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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

arXiv:1703.09278v3quant-ph

TL;DR

Practical CV-QKD security analysis is complicated by varied assumptions and hardware imperfections. This paper pedagogically derives the theory, analyzes attacks and parameter estimation, and develops practical noise models, showing that lower excess noise improves loss tolerance and transmission distance.

  • Problem

    CV-QKD implementations face difficult security analysis because performance depends on varied assumptions and multiple experimental noise sources.

  • Method

    The paper develops a self-contained theoretical treatment, deriving security relations and analyzing attacks, parameter estimation, and experimentally relevant noise.

  • Results

    Lower excess noise permits greater loss tolerance and therefore longer transmission distances in CV-QKD.

  • Takeaways & Limitations

    CV-QKD setup performance is primarily characterized by transmittance, excess noise, and reconciliation efficiency.

  • Takeaways & Limitations

    Unconditional security against coherent attacks remains established only for infinitely or impractically long keys, apart from a recent practical-block-size proof.

Abstract

from arXiv · show

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 tutorial presents Gaussian-modulated coherent-state CV-QKD, deriving its protocol and security foundations while addressing experimental implementation. It motivates this focus through CV-QKD’s deployment promise, difficult cross-study comparisons, and less-developed security proofs.

  • 1. Introduction: The tutorial derives CV-QKD with Gaussian modulation from scratch, assuming little prior knowledge and covering protocol security and experimental implementation.Its stated goal is to provide a detailed, self-contained overview of the protocol and its security against eavesdropping.
  • 1. Introduction: CV-QKD with Gaussian modulation of coherent states is presented as an auspicious candidate for deployment in widely used applications.
  • 1. Introduction: QKD generates a shared secret key by transmitting non-orthogonal quantum states and requires authenticated classical communication to prevent man-in-the-middle attacks.Strictly speaking, authentication makes QKD a key-growing protocol.
  • 1. Introduction: Although many experiments demonstrate CV-QKD feasibility [23] [24], differing settings and assumptions make performance comparisons difficult.Secure-key rate over a given transmission distance is identified as the primary evaluation criterion.
  • 1. Introduction: Security proofs for Gaussian-modulated coherent-state CV-QKD remain less advanced than those for discrete-variable QKD, despite existing reviews.The cited review provides an overview of GM CV-QKD and protocol security but does not provide detailed calculations.

2. Notions of Security and Secure-Key Rate

The section distinguishes individual, collective, and coherent attacks, then restricts the tutorial to asymptotic secure-key rates under collective attacks with non-asymptotic reconciliation effects included. It defines the reverse-reconciliation secret fraction through Alice–Bob mutual information, Eve’s Holevo information, and practical implementation penalties.

  • Attack models: Coherent attacks are most general, allowing globally correlated ancilla modes and collective measurement after classical post-processing, whereas individual and collective attacks impose i.i.d. interactions.Individual attacks measure stored ancillas independently; collective attacks defer an optimal joint measurement until after post-processing.
  • Related security proofs: Security proofs progressed from individual to collective attacks, while de Finetti reduction and Gaussian-attack optimality enabled security arguments against coherent attacks [40].For a fixed covariance matrix, Gaussian attacks provide the relevant optimality principle used in these arguments.
  • Security scope: Security may be analyzed asymptotically or for finite transmissions; this tutorial uses asymptotic collective-attack rates while retaining non-asymptotic information-reconciliation behavior.The asymptotic limit is easier to derive and upper-bounds the corresponding finite-size result, but does not directly describe realistic finite systems.
  • Secure-key rate: The asymptotic secret fraction is lower-bounded by I_AB minus Eve’s Holevo information, with χ_EA for direct reconciliation and χ_EB for reverse reconciliation.The review restricts subsequent analysis to reverse reconciliation, where Bob’s key remains unmodified and Alice corrects hers according to Bob’s data.
  • Secure-key rate: Practical reconciliation reduces the secret fraction through frame failures, efficiency below the Shannon limit, and symbols disclosed for covariance-matrix estimation.FER is the frame-error rate, β is reconciliation efficiency, and ν is the disclosed-symbol fraction; the resulting expression is a lower bound because χ_EB is itself an upper bound.

3. Basics of a Coherent-State Protocol

The coherent-state CV-QKD protocol uses Gaussian-modulated coherent states sent through a Gaussian channel and measured by homodyne or heterodyne detection. Classical post-processing estimates channel parameters, reconciles correlated data, verifies agreement, extracts a secure key, and authenticates communication.

