKD performance is primarily characterized by transmittance T and noise, which lower signal-to-noise ratio and increase a potential eavesdropper’s advantage.The paper distinguishes channel and detection noise rather than combining transmission and noise into one parameter.Source: §4 · p. 12
- The paper defines excess noise where it is observed, at the channel output in Bob’s laboratory, and incorporates electronic noise into that quantity.This differs from conventions that define excess noise at the channel input.Source: §4 · pp. 12–13
- Realistic transmittance includes channel loss, detection loss, and coupling efficiency, while Bob’s measured variance combines transmitted signal variance, excess noise, vacuum noise, and electronic noise.The resulting convention keeps hardware contributions tied to their observation point.Source: §4 · pp. 12–13
- Excess noise is modeled as a sum of independent contributions from modulation, Raman scattering, quantisation, phase fluctuations, detection, relative-intensity noise, and common-mode rejection.The additive decomposition is justified by stochastic independence of the noise sources.Source: §4 · pp. 12–13
- Analytical models for the excess-noise components estimate how hardware choices affect total noise and expected secure-key rate.The models are intended to guide design choices for lasers, modulators, detectors, and analog-to-digital converters.Source: §4 · p. 13
5. Covariance Matrix
The covariance-matrix formalism represents quadrature variances and correlations, enabling analysis of Gaussian-modulated CV-QKD and calculation of Eve’s Holevo information. The section relates homodyne and heterodyne measurements and establishes the equivalence between prepare-and-measure and entanglement-based descriptions.
5. Covariance Matrix
- Covariance matrices encode quadrature variances on the diagonal and mutual covariances off diagonal, and are vital for computing the Holevo information.Source: §5 · p. 14
- The prepare-and-measure protocol with displaced coherent states is equivalent to an entanglement-based protocol using a two-mode squeezed vacuum state.The entanglement-based description measures both quadratures of one mode and sends the other mode to Bob.Source: §5 · p. 14, §5.2 · p. 16
5.1. Homodyne and Heterodyne Detection
- Homodyne detection measures one quadrature at a time, whereas heterodyne detection splits the signal and measures q and p simultaneously.Homodyne uses a randomly selected local-oscillator phase; heterodyne uses a balanced beamsplitter followed by two homodyne measurements.Source: §5.1 · p. 15, Figure 5.1 · p. 16
- Heterodyne detection doubles mutual information per symbol but introduces an additional 3 dB loss from the beamsplitter.Its covariance matrix is equivalent to the homodyne case with a factor of 1/2 applied to transmission and excess noise, while the Holevo information can be derived before Bob’s measurement apparatus.Source: §5.1 · p. 15
5.2. Prepare-and-Measure vs. Entanglement-Based Protocols
- Alice’s prepare-and-measure data must be rescaled to obtain the entanglement-based covariance matrix used by the simpler security analysis.This rescaling is crucial for determining the Holevo bound and must be included in parameter estimation.Source: §5.2 · pp. 17–18
6. Signal-to-Noise Ratio and Mutual Information
The section derives the signal-to-noise ratio and mutual information for homodyne and heterodyne CV-QKD, then relates these quantities to code rate and energy per bit. Heterodyne detection gains simultaneous access to both quadratures but reduces the SNR through its beamsplitter.
6. Signal-to-Noise Ratio and Mutual Information
- The SNR is determined by signal power divided by noise power, with transmittance, detection type, vacuum noise, and excess noise contributing to the received variance.The detection parameter is µ = 1 for homodyne and µ = 2 for heterodyne detection.Source: §6 · p. 19
- Mutual information between Alice and Bob gives the maximal code rate and is determined by the SNR.Source: §6 · p. 19
- Heterodyne detection changes the mutual-information expression from log2(1 + SNR)/2 to log2(1 + SNR), while reducing SNR by roughly a factor of 1/2.The reduction is not exactly 1/2 because the heterodyne beamsplitter also halves the excess noise.Source: §6 · p. 19
6.1. Relation to Normalised SNR
- The code rate relates raw bits to information bits and connects the SNR to the energies required to transmit raw and information bits.Source: §6.1 · p. 19
- The error-correction efficiency β satisfies 0 ≤ β < 1 and measures how closely a code of rate R achieves the mutual information IAB.Source: §6.1 · p. 19
7. Holevo Information
The section derives Eve’s Holevo information from covariance matrices and von Neumann entropies under concrete and universal purification-based analyses. It also distinguishes strict security assumptions, where receiver imperfections are attributed to Eve, from trusted-device assumptions.
