1. Intro. Cont. Var. Information Theory CVQKD XP Next
Continuous Variable
Quantum Cryptography
Towards High Speed Quantum Cryptography
Frédéric Grosshans
CNRS / ENS Cachan
Palacký University, Olomouc, 2011

2. Intro. Cont. Var. Information Theory CVQKD XP Next
1 Introduction
Prefect Secrecy and Quantum Cryptography
Various Secure Systems
2 Continuous variables
Field quadratures
Homodyne Detection : Theory
3 Information Theory
XXth century CVQKD
Where are the bits ?
4 Continuous Variable Quantum Key Distribution
Spying
Protocols
5 Experimental systems
1st and 2nd generation demonstrators
Key-Rates
Integration with classical cryptography
6 Open problems

3. Intro. Cont. Var. Information Theory CVQKD XP Next
Conditions for Perfect Secrecy
Alice sends a secret message to Bob

4. Intro. Cont. Var. Information Theory CVQKD XP Next
Conditions for Perfect Secrecy
Alice sends a secret message to Bob
through a channel observed by Eve.

5. Intro. Cont. Var. Information Theory CVQKD XP Next
Conditions for Perfect Secrecy
Alice sends a secret message to Bob
through a channel observed by Eve.
She encrypts the message with a secret key

6. Intro. Cont. Var. Information Theory CVQKD XP Next
Conditions for Perfect Secrecy
Alice sends a secret message to Bob
through a channel observed by Eve.
She encrypts the message with a secret key
as long as the message.

7. Intro. Cont. Var. Information Theory CVQKD XP Next
Quantum Key Distribution
Alice sends quantum objects to Bob

8. Intro. Cont. Var. Information Theory CVQKD XP Next
Quantum Key Distribution
Alice sends quantum objects to Bob
Eve’s Measurenents

9. Intro. Cont. Var. Information Theory CVQKD XP Next
Quantum Key Distribution
Alice sends quantum objects to Bob
Eve’s Measurenents ⇒ measurable perturbations
⇒ secret key generation

10. Intro. Cont. Var. Information Theory CVQKD XP Next
Unconditionnally Secure Systems . . .
Single Photon QKD
Long Range (∼ 100 km)
Low rate (kbit/s)

11. Intro. Cont. Var. Information Theory CVQKD XP Next
Unconditionnally Secure Systems . . .
Single Photon QKD
Long Range (∼ 100 km)
Low rate (kbit/s) maybe a few Mbit/s in the long run

12. Intro. Cont. Var. Information Theory CVQKD XP Next
Unconditionnally Secure Systems . . .
Single Photon QKD
Long Range (∼ 100 km)
Low rate (kbit/s) maybe a few Mbit/s in the long run
Classical One-Time-Pad
Very Long Range (Paris–Olomouc)
Not so small rate :

13. Intro. Cont. Var. Information Theory CVQKD XP Next
Unconditionnally Secure Systems . . .
Single Photon QKD
Long Range (∼ 100 km)
Low rate (kbit/s) maybe a few Mbit/s in the long run
Classical One-Time-Pad
Very Long Range (Paris–Olomouc)
Not so small rate :
1 CD / year = 180 bits/s

14. Intro. Cont. Var. Information Theory CVQKD XP Next
Unconditionnally Secure Systems . . .
Single Photon QKD
Long Range (∼ 100 km)
Low rate (kbit/s) maybe a few Mbit/s in the long run
Classical One-Time-Pad
Very Long Range (Paris–Olomouc)
Not so small rate :
1 CD / year = 180 bits/s
1 iPod (160 GB)/ year = 40 kbit/s

15. Intro. Cont. Var. Information Theory CVQKD XP Next
Unconditionnally Secure Systems . . .
Single Photon QKD
Long Range (∼ 100 km)
Low rate (kbit/s) maybe a few Mbit/s in the long run
Classical One-Time-Pad
Very Long Range (Paris–Olomouc)
Not so small rate :
1 CD / year = 180 bits/s
1 iPod (160 GB)/ year = 40 kbit/s
But the data has to stay here

16. Intro. Cont. Var. Information Theory CVQKD XP Next
. . . and Continuous Variable
Medium Range :∼ 25 km
Medium Rate :∼ a few kbit/s

17. Intro. Cont. Var. Information Theory CVQKD XP Next
. . . and Continuous Variable
Medium Range :∼ 25 km
Medium Rate :∼ a few kbit/s
Much less mature

18. Intro. Cont. Var. Information Theory CVQKD XP Next
. . . and Continuous Variable
Medium Range :∼ 25 km ; 80 km soon ?
Medium Rate :∼ a few kbit/s ; Mbits/s soon ?
Much less mature ⇒ Much room for improvements

