continuous variables quantum cryptography

Intro. Cont. Var. Information Theory CVQKD XP Next

Continuous VariableQuantum Cryptography

Towards High Speed Quantum Cryptography

Frédéric Grosshans

CNRS / ENS Cachan

Palacký University, Olomouc, 2011


1 IntroductionPrefect Secrecy and Quantum CryptographyVarious Secure Systems

2 Continuous variablesField quadraturesHomodyne Detection : Theory

3 Information TheoryXXth century CVQKDWhere are the bits ?

4 Continuous Variable Quantum Key DistributionSpyingProtocols

5 Experimental systems1st and 2nd generation demonstratorsKey-RatesIntegration with classical cryptography

6 Open problems


Conditions for Perfect Secrecy

Alice sends a secret message to Bob

through a channel observed by Eve.

She encrypts the message with a secret keyas long as the message.



Alice sends a secret message to Bobthrough a channel observed by Eve.





She encrypts the message with a secret key

as long as the message.


Quantum Key Distribution

Alice sends quantum objects to Bob

Eve’s Measurenents⇒measurable perturbations⇒ secret key generation




Eve’s Measurenents

⇒measurable perturbations⇒ secret key generation




Eve’s Measurenents⇒measurable perturbations⇒ secret key generation


Unconditionnally Secure Systems . . .

Single Photon QKD

I Long Range (∼ 100 km)I Low rate (kbit/s)

maybe a few Mbit/s in the long run

Classical One-Time-PadI Very Long Range (Paris–Olomouc)I Not so small rate :

1 CD / year = 180 bits/s1 iPod (160 GB)/ year = 40 kbit/s

I But the data has to stay here



Single Photon QKD

I Long Range (∼ 100 km)I Low rate (kbit/s) maybe a few Mbit/s in the long run






Single Photon QKD



1 CD / year = 180 bits/s

1 iPod (160 GB)/ year = 40 kbit/s




Single Photon QKD






. . . and Continuous Variable

I Medium Range :∼ 25 km

; 80 km soon ?

I Medium Rate :∼ a few kbit/s

; Mbits/s soon ?I Much less mature

⇒ Much room for improvements



I Medium Range :∼ 25 km

; 80 km soon ?

I Medium Rate :∼ a few kbit/s

; Mbits/s soon ?

I Much less mature

⇒ Much room for improvements



I Medium Range :∼ 25 km ; 80 km soon ?I Medium Rate :∼ a few kbit/s ; Mbits/s soon ?I Much less mature⇒ Much room for improvements







6 Open problems


Field quadratures

Classical fieldElectromagnetic fielddescribed by QA and PAE(t) = QA cosωt + PA sinωt

Quantum descriptionQ and P do not commute:

[Q,P] ∝ i~.Add a

“quantum noise”:Q = QA + BQ et P = PA + BP

Heisenberg =⇒ ∆BQ∆BP ≥ 1


Homodyne Detection : Theory

Photocurrents:

i± ∝ (Eosc.(t) ± Esignal(t))2

∝ Eosc.(t)2± 2Eosc.(t)Esignal(t)

after substraction:

δi ∝ Eosc.(t)Esignal(t)

∝ Eosc.(Qsignal cosϕ + Psignal sinϕ)







6 Open problems


XXth century CVQKD

At the end of XXth century it was obvious that ageneralization of QKD to continuous variables could beinteresting.Problem : discrete bits , continuous variable


XXth century CVQKD


Adapting BB84?Mark Hillery, “Quantum Cryptography withSqueezed States”,arXiv:quant-ph/9909006/PRA 61 022309


XXth century CVQKD


Natural modulation + information theory!Nicolas J. Cerf, Marc Lévy, Gilles VanAssche : “Quantum distribution of Gaussiankeys using squeezed states”,arXiv:quant-ph/0008058/PRL 63 052311


Where are the bits ?

Quite frequent discussion with discrete quantumcryptographers :

DQC : How do you encode a 0 or a 1 in CVQKD?Me : I don’t care, C. E. Shannon tells me

“∀ε > 0,∃ code of rate I − ε.”

Computation of the ideal code performance is easy !


Where are the bits ?

Quite frequent discussion with discrete quantumcryptographers :

DQC : How do you encode a 0 or a 1 in CVQKD?Me : Gilles/Jérôme/Anthony/Sébastien developed

a really efficient code, using slicedreconciliation/LDPC matrices/R8 rotations andoctonions. Only he knows how it works.

Computation of the ideal code performance is easy !


They’re hidden

Availaible informationin a continuous signal

with noise ?

Differential entropy

H(X) = −∑P(x) dx logP(x) dx

'

∫dxP(x) logP(x)︸︷︷︸

H(X)

− log dx︸︷︷︸constante


They’re hidden

Availaible informationin a continuous signalwith noise ?



'


H(X)


H(X) = log ∆X + constante


They’re hidden

Availaible informationin a continuous signalwith noise ?



'


H(X)


Mutual information

I(X : Y) = H(Y) −H(Y|X)= H(Y) −H(Y|X)

= 12 log

∆Y2

∆Y2|X







6 Open problems


The spy’s power

Heisenberg :∆BEve∆BBob ≥ 1

⇒ ∆BBob gives I IEve

I IBob


Quantum Key Distribution Protocols

Channel Evauation (noise measure)

Alice&Bob evaluate IEve

Reconciliation (error correction)

Alice&Bob share IBob identical bits.Ève knows IEve.

Privacy AmplificationAlice&Bob share IBob − IEve identical bits.

Ève knows ∼ 0.


Theoretical Progresses in the last 10 years

We went from a protocolI using squeezed states,I insecure beyond 50% losses (15 km),I proved secure against Gaussian individual attack

to a protocol

I using coherent statesI with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacksI and experimentally working




to a protocolI using coherent states

I with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacksI and experimentally working




to a protocolI using coherent statesI with no fundamental range limitI proved secure against collective attacks

I likely secure against coherent attacksI and experimentally working




to a protocolI using coherent statesI with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacks

I and experimentally working




to a protocolI using coherent statesI with no fundamental range limitI proved secure against collective attacksI likely secure against coherent attacksI and experimentally working







6 Open problems


1st generation demonstratorF. Grosshans et. al., Nature (2003) & Brevet US

m

Key rate I 75 kbit/s 3.1 dB (51%) lossesI 1.7 Mbit/s without losses


Integrated system


Key-Rates


Key-Rates

Losses onlyExcess noise95% efficient code

90% efficient codeSlow code

100 kb/s

10 kb/s

1 kb/s 0 km

10 km

20 km

30 km

40 km

50 km

SECOQC Performance(2008)


Key-Rates


Key-Rates

Losses onlyExcess noise95% efficient code

90% efficient codeSlow code

100 kb/s

10 kb/s

1 kb/s 0 km

10 km

20 km

30 km

40 km

50 km

SECOQC Performance(2008)

use GPUsIncr

ease

s m

odul

atio

n ra

te :

×10

easy

, ×10

0 do

able

use modern codes :ocotonion based protocol+multi-edge LDPC codes

+ repetition codes


Integration with classical cryptography







6 Open problems


Open Problems

I Finite size effectsI Link with post-selection based protocols (.de, .au)I Side-channels and quantum hackingI Other cryptographic applications

continuous variables quantum cryptography

Technology

information theorycvqkd

kmlow rate kbits

medium rate

small rate

single photon qkdlong

long run classical

quantum objects

secret message