7. Holevo Information
- The Holevo information χEB is the difference between Eve’s entropy before Bob’s measurement and her conditional entropy afterward.These entropies are determined from symplectic eigenvalues of the relevant covariance matrices.Source: §7 · p. 20
- Because von Neumann entropy is invariant under unitary transformations, any purification of Alice and Bob’s state gives the same Holevo information.This freedom allows the security calculation to avoid specifying which purification Eve holds.Source: §7 · p. 20
7.1. Entangling Cloner Attack
- The entangling-cloner analysis models Eve’s attack with a two-mode squeezed vacuum and beamsplitter, then computes her information from symplectic eigenvalues.Choosing Eve’s variance appropriately produces a covariance matrix representing lossy transmission with excess noise.Source: §7.1 · pp. 20–22
7.1.1. Homodyne Detection
- For homodyne and heterodyne detection, Bob’s measurement transforms Eve’s conditional covariance matrix, whose symplectic eigenvalues determine the conditional entropy.Source: §7.1.1 · pp. 22–23, §7.1.2 · p. 23
7.2. Universal Analysis of Purification Attacks
- Assuming Eve holds a purification makes her entropy equal to the entropy of Alice and Bob’s joint state, simplifying the Holevo calculation.The joint entropy is obtained from Alice and Bob’s covariance matrix, while the conditional entropy still depends on whether Bob uses homodyne or heterodyne detection.Source: §7.2 · p. 24
7.3. Strict and Loose Security Assumptions
- Under trusted-device assumptions, receiver noise and detection inefficiency need not contribute to Eve’s Holevo information, although they still affect Bob’s SNR and mutual information.Under strict assumptions, the entire noise and transmittance are attributed to Eve.Source: §7.3 · pp. 25–26
- Treating preparation noise as trusted is excluded because it was shown to open the door to severe side-channel attacks.Source: §7.3 · p. 26
8. Practical Parameter Estimation
The practical parameter-estimation procedure calibrates Bob’s measurements, estimates channel and covariance parameters, and converts prepare-and-measure data into the entanglement-based form required for security analysis. The resulting parameters determine the SNR, mutual information, and Holevo bound.