19. Intro. Cont. Var. Information Theory CVQKD XP Next
1 Introduction
Prefect Secrecy and Quantum Cryptography
Various Secure Systems
2 Continuous variables
Field quadratures
Homodyne Detection : Theory
3 Information Theory
XXth century CVQKD
Where are the bits ?
4 Continuous Variable Quantum Key Distribution
Spying
Protocols
5 Experimental systems
1st and 2nd generation demonstrators
Key-Rates
Integration with classical cryptography
6 Open problems

20. Intro. Cont. Var. Information Theory CVQKD XP Next
Field quadratures
Classical field
Electromagnetic field
described by QA and PA
E(t) = QA cos ωt + PA sin ωt

21. Intro. Cont. Var. Information Theory CVQKD XP Next
Field quadratures
Classical field
Electromagnetic field
described by QA and PA
E(t) = QA cos ωt + PA sin ωt
Quantum description
Q and P do not commute:
[Q, P] ∝ i .
Add a
“quantum noise”:
Q = QA + BQ et P = PA + BP
Heisenberg =⇒ ∆BQ ∆BP ≥ 1

22. Intro. Cont. Var. Information Theory CVQKD XP Next
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 cos ϕ + Psignal sin ϕ)

23. Intro. Cont. Var. Information Theory CVQKD XP Next
1 Introduction
Prefect Secrecy and Quantum Cryptography
Various Secure Systems
2 Continuous variables
Field quadratures
Homodyne Detection : Theory
3 Information Theory
XXth century CVQKD
Where are the bits ?
4 Continuous Variable Quantum Key Distribution
Spying
Protocols
5 Experimental systems
1st and 2nd generation demonstrators
Key-Rates
Integration with classical cryptography
6 Open problems

24. Intro. Cont. Var. Information Theory CVQKD XP Next
XXth century CVQKD
At the end of XXth century it was obvious that a
generalization of QKD to continuous variables could be
interesting.
Problem : discrete bits continuous variable

25. Intro. Cont. Var. Information Theory CVQKD XP Next
XXth century CVQKD
At the end of XXth century it was obvious that a
generalization of QKD to continuous variables could be
interesting.
Problem : discrete bits continuous variable
Adapting BB84?
Mark Hillery, “Quantum Cryptography with
Squeezed States”,
arXiv:quant-ph/9909006/PRA 61 022309

26. Intro. Cont. Var. Information Theory CVQKD XP Next
XXth century CVQKD
At the end of XXth century it was obvious that a
generalization of QKD to continuous variables could be
interesting.
Problem : discrete bits continuous variable
Natural modulation + information theory!
Nicolas J. Cerf, Marc Lévy, Gilles Van
Assche : “Quantum distribution of Gaussian
keys using squeezed states”,
arXiv:quant-ph/0008058/PRL 63 052311

27. Intro. Cont. Var. Information Theory CVQKD XP Next
Where are the bits ?
Quite frequent discussion with discrete quantum
cryptographers :
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.”

28. Intro. Cont. Var. Information Theory CVQKD XP Next
Where are the bits ?
Quite frequent discussion with discrete quantum
cryptographers :
DQC : How do you encode a 0 or a 1 in CVQKD?
Me : Gilles/Jérôme/Anthony/Sébastien developed
a really efficient code, using sliced
reconciliation/LDPC matrices/R8 rotations and
octonions. Only he knows how it works.

29. Intro. Cont. Var. Information Theory CVQKD XP Next
Where are the bits ?
Quite frequent discussion with discrete quantum
cryptographers :
DQC : How do you encode a 0 or a 1 in CVQKD?
Me : Gilles/Jérôme/Anthony/Sébastien developed
a really efficient code, using sliced
reconciliation/LDPC matrices/R8 rotations and
octonions. Only he knows how it works.
Computation of the ideal code performance is easy !