  • Gaussian modulation: Alice independently Gaussian-modulates both quadratures to prepare a sequence of displaced coherent states with modulation variance ˜Vmod.The ensemble is represented in phase space by independent, identically distributed, zero-centered normal variables for q and p.
  • Gaussian modulation: Each quadrature carries total variance V = Vmod + 1 because modulation variance adds to one shot-noise unit.Here Vmod = 4˜Vmod, while the unit shot-noise contribution remains even without modulation.
  • Quantum transmission and detection: Alice transmits each coherent state through a Gaussian quantum channel, and Bob measures one or both quadratures using homodyne or heterodyne detection.The quantum transmission occurs over a potentially insecure channel, with measurement results later used in classical processing.
  • Classical post-processing: Parameter estimation uses revealed sample data to estimate transmission and excess noise, compute IAB and bound χ, and abort when χ exceeds βIAB.If βIAB > χ, Alice and Bob proceed to information reconciliation.
  • Classical post-processing: Information reconciliation uses direct or reverse reconciliation, with slice or multidimensional schemes and potentially LDPC codes, followed by confirmation, privacy amplification, and authentication.Confirmation compares universal-hash values, privacy amplification applies a seeded randomness extractor, and authentication protects the classical channel against man-in-the-middle attacks.

4. Transmittance and Noise

The section characterizes CV-QKD performance through transmittance and noise, which reduce the signal-to-noise ratio and increase an eavesdropper’s potential advantage. It defines channel-output excess noise by combining independent contributions, including detector noise, and accounts for channel, detection, and coupling losses in the total transmittance.

  • System characterization: Transmittance and noise are the primary system parameters because both degrade signal-to-noise performance and increase a potential eavesdropper’s advantage.Transmittance includes channel transmission, detection efficiency, and coupling losses.
  • Noise composition: Independent excess-noise sources add through their variances, including modulation, Raman-scattering, quantisation, phase, detector, relative-intensity, and common-mode-rejection contributions.The article writes ξ = ξmodul + ξRaman + ξquant + ξphase + ξdet + ξRIN + ξCMRR + . . . .
  • Effective transmittance and variance: The total transmittance T combines channel and detection losses with coupling efficiency, while detection noise is referred to the channel input when combined with channel noise.Bob’s measured variance is consequently expressed using the transmitted signal variance, excess noise, vacuum noise, and electronic noise.
  • Noise definition: The article defines excess noise where it is observed, at the channel output in Bob’s lab, unlike most CV-QKD literature.Detector electronic noise is incorporated into the excess-noise description as an additional variance contribution.
  • Practical modeling: Section 9 derives analytic models for the various excess-noise constituents.These models support estimating the performance of realistic hardware implementations.

5. Covariance Matrix

Covariance matrices encode quadrature variances and correlations and are vital for calculating the Holevo bound in CV-QKD. The section develops their PM and EB descriptions, transmission and detection transformations, and the rescaling needed to make practical security analysis valid.

  • Covariance-matrix formalism: Covariance matrices represent quadrature variances on the diagonal and mutual quadrature covariances off-diagonal, providing the basis for analyzing bosonic modes and CV-QKD security.Knowledge of the covariance matrix and its transformations is vital for computing the Holevo bound.
  • Prepare-and-measure and entanglement-based descriptions: The prepare-and-measure protocol with Gaussian-modulated coherent states is equivalent to an entanglement-based protocol using a two-mode squeezed vacuum, with Alice measuring one mode and sending the other to Bob.This equivalence enables a simplified security analysis based on the entanglement-based covariance matrix.
  • Covariance-matrix transformations: The covariance description incorporates Gaussian quadrature modulation, whose variance combines Alice’s modulation variance with vacuum shot noise, while transmission and noise transform the matrix.Other alphabets such as PSK and QAM require distinct covariance matrices, differing in the off-diagonal covariance structure.
  • Detection protocols: Homodyne detection measures one randomly selected quadrature, whereas heterodyne detection splits the signal and measures both quadratures simultaneously; the latter changes the covariance matrix and introduces a factor of 1/2 at transmission and excess noise.Despite this change, the Holevo information can be derived from the covariance matrix before Bob’s measurement apparatus acts.
  • Prepare-and-measure and entanglement-based descriptions: Alice’s rescaling of heterodyne outcomes simulates the prepare-and-measure scenario without Bob or Eve noticing, while inverse rescaling of prepared quadratures maps the practical protocol to the entanglement-based one.This transformation makes the security analysis significantly simpler and is crucial for determining the Holevo bound and parameter estimation.