8.1. Introduction
- Parameter estimation uses measured data to obtain the SNR and Holevo bound needed for estimating the secret-key rate.The SNR depends on transmittance, modulation variance, excess noise, and the homodyne or heterodyne choice, while the Holevo bound comes from the covariance matrix.Source: §8 · p. 27, §8.1 · p. 27
8.2. Calibration of the Measurement Setup
- Bob must calibrate his apparatus by mapping voltages to shot-noise units before estimating channel parameters.The conversion procedure should be repeated during key exchange to track changing calibration parameters.Source: §8.1 · p. 27, §8.2 · p. 27
- Receiver noise is separated from channel excess noise, and the measured voltage variance is related to shot-noise units through the setup-specific factor φ.The voltage-variance estimator is based on the measured samples; the asymptotic treatment ignores deviations from the maximum-likelihood estimate.Source: §8.2 · pp. 27–28
- The conversion factor φ is determined experimentally by measuring vacuum with the signal input disconnected, rather than relying only on a receiver model.Bob divides measured voltages by √φ and variances by φ to express the data in shot-noise units.Source: §8.2 · p. 28
8.3. Estimation of the Covariance Matrix
- Alice and Bob reveal a subset of raw-key samples to estimate covariance coefficients and rescale the prepare-and-measure covariance matrix into the entanglement-based form.The coefficients aEB, bEB, and cEB suffice to compute the Holevo information; transmittance and excess noise can also be recovered.Source: §8.3 · pp. 29–30
- The heterodyne transformation changes Bob’s variance from bPM = TVmod/2 + 1 + ξ/2 to bEB = TVmod + 1 + ξ, while homodyne leaves it unchanged.Source: §8.3 · p. 29
- The SNR can be estimated either from transmittance and excess noise or from Bob’s total and conditional variances.The conditional variance is Bob’s measurement variance conditioned on Alice’s data.Source: §8.3 · p. 30
9. Noise Models
This section develops implementation-oriented models for excess noise in CV-QKD, treating contributions from signal and local-oscillator RIN, modulation, phase recovery, Raman noise, common-mode rejection, receiver detection, and ADC quantisation. It derives their expressions in shot-noise units, including the effects of two-detector or two-ADC heterodyne configurations, and notes the need for fundamental security treatment of discretized Gaussian modulation.
9. Noise Models
- Independent imperfections add their quadrature variances to excess noise ξ, enabling hardware-specific estimates of total noise and expected secure-key rate.Contributors include detection, preparation, laser intensity fluctuations, imperfect modulation, measurement quantization, and Raman scattering.Source: §9 · p. 32
9.1. Relative Intensity Noise – Signal
- Discretized Gaussian modulation has effects beyond quantization noise and must be incorporated more fundamentally because standard CV-QKD security analysis assumes continuous Gaussian states.The chapter’s signal-RIN model relates laser-power variance to quadrature variance under variance symmetry.Source: §9.1 · pp. 32–33
- Signal-laser RIN is converted into transmitter quadrature variance and then scaled by total channel transmittance when excess noise is defined at the output.The derivation uses ⟨nsig⟩ = Vmod/2 to obtain the final output-referenced expression.Source: §9.1 · p. 33
9.2. Relative Intensity Noise – Local Oscillator
- Local-oscillator power fluctuations add excess noise because homodyne difference measurements scale with the local-oscillator amplitude, yielding a final RIN-dependent variance expression.The derivation treats signal quadrature and local-oscillator amplitude as independent random variables and relates amplitude variance to mean photon number.Source: §9.2 · pp. 33–35
9.3. Modulation Noise
- Modulation-voltage noise is propagated through the DAC, amplification, modulator, attenuation, and channel to derive quadrature excess noise, with a simplified final expression for QPSK.The general derivation uses a Taylor expansion and provides a generous upper bound valid for any modulation scheme, phase angle, and quadrature; QPSK removes the first-order sine term.Source: §9.3 · pp. 36–37, §9.4 · p. 37
9.4. Phase-Recovery Noise
- Pilot-tone phase recovery converts residual phase and pilot-measurement noise into quantum-signal phase noise, while averaging N pilot measurements reduces the relevant variance.Pilot noise includes modulation, phase, Raman, detection, and ADC components, and the final relation depends on pilot and quantum-signal mean photon numbers.Source: §9.4 · pp. 37–38
9.5. Raman Noise
- Raman photons from wavelength-multiplexed classical channels are modeled thermally, and their excess noise is expressed in shot-noise units from spectral noise density and filter bandwidth.The derivation assumes equal q- and p-quadrature variances and gives an example filter bandwidth of ∆λ ≈8 pm at B = 1 GHz and f = 193.4 THz for λ = 1550 nm.Source: §9.5 · pp. 39–40
