Continuous‐Variable Quantum Key Distribution with Gaussian Modulation—The Theory of Practical Implementations

Laudenbach, Fabian; Pacher, Christoph; Fung, Chi-Hang Fred; Poppe, Andreas; Peev, Momtchil; Schrenk, Bernhard; Hentschel, Michael; Walther, Philip; Hübel, Hannes

doi:10.1002/qute.201800011

Cited by 300 publications

(338 citation statements)

References 94 publications

(213 reference statements)

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“…As such it can be written as a Schmidt decomposition

| Ψ_{A B E} ⟩ = \sum_{j} λ_{j} {false| ψ_{i} false⟩}_{A B} {false| ϕ_{i} false⟩}_{E}

Sharing the coefficients λ, both subsystems have the same entropy. Thus we have

S_{E} = S_{A B}

and

S_{A B}

is obtained from the symplectic eigenvalues of a covariance matrix of the form (e.g., derived in [], Appendix C)

{normalΣ}_{A B} = (\begin{matrix} V 1_{2} & \sqrt{T false(V^{2} - 1 false)} σ_{z} \\ \sqrt{T false(V^{2} - 1 false)} σ_{z} & (T false(V - 1 false) + 1 + ξ) 1_{2} \end{matrix})

If the entire transmission T and excess noise ξ are attributed to Eve, the parameters above comprise both the contributions from the channel as well as from the receiver, that is,

T = T_{ch} T_{rec}

and

ξ = T_{rec} ξ_{ch} + ξ_{rec}

, where

ξ_{ch}

is the channel noise as received by Bob.…”

Section: Purification Ansatzmentioning

confidence: 99%

“…Although in the trusted‐receiver model

T_{rec}

and

ξ_{rec}

do not contribute to

S_{E}

, they still influence Bob's measurement and therefore also Eve's entropy conditioned on Bob's measurement. The computation of

S_{E false| B}

is therefore a bit more elaborate than just omitting

T_{rec}

and

ξ_{rec}

in the calculations (and has not been discussed in our review paper).…”

Section: Purification Ansatzmentioning

confidence: 99%

“…In particular, since coherent states can be both reliably produced at high rates and efficiently measured using standard coherent detection, continuous‐variable quantum key distribution (CV‐QKD) with displaced thermal states (i.e., noisy coherent states) is valued for its modest technological requirements. Our recent review article provides a detailed introduction into the mathematical tools and methods required for the security analysis, noise‐modeling, and parameter estimation in CV‐QKD. The so‐called trusted‐device scenario, however, is only mentioned briefly and incompletely in the above mentioned review (Section 7.3).…”

Section: Introductionmentioning

confidence: 99%

“…In CV‐QKD the lower bound on the asymptotic secure‐key rate for reverse reconciliation, assuming collective attacks, is described by

K = f_{sym} false(1 - FER false) false(1 - ν false) false(β I_{A B} - χ_{E B} false)

where

f_{sym}

is the symbol rate, FER is the frame‐error rate during error correction, ν is the fraction of the raw key consumed by parameter estimation, β is the reconciliation efficiency,

I_{A B}

is the mutual information between Alice and Bob, and

χ_{E B}

is an upper bound for the mutual information between Eve and Bob, also referred to as Holevo bound. In the case of Gaussian modulation of coherent states, the mutual information between Alice and Bob reads

\begin{matrix} true \\ right I_{A B} & left = \frac{μ}{2} \log_{2} (() 1 + SNR) \\ left = \frac{μ}{2} \log_{2} (() 1 + \frac{T_{tot} false(V - 1 false)}{μ + ξ_{tot}}) \end{matrix}

where

T_{tot}

is the total attenuation (channel and detection), V is the variance of the two‐mode‐squeezed vacuum state (in SNU) that Alice and Bob share,

ξ_{tot}

is the total excess noise as measured by Bob (in SNU) and μ indicates whether homodyne detection (measurement of q or p ;

μ = 1

) or heterodyne detection (simultaneous measurement of q and p ;