6. Signal-to-Noise Ratio and Mutual Information

This section defines the channel signal-to-noise ratio from output signal and noise powers, relates it to transmittance, detection scheme, and excess noise, and derives the Alice–Bob mutual information. Heterodyne detection appears to double the mutual-information expression but reduces SNR by roughly half, while code rate and error-correction efficiency determine practical transmission rates.

  • Signal-to-Noise Ratio: SNR is determined by the total signal power divided by the total noise power at the channel output.The signal power is the damped modulation variance, while vacuum and excess noise contribute to total noise.
  • Signal-to-Noise Ratio: Total transmittance includes coupling and detection efficiency, while μ equals 1 for homodyne and 2 for heterodyne detection.Bob’s quadrature variance and the signal/noise decomposition depend on the detection scheme and excess noise ξ.
  • Mutual Information: Heterodyne detection changes mutual information from log2(1 + SNR)/2 to log2(1 + SNR), but its SNR decreases by roughly a factor of 1/2.The reduction is not exactly one-half because the heterodyning beamsplitter also halves excess noise ξ.
  • Code Rate: The code rate is the fraction of raw bits carrying information bits, and error-correction efficiency β satisfies 0 ≤ β < 1 relative to mutual information IAB.The section relates code rate to the energies required for raw and information bits and rewrites SNR in terms of energy per bit.

7. Holevo Information

The Holevo bound χEB is determined by the difference between Eve’s unconditional and measurement-conditioned von Neumann entropies, computed from symplectic eigenvalues of covariance matrices. Purification invariance enables an attack-independent calculation, while trusted-device assumptions can reduce Eve’s attributed noise but do not justify treating preparation noise as trusted.

  • Holevo information: χEB under reverse reconciliation is obtained from Eve’s von Neumann entropy before and after Bob’s homodyne or heterodyne measurement.These entropies are determined by the symplectic eigenvalues of the corresponding covariance matrices, whose quantum state depends on the assumed attack.
  • Purification-based calculation: Any purification of ρAB gives the same Eve entropy and therefore the same Holevo information χEB.This follows from entropy invariance under unitary transformations and the shared Schmidt coefficients of ρAB and Eve’s reduced state.
  • Entangling cloner attack: The entangling-cloner construction reproduces the covariance matrix of a lossy channel with excess noise ξ, and its Eve-state symplectic eigenvalues determine SE.Conditional entropies are then obtained from Eve’s covariance matrix after Bob performs homodyne or heterodyne detection.
  • Attack-independent calculation: The purification method computes χEB from Alice and Bob’s covariance matrix without specifying how Eve implements the attack.The unconditional entropy uses the joint covariance matrix, while the conditional entropy depends on whether Bob uses homodyne or heterodyne detection.
  • Trusted-device scenario: Under the trusted-device scenario, detection and quantisation noise and detection inefficiency need not contribute to Eve’s Holevo information, whereas preparation noise remains excluded because it enables severe side-channel attacks.The trusted-device assumption attributes only channel-accessible imperfections to Eve, while Bob’s measurement results are still affected by trusted receiver imperfections.

8. Practical Parameter Estimation

This section presents practical parameter estimation for CV-QKD with a true or local local oscillator, emphasizing calibration from voltages to shot-noise units and covariance-based estimation of security parameters. The procedure yields the SNR, mutual information, and Holevo bound needed to estimate the secret-key rate.

  • Scope: The analysis avoids attacks on a co-transmitted local oscillator and focuses on estimating channel parameters for true or local local-oscillator implementations.Channel-parameter estimation is identified as essential for QKD security analysis.
  • Rate estimation: The mutual information IAB can be computed from T and ξ or from Bob’s total and conditional variances, while the SNR follows from the same estimated measurement statistics.The Holevo bound is computed from covariance-matrix coefficients, and IAB depends uniquely on the signal-to-noise ratio.
  • Statistical assumptions: The asymptotic procedure uses a maximum-likelihood voltage-variance estimator and ignores deviations from the estimator, while confidence-interval methods are referenced for detailed finite-sample estimation.This assumption applies to the asymptotic analysis described here.
  • Calibration: Bob experimentally calibrates the conversion factor φ by measuring vacuum with the signal input disconnected, then converts voltages and variances into shot-noise units.Calibration should be repeated during key exchange to track varying parameters; the resulting factor depends on the measurement setup and local-oscillator power.
  • Covariance-matrix estimation: Alice and Bob disclose n raw-key samples to estimate the covariance matrix, from which the channel transmittance T, excess noise ξ, and Holevo bound χEB are obtained.The disclosed samples are omitted from secure-key generation, and the prepare-and-measure protocol uses randomly distributed coherent states.