9.6. Common-Mode Rejection Ratio
- A realistic differential amplifier also amplifies part of the homodyne currents’ mean value, motivating the common-mode rejection ratio g/gs as a detector-noise parameter.The model starts from PIN-diode currents whose difference carries the input quadrature and defines CMRR through differential and common-mode gains.Source: §9.6 · pp. 40–41
- Common-mode rejection contributes to excess-noise variance only when signal or local-oscillator photon numbers fluctuate; otherwise it produces a removable voltage offset.The variance expression includes the CMRR term through the variances of the signal and local-oscillator photon numbers, while constant photon numbers leave only an offset that must be subtracted before post-processing.Source: §9.6 · p. 41
- In heterodyne detection, the CMRR and detection-noise contributions do not halve because separate noisy receivers serve the two beamsplitter outputs, requiring a factor µ = 2.For homodyne detection, µ = 1; for heterodyne detection, µ = 2, and the CMRR noise parameter must be doubled relative to the per-detector expression.Source: §9.6 · p. 42, §9.7 · p. 44
9.7. Detection Noise
- Detection noise is back-transformed from receiver voltage noise through amplifier gain, diode responsivity, optical power, and bandwidth into an input excess-noise parameter in shot-noise units.It can alternatively be expressed using the clearance between experimentally measured shot-noise variance and receiver electronic-noise variance; the local-oscillator power is limited by PIN-diode saturation.Source: §9.7 · pp. 43–44
9.8. ADC Quantisation Noise
- ADC quantisation noise is modeled by back-transforming output voltage error into photon-number-difference variance and separating intrinsic ADC noise from quantisation error.For a uniformly distributed quantisation error over one interval, the interval width is set by the full voltage range divided by 2^n, and heterodyne detection accounts for two ADCs.Source: §9.8 · pp. 44–45
10. Experimental Implications
The section connects CV-QKD performance to transmission, excess noise, reconciliation, modulation, and hardware choices, while introducing Gaussian-state tools for the security analysis.
10. Experimental Implications
- At 40 km, the excess-noise threshold is below 1% of shot noise, making low-noise operation a major experimental challenge.Lower excess noise permits greater loss tolerance and longer transmission distances.Source: §10 · p. 46
- Detection noise dominates a typical setup, while Raman, phase-recovery, and ADC quantisation noise are the other major contributors.The example gives ξtot = 0.0653 SNU and ξdet = 0.0396 SNU, about 60% of the total.Source: §10 · p. 46, Figure 10.2 · p. 48
- Trusted-receiver assumptions substantially relax detector and ADC requirements, whereas untrusted detection makes low NEP and adequate PIN-diode saturation limits critical.Higher local-oscillator power can mitigate detector noise, but conventional PIN diodes typically saturate near 10 mW.Source: §10 · pp. 46–47
- Six ADC bits suffice for a non-zero key at T = 0.1, while increasing ADC resolution reduces quantisation noise by factors of 4 per added bit.The figure also relates Raman noise to the condition R = 0.95IAB > χEB.Source: §10 · p. 47, Figure 10.4 · p. 49
- Efficient reconciliation becomes increasingly important with excess noise and distance, and lower β or higher ξ narrows the modulation-variance interval supporting a non-zero key.For T = 0.1 and ξ = 0.01 SNU, reconciliation efficiency must exceed 0.96.Source: §10 · p. 47, Figure 10.5 · p. 50, Figure 10.6 · p. 50
A. Prerequisites on Coherent States
- Coherent-state quadratures have equal minimum uncertainty, and their mean photon number is ⟨n⟩ = |α|^2 = q^2 + p^2.Gaussian modulation relates quadrature variance to modulation variance through V(ˆq) = 4Ṽmod + 1.Source: A. Prerequisites on Coherent States · pp. 52–53
B.1. Displacement Vector and Covariance Matrix
- Gaussian states are represented by displacement vectors and covariance matrices, with covariance entries encoding quadrature correlations and separability when cross terms vanish.The displacement vector has dimension 2N and the covariance matrix has dimension 2N × 2N.Source: §B.1 · p. 54
B.2. Symplectic Operations
- Symplectic operations represent Gaussian-state transformations, with the beamsplitter identified as a central operator for this analysis.A symplectic operation transforms a Gaussian state through its displacement vector and covariance matrix.Source: §B.2 · p. 55
B.5. Partial Measurements
This section derives how measuring one mode changes the covariance matrix of the remaining modes, distinguishing homodyne from heterodyne detection. A balanced-beamsplitter construction shows that heterodyne conditioning includes vacuum shot noise and reproduces the general result for coherent states.