μ = 2

) is performed.…”

Section: Introductionmentioning

confidence: 99%

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Analysis of the Trusted‐Device Scenario in Continuous‐Variable Quantum Key Distribution

Laudenbach

Pacher

2019

Adv Quantum Tech

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The assumption that detection and/or state preparation devices used for continuous‐variable quantum key distribution are beyond influence of potential eavesdroppers leads to a significant performance enhancement in terms of achievable key rate and transmission distance. A detailed and comprehensible derivation of the Holevo bound in this so‐called trusted‐device scenario is provided. Modeling an entangling‐cloner attack and using some basic algebraic matrix transformations, it is shown that the computation of the Holevo bound can be reduced to the solution of a quadratic equation. As an advantage of our derivation, the mathematical complexity of our solution does not increase with the number of trusted‐noise sources. Finally, a numerical evaluation of our results is provided, illustrating the counter‐intuitive fact that an appropriate amount of trusted‐receiver loss and noise can even be beneficial for the key rate.

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“…As such it can be written as a Schmidt decomposition

| Ψ_{A B E} ⟩ = \sum_{j} λ_{j} {false| ψ_{i} false⟩}_{A B} {false| ϕ_{i} false⟩}_{E}

Sharing the coefficients λ, both subsystems have the same entropy. Thus we have

S_{E} = S_{A B}

and

S_{A B}

is obtained from the symplectic eigenvalues of a covariance matrix of the form (e.g., derived in [], Appendix C)

{normalΣ}_{A B} = (\begin{matrix} V 1_{2} & \sqrt{T false(V^{2} - 1 false)} σ_{z} \\ \sqrt{T false(V^{2} - 1 false)} σ_{z} & (T false(V - 1 false) + 1 + ξ) 1_{2} \end{matrix})

If the entire transmission T and excess noise ξ are attributed to Eve, the parameters above comprise both the contributions from the channel as well as from the receiver, that is,

T = T_{ch} T_{rec}

and

ξ = T_{rec} ξ_{ch} + ξ_{rec}

, where

ξ_{ch}

is the channel noise as received by Bob.…”

Section: Purification Ansatzmentioning

confidence: 99%

“…Although in the trusted‐receiver model

T_{rec}

and

ξ_{rec}

do not contribute to

S_{E}

, they still influence Bob's measurement and therefore also Eve's entropy conditioned on Bob's measurement. The computation of

S_{E false| B}

is therefore a bit more elaborate than just omitting

T_{rec}

and

ξ_{rec}

in the calculations (and has not been discussed in our review paper).…”

Section: Purification Ansatzmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In CV‐QKD the lower bound on the asymptotic secure‐key rate for reverse reconciliation, assuming collective attacks, is described by

K = f_{sym} false(1 - FER false) false(1 - ν false) false(β I_{A B} - χ_{E B} false)

where

f_{sym}

is the symbol rate, FER is the frame‐error rate during error correction, ν is the fraction of the raw key consumed by parameter estimation, β is the reconciliation efficiency,

I_{A B}

is the mutual information between Alice and Bob, and

χ_{E B}

\begin{matrix} true \\ right I_{A B} & left = \frac{μ}{2} \log_{2} (() 1 + SNR) \\ left = \frac{μ}{2} \log_{2} (() 1 + \frac{T_{tot} false(V - 1 false)}{μ + ξ_{tot}}) \end{matrix}

where

T_{tot}

is the total attenuation (channel and detection), V is the variance of the two‐mode‐squeezed vacuum state (in SNU) that Alice and Bob share,

ξ_{tot}

is the total excess noise as measured by Bob (in SNU) and μ indicates whether homodyne detection (measurement of q or p ;

μ = 1

) or heterodyne detection (simultaneous measurement of q and p ;

μ = 2

) is performed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Analysis of the Trusted‐Device Scenario in Continuous‐Variable Quantum Key Distribution