9. Noise Models

This section models excess noise from experimental imperfections in CV-QKD, decomposing it into detector, laser, quantisation, modulation, and related contributions. The models estimate hardware impacts on total noise and expected secure-key rates, while noting limitations of the simplified treatments.

  • Noise composition: Excess noise ξ comprises detector, relative-intensity, quantisation, Raman, and other hardware contributions that add to shot noise and degrade system performance.The decomposition is ξ = ξdet + ξRIN + ξquant + ξRam + … (9.1).
  • Noise composition: The proposed mathematical models estimate how lasers, modulators, detectors, and analog-to-digital converters affect total noise and the expected secure-key rate.They are intended as design guidance for practical CV-QKD implementations and motivate hardware-specific conclusions in Section 10.
  • Relative-intensity noise: Relative-intensity noise from signal and local-oscillator lasers is modeled as quadrature noise, with output-referenced excess noise scaled by total channel transmittance.The local oscillator contributes an additional RIN term because balanced detection scales the signal quadrature by the local-oscillator amplitude.
  • Modulation and phase noise: Alice’s modulation-voltage noise is propagated through the DAC or waveform generator, amplifier, and q/p-modulator to derive excess-noise expressions for QPSK and quantum-signal phase noise.The derivation uses a Taylor expansion in the voltage deviation and relates the final phase-noise expression to ⟨n⟩ = Vmod/2.
  • Modulation and phase noise: The modulation analysis excludes DAC quantisation, q/p-modulator phase deviations, electro-optic ripple, and detection-response noise, while its generous bound covers multiple modulator schemes.For an applied phase π, the first-order sin A term vanishes, leaving dependence on second and higher orders of λ.

10. Experimental Implications … B.4. Von Neumann entropy

The paper connects CV-QKD’s practical performance to symbol rate, transmittance, excess noise, reconciliation, and hardware choices, then supplies Gaussian-state tools for modeling states, transformations, correlations, entanglement, and entropy.

  • 10. Experimental Implications: Experimental Implications: State-of-the-art telecom hardware makes symbol rates of approximately 10 Gbaud feasible, but increasing symbol rate alone cannot extend achievable transmission distance.Higher symbol rates can raise key throughput, whereas transmission distance remains limited by the channel’s noise and loss behavior.
  • 10. Experimental Implications: Experimental Implications: Secure-key performance is governed primarily by transmittance T, excess noise ξ, and reconciliation efficiency, while modulation variance Vmod can be adjusted to security requirements.The secret fraction depends on modulation variance, transmittance, excess noise, and reconciliation efficiency; optical attenuation allows Vmod to be tuned to the setup.
  • 10. Experimental Implications: Experimental Implications: Practical bottlenecks also include ADC quantisation, Raman crosstalk from wavelength-multiplexed classical channels, and inefficient reconciliation software.Adding one ADC bit reduces quantisation noise by a factor of 4, while the example requires reconciliation efficiency above 0.96 at T = 0.1 and ξ = 0.01 SNU.
  • 10. Experimental Implications: Experimental Implications: Detection noise is the dominant component in the example setup, and trusted versus untrusted receiver assumptions substantially change detector NEP requirements.Under strict assumptions, low NEP is critical for a non-zero key; with trusted detectors, NEP requirements are more relaxed, and higher local-oscillator power can mitigate detection noise subject to saturation limits.
  • A. Prerequisites on Coherent States: A. Prerequisites on Coherent States: Coherent states represent quadrature amplitudes through the real and imaginary parts of α, have equal minimal quadrature uncertainty, and satisfy ⟨n̂⟩ = |α|^2 = q^2 + p^2.In shot-noise units, the coherent-state uncertainty establishes the reference variance used for Gaussian-modulated CV-QKD.
  • B.1. Displacement Vector and Covariance Matrix: B.1. Displacement Vector and Covariance Matrix: An N-mode Gaussian state is represented by a 2N-dimensional quadrature displacement vector and covariance matrix encoding variances and mutual correlations.Vanishing cross terms between two modes indicate that their quadratures are uncorrelated and the modes are separable.
  • B.2. Symplectic Operations: B.2. Symplectic Operations: Gaussian-state transformations are represented by symplectic matrices, with the beamsplitter serving as the principal operation used in the paper.Symplectic operators correspond to unitary Hilbert-space operations in the Gaussian-information formalism.
  • B.3. Composite States / B.4. Von Neumann entropy: B.3. Composite States and B.4. Von Neumann entropy: Composite Gaussian states combine subsystem displacement vectors and covariance matrices, while entanglement appears through nonzero off-diagonal covariance blocks and entropy is computed from symplectic eigenvalues.A 2N × 2N covariance matrix has N symplectic eigenvalues, which determine the Gaussian state’s von Neumann entropy.

