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Quantum Cryptography Approaching the Classical Limit
Christian Weedbrook, Stefano Pirandola, Seth Lloyd, Timothy C. Ralph
TL;DR
Deriving the paper’s main-text results requires Gaussian formalisms for continuous-variable quantum information. The paper provides a self-contained treatment and uses symplectic eigenvalues to characterize Gaussian systems, with a simplified expression for a specified two-mode covariance matrix.
Problem
Deriving the paper’s main-text results requires Gaussian formalisms for continuous-variable quantum information.
Source: Introduction to Gaussian Formalisms · p. 4Method
The paper gives a self-contained treatment of Gaussian tools and models Bob’s received output with a beamsplitter channel of transmission T and excess noise N.
Source: Introduction to Gaussian Formalisms · p. 4, Reverse Reconciliation · p. 6Results
Symplectic eigenvalues provide a tool for determining important features of Gaussian systems, with a simplified expression for the specified two-mode covariance matrix.
Source: Introduction to Gaussian Formalisms · p. 5Takeaways & Limitations
Covariance-matrix diagonal terms contain quadrature variances, while off-diagonal terms contain correlations.
Source: Introduction to Gaussian Formalisms · p. 5
Abstract
from arXiv · showhide
We consider the security of continuous-variable quantum cryptography as we approach the classical-limit, i.e., when the unknown preparation noise at the sender's station becomes significantly noisy or thermal (even by as much as 10,000 times the variance of the vacuum mode). We show that, provided the channel transmission losses do not exceed 50%, the security of quantum cryptography is not dependent on the channel transmission, and is therefore, incredibly robust against significant amounts of excess preparation noise. We extend these results to consider for the first time quantum cryptography at wavelengths considerably longer than optical and find that regions of security still exist all the way down to the microwave.
Introduction to Gaussian Formalisms
This section introduces a self-contained Gaussian formalism for analyzing continuous-variable quantum systems. It uses displacement and correlation matrices to characterize states, with symplectic eigenvalues supporting entropy and other system calculations.
Introduction to Gaussian Formalisms
- The section provides the Gaussian tools needed to understand and derive the paper’s main-text results.Source: Introduction to Gaussian Formalisms · p. 4
- A Gaussian bosonic state is characterized by its displacement and correlation matrix (CM).Source: Introduction to Gaussian Formalisms · p. 5
- CM diagonal entries contain variances, while off-diagonal entries contain correlations; for the vacuum example, the quadratures have zero correlation.Source: Introduction to Gaussian Formalisms · p. 5
- The symplectic spectrum is obtained from the eigenvalues of iΩV by taking their absolute values, and helps determine features of a Gaussian system.The passage also states that ν ≥ 1.Source: Introduction to Gaussian Formalisms · p. 5
Reverse Reconciliation
Reverse reconciliation with homodyne detection defines the key rate from Alice–Bob information minus Bob–Eve information. The channel model incorporates transmission and excess noise into Bob’s received quadratures.
Reverse Reconciliation
- The reverse-reconciliation key rate is Alice–Bob mutual information minus Bob–Eve information.Source: Reverse Reconciliation · pp. 5–6
- A secure key is associated with Eve having less information than Alice and Bob.Source: Reverse Reconciliation · p. 6
- The channel is modeled with transmission T and excess noise, and Bob’s received output is given by a beamsplitter equation.Source: Reverse Reconciliation · p. 6
- Bob’s two quadratures have variance (1 − T)W + TV, with W denoting the excess-noise variance.Source: Reverse Reconciliation · p. 6
V = VS + V0, (31)
This section calculates Alice–Bob mutual information and Eve’s information using conditional variances and covariance matrices. It then obtains reverse-reconciliation key rates numerically for different preparation-noise values.
V = VS + V0, (31)
- Bob’s estimate of Alice’s signal uses the signal quadrature rather than Alice’s signal-plus-noise variable.The resulting conditional variance is (1 − T)W + TV0.Source: V = VS + V0, (31) · p. 6
- Eve’s information is calculated with the Holevo quantity from Eve’s and conditional Eve covariance matrices and their symplectic spectra.Source: V = VS + V0, (31) · pp. 6–7
- The reverse-reconciliation key rate is calculated numerically for different preparation-noise values.Source: V = VS + V0, (31) · p. 7
Direct Reconciliation
Direct reconciliation with homodyne detection defines the key rate using Alice–Bob mutual information and Eve–Alice information. The Eve term is calculated from a conditional covariance matrix.
Direct Reconciliation
- The direct-reconciliation key rate subtracts Eve–Alice information from Alice–Bob mutual information.Source: Direct Reconciliation · p. 7
- The passage states that I(XA : XB) has already been calculated in Eq. (34), but does not provide the subsequent expression defining Eve’s contribution.Source: Direct Reconciliation · p. 7
- For homodyne detection, Eve’s conditional covariance matrix is used to calculate S(E|XA).Source: Direct Reconciliation · p. 7
VE|QA = VE(V0, V ), (51)
This section gives the conditional Eve-mode symplectic spectrum and the expressions used to calculate the direct-reconciliation secret key rate numerically.
VE|QA = VE(V0, V ), (51)
- The conditional symplectic spectrum νE|XA is specified using VE, defined in Eq. (36).The section identifies F and G as coefficients in the spectrum expressions.Source: VE|QA = VE(V0, V ), (51) · p. 7
- The final direct-reconciliation secret key rate is calculated numerically using Eq. (49).The calculation is repeated for various preparation-noise values.Source: VE|QA = VE(V0, V ), (51) · p. 7