B.5. Partial Measurements
- A measurement on mode B updates the covariance matrix of the remaining modes according to their correlations with B.The total covariance matrix is partitioned into ΣA, ΣB, and ΣC for the unmeasured modes, measured mode, and their correlations.Source: §B.5 · p. 56
- Homodyne detection measures one quadrature of B, using Πq or Πp to select the measured component.The q- and p-measurement cases are represented by Πq = diag(1, 0) and Πp = diag(0, 1).Source: §B.5 · p. 56
- The pseudoinverse in the partial-measurement expression reduces to elementwise inversion when the relevant matrix is diagonal.This gives a convenient form for rewriting the conditional covariance update.Source: §B.5 · p. 56
- Heterodyne detection updates the remaining modes after both quadratures of B are measured.The derivation models this as two sequential partial homodyne measurements after splitting B on a balanced beamsplitter.Source: §B.5 · pp. 56–57
- For a coherent state with equal q- and p-variances, the heterodyne-conditioning expression simplifies and coincides with the previously derived result.The balanced beamsplitter introduces an additional vacuum contribution before the two homodyne measurements.Source: §B.5 · pp. 56–57
C. Derivation of the Covariance Matrix
The paper derives the covariance matrix of a two-mode squeezed vacuum state from its ket representation, expectation values, normalization, and quadrature variances. The componentwise calculation establishes the diagonal, local off-diagonal, and intermode entries.
C. Derivation of the Covariance Matrix
- The derivation starts from the ket representation of a two-mode squeezed vacuum state and computes expectation values needed for its covariance matrix.The state is checked for normalization before the covariance entries are evaluated.Source: C. Derivation of the Covariance Matrix · p. 58
C.1. Initial Covariance Matrix
- A two-mode squeezed vacuum has a 4×4 covariance matrix whose 16 components are derived individually.The matrix is organized into diagonal variances, local quadrature covariances, and intermode correlations.Source: §C.1 · p. 58, §C.1.1 · p. 58
C.1.1. Σ11, Σ22, Σ33, Σ44
- The four diagonal covariance entries are equal to V: Σ11 = Σ22 = Σ33 = Σ44 = V.The diagonal terms are obtained from variances of the four quadrature operators.Source: §C.1.1 · pp. 58–59
C.1.2. Σ12, Σ21, Σ34, Σ43
- The local q-p covariance entries vanish, giving Σ34 = Σ43 = 0 and analogously the corresponding local terms.These entries are evaluated from symmetrized products of quadrature operators.Source: §C.1.2 · p. 59
C.1.3. Σ13, Σ31, Σ24, Σ42
- The derivation computes required quadrature-operator expectation values by reexpressing them in creation and annihilation operators and applying a geometric-series calculation for q < 1.With q = (V − 1)/(V + 1) = 1 − 2/(V + 1), the calculation gives q(1 − q)−2 = 1/4(V − 1)(V + 1).Source: §C.1.2 · p. 59, §C.1.3 · p. 59
C.1.4. Σ14, Σ41, Σ32, Σ23
- The remaining cross-quadrature entries satisfy Σ32 = Σ23 = 0.This completes the vanishing off-diagonal components reported for the covariance matrix.Source: §C.1.4 · p. 60
C.1.5. Final Σ
The paper completes the two-mode squeezed-vacuum covariance matrix and models transmission of one mode through a lossy channel. The channel is represented by a beamsplitter with an added vacuum input, followed by a symplectic covariance transformation.