B.5. Partial Measurements

Partial measurements update the covariance matrix of the unmeasured modes according to their correlations with the measured mode. Homodyne and heterodyne detection produce distinct conditional updates, with heterodyne’s added shot noise arising from vacuum entering a balanced beamsplitter.

  • Partial Measurements: A Gaussian state’s covariance matrix can be partitioned into the unmeasured modes, measured mode, and their cross-correlations, which determine the post-measurement covariance.The measured mode is denoted B, the remaining modes A, and their correlations C.
  • Partial Measurements: Homodyne detection updates the remaining modes differently depending on whether the measured quadrature is q or p.The corresponding projectors select one quadrature, with Πq = diag(1, 0) and Πp = diag(0, 1).
  • Partial Measurements: The pseudoinverse required for homodyne conditioning reduces to elementwise inversion when the projected covariance matrix is diagonal.For a diagonal matrix, the pseudoinverse is obtained by inverting its diagonal elements.
  • Partial Measurements: Heterodyne detection updates both quadratures and adds the unit-matrix contribution associated with shot noise from the beamsplitter’s vacuum input.The heterodyne transformation can be derived by splitting the measured mode on a balanced beamsplitter and applying successive homodyne measurements.
  • Partial Measurements: For a coherent measured mode with equal q- and p-quadrature variances, the heterodyne conditional covariance simplifies accordingly.The simplifying condition is V(q_B) = V(p_B) = V_B.

C. Derivation of the Covariance Matrix … C.1.4. Σ14, Σ41, Σ32, Σ23

The paper derives the initial 4×4 covariance matrix of a two-mode squeezed vacuum state by evaluating quadrature expectation values from ladder-operator expressions. Its component-wise calculation establishes the diagonal variances and vanishing off-diagonal terms reported across the subsections.

  • C. Derivation of the Covariance Matrix: The derivation uses orthogonality of Fock states and ladder-operator representations of quadratures to evaluate the required expectation values.These relations support the calculation of quadrature means and products for the covariance entries.
  • C. Derivation of the Covariance Matrix: The covariance matrix Σ of a two-mode squeezed vacuum state is 4×4, and its 16 components are derived individually.The state is represented in ket notation, with expectation values and normalization checked before computing matrix elements.
  • C.1.1. Σ11, Σ22, Σ33, Σ44: The diagonal entries are obtained from quadrature variances, expressed through operator expectation values and their squares.The quadrature operators q_A, p_A, q_B, and p_B are reexpressed using creation and annihilation operators.
  • C.1.1. Σ11, Σ22, Σ33, Σ44: Σ22 = Σ33 = Σ44 = V, completing the analogous diagonal variance results after evaluating the relevant series.The derivation uses q = (V −1)/(V +1) and the geometric-series normalization relation.
  • C.1.2. Σ12, Σ21, Σ34, Σ43: The within-mode covariance terms are calculated from symmetrized products of quadrature operators, with the analogous result Σ34 = Σ43 = 0.Expectation values are again computed by rewriting quadratures in terms of creation and annihilation operators.
  • C.1.4. Σ14, Σ41, Σ32, Σ23: The remaining covariance components satisfy Σ32 = Σ23 = 0.This completes the stated component-wise derivation of the initial covariance matrix.

C.1.5. Final Σ · C.2. Channel Transmission

The section collects the previously derived components into the final covariance matrix of a two-mode squeezed vacuum state and models transmission of Bob’s mode through a lossy channel using a beamsplitter and vacuum input.