Laudenbach

Pacher

2019

Adv Quantum Tech

Self Cite

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show abstract

“…Key areas include quantum‐based communication, computation, control, engineering, information, metrology, optics, sensing and simulation, as well as adjacent areas such as nanophotonics, quasiparticle excitations, topological materials, superconductors, micro‐ and nano‐electromechanical systems, ultracold atoms and others. The first three issues published in 2018 already provided a glimpse of our intended broad topical spectrum, with Reviews on continuous‐variable quantum key distribution , non‐equilibrium Bose–Einstein‐like condensation and integrated quantum photonics based on diamond , accompanied by original research contributions as Communications or Full Papers.…”

mentioning

confidence: 99%

Quantum Technologies are (in) QUTE

Hildebrandt¹,

Stournara²,

Wang³

et al. 2019

Adv Quantum Tech

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Secret key rate proof of multicarrier continuous‐variable quantum key distribution

Gyöngyösi

Imre

2019

Int J Communication

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We prove the secret key rate formulas and derive security threshold parameters of multicarrier continuous-variable quantum key distribution CVQKD. In a multicarrier CVQKD scenario, the Gaussian input quantum states of the legal parties are granulated into Gaussian subcarrier continuous variables (CVs). The multicarrier communication formulates Gaussian subchannels from the physical quantum channel, each dedicated to the transmission of a subcarrier CV.The Gaussian subcarriers are decoded by a unitary CV operation, which results in the recovered single-carrier Gaussian CVs. We derive the formulas through the adaptive multicarrier quadrature division (AMQD) scheme, the singular value decomposition (SVD)-assisted AMQD, and the multiuser AMQD multiuser quadrature allocation (MQA). We prove that the multicarrier CVQKD leads to improved secret key rates and higher tolerable excess noise in comparison with single-carrier CVQKD. We derive the private classical capacity of a Gaussian subchannel and the security parameters of an optimal Gaussian collective attack in the multicarrier setting. We reveal the secret key rate formulas for one-way and two-way multicarrier CVQKD protocols, assuming homodyne and heterodyne measurements and direct and reverse reconciliation. The results reveal the physical boundaries of physically allowed Gaussian attacks in a multicarrier CVQKD scenario and confirm that the improved transmission rates lead to enhanced secret key rates and security thresholds. INTRODUCTIONContinuous-variable quantum key distribution (CVQKD) allows for legal parties to transmit information with unconditional security over the currently established telecommunication networks. [33][34][35][36][37][38] The CVQKD systems, in contrast to discrete variable (DV) QKD protocols, do not require single-photon sources and can be implemented by standard devices and modulation techniques, allowing an efficient signal processing in practical scenarios. The CVQKD schemes in general are based on Gaussian modulation, which is a well-applicable practical finding in the experiment. 29 In CVQKD, the information is carried via a Gaussian-modulated position and momentum quadratures in the phase space. The Gaussian quantum states (referred to as single carriers throughout) are sent through a noisy quantum channel 39-50 by the sender, Alice. The quantum channel is attacked by an eavesdropper (Eve), and the receiver (Bob) gets a noisy system. The optimal attack against CVQKD is a Gaussian attack; thus, the noise of the quantum channel can be provably modeled as an additive white Gaussian noise. 1-3,7-13,16-19 The security of CVQKD has been already proven against several of the most Int J Commun Syst. 2019;32:e3865.wileyonlinelibrary.com/journal/dac powerful optimal Gaussian collective attacks, [8][9][10][11][12][13][16][17][18][19] where the eavesdropper is allowed to use quantum memory and to perform a collective (joint) measurement on her quantum register at the end of the protocol run.Besides the attractive properties of CVQKD, in compari...

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Continuous‐Variable Quantum Key Distribution with Gaussian Modulation—The Theory of Practical Implementations

Cited by 300 publications

References 94 publications

Analysis of the Trusted‐Device Scenario in Continuous‐Variable Quantum Key Distribution

Analysis of the Trusted‐Device Scenario in Continuous‐Variable Quantum Key Distribution

Quantum Technologies are (in) QUTE

Secret key rate proof of multicarrier continuous‐variable quantum key distribution

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