C.1.5. Final Σ
- The componentwise derivation is collected into the final covariance matrix of the two-mode squeezed vacuum state.This final expression consolidates the diagonal and correlation entries derived previously.Source: §C.1.5 · p. 60
C.2. Channel Transmission
- Transmission loss is modeled by sending Bob’s mode through a beamsplitter with transmittance T and adding a vacuum mode.The vacuum input is represented by a 2×2 unit matrix and included in the enlarged covariance matrix.Source: §C.2 · p. 61
- The beamsplitter acts on Bob’s mode and the vacuum input while leaving Alice’s mode unchanged.The corresponding symplectic transformation maps the initial covariance matrix to the post-transmission state.Source: §C.2 · p. 61
- The resulting covariance matrix yields the post-channel Alice-Bob substate after transmission through the lossy channel.The passage identifies the transformed state and its Alice and Bob substates.Source: §C.2 · p. 61
C.3. Channel Transmission including Excess Noise
The paper extends channel transmission to excess noise and then models Bob’s heterodyne measurement with a balanced beamsplitter and vacuum input. These transformations determine Bob’s noisy covariance and simultaneous q- and p-quadrature detection.
C.3. Channel Transmission including Excess Noise
- Excess noise is modeled by replacing the channel beamsplitter’s vacuum input with a noisy input whose quadratures have variance N.Applying the beamsplitter transformation to the two-mode squeezed state and noisy input gives the transmitted covariance matrix.Source: §C.3 · p. 62
- Bob’s transmitted quadrature variance is expressed using the channel transmittance T, initial variance V, and noise variance N.The noise definition relates TV + (1 − T)N to T(V − 1) + 1 + ξ.Source: §C.3 · p. 62
- The excess-noise model produces a final covariance matrix for the channel output.This is the stated endpoint of the noisy transmission calculation.Source: §C.3 · p. 62
C.4. Bob Heterodyning
- Bob heterodynes by splitting his signal with a balanced beamsplitter and using an additional vacuum mode as the second input.The beamsplitter acts on Bob’s mode and vacuum while leaving Alice’s mode unchanged.Source: §C.4 · pp. 62–63
- The heterodyne transformation produces covariance-matrix components for Alice and Bob after simultaneous quadrature detection.The supplied expressions include the added vacuum contribution and the transformed q-quadrature terms.Source: §C.4 · p. 63
D. Quantum Homodyne Detection
The paper derives homodyne detection from coherent-state and beamsplitter transformations, then describes a double-MZI q/p-modulator and distinguishes SNU, NU, and SI conventions.
D. Quantum Homodyne Detection
- The homodyne number-difference operator selects the q or p quadrature according to the local-oscillator phase θ.The derivation proceeds from coherent-state quadratures through balanced-beamsplitter outputs and photon-number subtraction.Source: D. Quantum Homodyne Detection · p. 64
E. Principles of a q/p-Modulator
- A double Mach-Zehnder interferometer uses nested inner MZIs, voltage-controlled phase rotations, and selective output recombination to implement q/p modulation.The applied voltages change waveguide refractive indices through the electro-optic effect, producing phases ±ϕ1 and ±ϕ2 before recombination.Source: E. Principles of a q/p-Modulator · pp. 65–66
- The q/p-modulator begins by splitting a coherent state with vacuum and then splitting each outer-MZI arm again into the two inner MZIs.Only selected beamsplitter outputs are retained in the modeled implementation.Source: E. Principles of a q/p-Modulator · pp. 65–66
F. SI-, Natural- and Shot-Noise Units
- Quadrature conventions determine the commutator, photon-number operator, and minimum uncertainty product, motivating direct comparison of SNU, NU, and SI units.The paper uses shot-noise units and refers to a comparison table for the three conventions.Source: F. SI-, Natural- and Shot-Noise Units · p. 67, Table F.1 · p. 67