  • C.1.5. Final Σ: The preceding components are collected into the final covariance-matrix expression for the TMSVS.
  • C.2. Channel Transmission: Channel transmission is analyzed by rewriting the TMSVS covariance matrix before applying the loss transformation.
  • C.2. Channel Transmission: Loss is modeled with a beamsplitter of transmittance T, while an additional vacuum mode accounts for the beamsplitter’s vacuum input.
  • C.2. Channel Transmission: The vacuum mode is incorporated into the total covariance matrix by direct summation, using a 2 × 2 unit matrix.
  • C.2. Channel Transmission: The beamsplitter is expressed symplectically and extended to the covariance-matrix dimensions by including its action on Alice’s mode.
  • C.2. Channel Transmission: The resulting symplectic transformation acts on Bob’s mode and the vacuum input while leaving Alice’s mode unchanged.
  • C.2. Channel Transmission: Under a symplectic operator S, the Gaussian covariance matrix transforms to a post-transmission state, yielding the transmitted covariance matrix.

C.3. Channel Transmission including Excess Noise · C.4. Bob Heterodyning · D. Quantum Homodyne Detection

The merged sections model channel excess noise, derive Bob’s heterodyne transformation, and explain quantum homodyne detection through balanced-beamsplitter interference with a local oscillator.

  • C.3. Channel Transmission including Excess Noise: Channel excess noise is modeled by replacing the channel beamsplitter’s vacuum input with a noisy input whose quadratures have variance N.The resulting covariance-matrix treatment leads to Bob’s output variance and a final covariance-matrix form.
  • C.3. Channel Transmission including Excess Noise: The excess-noise derivation applies the beamsplitter transformation to the joint covariance matrix of the two-mode squeezed vacuum and noisy input.This construction precedes the stated expressions for Bob’s quadrature variance and the final covariance matrix.
  • C.3. Channel Transmission including Excess Noise: The noise parameter is defined through TV + (1 −T)N = T(V −1) + 1 + ξ, linking the noisy-input model to excess noise ξ.This relation specifies how the channel input variance, transmission, noise variance, and excess noise are connected.
  • C.4. Bob Heterodyning: Bob measures both quadratures simultaneously by inserting a balanced beamsplitter that splits his mode and introduces an additional vacuum mode.The beamsplitter acts on Bob’s mode and the vacuum mode, not on Alice’s mode.
  • C.4. Bob Heterodyning: The heterodyne transformation is represented by applying the balanced-beamsplitter operator to the total state containing Bob’s signal and the additional vacuum input.The supplied derivation records the resulting transformed quadrature structure.
  • D. Quantum Homodyne Detection: Quantum homodyne detection begins with coherent-state quadrature operators and a classical local oscillator entering a balanced beamsplitter.The derivation is presented as leaning toward.
  • D. Quantum Homodyne Detection: Subtracting the two output photon-number operators produces a number-difference operator whose value depends on the local-oscillator phase θ.The phase-dependent difference is proportional to a quadrature operator, as established by substituting the coherent-state and local-oscillator expressions.

E. Principles of a q/p-Modulator · F. SI-, Natural- and Shot-Noise Units

Section E describes a nested-MZI q/p-modulator that splits, phase-modulates, recombines, and selects coherent-state outputs. Section F explains that quadrature conventions determine operator relations and compares shot-noise, natural, and SI units.

  • E. Principles of a q/p-Modulator: The q/p-modulator uses a double Mach–Zehnder interferometer, with a nested MZI in each arm, to process an input coherent state and vacuum.The schematic is identified as a possible implementation of this architecture.
  • E. Principles of a q/p-Modulator: A coherent state α and vacuum first enter a balanced beamsplitter, whose outputs form the two arms of the outer MZI.Each outer-MZI arm is then mixed with vacuum by another balanced beamsplitter.
  • E. Principles of a q/p-Modulator: Balanced beamsplitters recombine the inner-MZI arms, after which the construction retains only one output port from each beamsplitter.The omitted components are not included in the remaining state vector.
  • E. Principles of a q/p-Modulator: The two inner MZIs apply opposite phase rotations, +ϕ1/−ϕ1 and +ϕ2/−ϕ2, produced by voltage-driven electro-optic refractive-index changes.The phase rotation is proportional to the applied voltage.
  • E. Principles of a q/p-Modulator: A π/2 phase is added to the second outer-MZI arm before the two outer arms are recombined and one final beamsplitter output is selected.This sequence produces the final retained state.
  • F. SI-, Natural- and Shot-Noise Units: Coherent states may be represented with annihilation and creation operators, while the definitions of q̂ and p̂ relative to ladder operators remain a notational choice.That choice changes the corresponding commutator, photon-number operator, and minimal uncertainty-product expressions.
  • F. SI-, Natural- and Shot-Noise Units: Table F.1 compares shot-noise units, natural units, and SI units as alternative conventions for expressing the relevant quantities.The supplied passage identifies the comparison scope but does not provide the table’s individual values.
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