30. Intro. Cont. Var. Information Theory CVQKD XP Next
They’re hidden
Availaible informationin a continuous signal

31. Intro. Cont. Var. Information Theory CVQKD XP Next
They’re hidden
Availaible informationin a continuous signal
Differential entropy
H(X) = − P(x) dx log P(x) dx
dx P(x) log P(x) − log dx
H(X) constante

32. Intro. Cont. Var. Information Theory CVQKD XP Next
They’re hidden
Availaible informationin a continuous signal
with noise ?
Differential entropy
H(X) = − P(x) dx log P(x) dx H(X) = log ∆X + constante
dx P(x) log P(x) − log dx
H(X) constante

33. Intro. Cont. Var. Information Theory CVQKD XP Next
They’re hidden
Availaible informationin a continuous signal
with noise ?
Differential entropy Mutual information
I(X : Y) = H(Y) − H(Y|X)
H(X) = − P(x) dx log P(x) dx
= H(Y) − H(Y|X)
dx P(x) log P(x) − log dx
∆Y2
= 1
log
H(X) constante 2
∆Y2 |X

34. Intro. Cont. Var. Information Theory CVQKD XP Next
1 Introduction
Prefect Secrecy and Quantum Cryptography
Various Secure Systems
2 Continuous variables
Field quadratures
Homodyne Detection : Theory
3 Information Theory
XXth century CVQKD
Where are the bits ?
4 Continuous Variable Quantum Key Distribution
Spying
Protocols
5 Experimental systems
1st and 2nd generation demonstrators
Key-Rates
Integration with classical cryptography
6 Open problems

35. Intro. Cont. Var. Information Theory CVQKD XP Next
The spy’s power
Heisenberg :
∆BEve ∆BBob ≥ 1

36. Intro. Cont. Var. Information Theory CVQKD XP Next
The spy’s power
Heisenberg :
∆BEve ∆BBob ≥ 1
⇒ ∆BBob gives IEve
IBob

37. Intro. Cont. Var. Information Theory CVQKD XP Next
Quantum Key Distribution Protocols
Channel Evauation (noise measure)
Alice&Bob evaluate IEve

38. Intro. Cont. Var. Information Theory CVQKD XP Next
Quantum Key Distribution Protocols
Channel Evauation (noise measure)
Alice&Bob evaluate IEve
Reconciliation (error correction)
Alice&Bob share IBob identical bits.
Ève knows IEve .

39. Intro. Cont. Var. Information Theory CVQKD XP Next
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 Amplification
Alice&Bob share IBob − IEve identical bits.
Ève knows ∼ 0.

40. Intro. Cont. Var. Information Theory CVQKD XP Next
Theoretical Progresses in the last 10 years
We went from a protocol
using squeezed states,
insecure beyond 50% losses (15 km),
proved secure against Gaussian individual attack

41. Intro. Cont. Var. Information Theory CVQKD XP Next
Theoretical Progresses in the last 10 years
We went from a protocol
using squeezed states,
insecure beyond 50% losses (15 km),
proved secure against Gaussian individual attack
to a protocol
using coherent states

42. Intro. Cont. Var. Information Theory CVQKD XP Next
Theoretical Progresses in the last 10 years
We went from a protocol
using squeezed states,
insecure beyond 50% losses (15 km),
proved secure against Gaussian individual attack
to a protocol
using coherent states
with no fundamental range limit
proved secure against collective attacks

43. Intro. Cont. Var. Information Theory CVQKD XP Next
Theoretical Progresses in the last 10 years
We went from a protocol
using squeezed states,
insecure beyond 50% losses (15 km),
proved secure against Gaussian individual attack
to a protocol
using coherent states
with no fundamental range limit
proved secure against collective attacks
likely secure against coherent attacks

44. Intro. Cont. Var. Information Theory CVQKD XP Next
Theoretical Progresses in the last 10 years
We went from a protocol
using squeezed states,
insecure beyond 50% losses (15 km),
proved secure against Gaussian individual attack
to a protocol
using coherent states
with no fundamental range limit
proved secure against collective attacks
likely secure against coherent attacks
and experimentally working

45. Intro. Cont. Var. Information Theory CVQKD XP Next
1 Introduction
Prefect Secrecy and Quantum Cryptography
Various Secure Systems
2 Continuous variables
Field quadratures
Homodyne Detection : Theory
3 Information Theory
XXth century CVQKD
Where are the bits ?
4 Continuous Variable Quantum Key Distribution
Spying
Protocols
5 Experimental systems
1st and 2nd generation demonstrators
Key-Rates
Integration with classical cryptography
6 Open problems

46. Intro. Cont. Var. Information Theory CVQKD XP Next
1st generation demonstrator
F. Grosshans et. al., Nature (2003) & Brevet US
m
Key rate 75 kbit/s 3.1 dB (51%) losses
1.7 Mbit/s without losses

47. Intro. Cont. Var. Information Theory CVQKD XP Next
Integrated system

48. Intro. Cont. Var. Information Theory CVQKD XP Next
Integrated system

49. Intro. Cont. Var. Information Theory CVQKD XP Next
Integrated system

50. Intro. Cont. Var. Information Theory CVQKD XP Next
Key-Rates

51. Intro. Cont. Var. Information Theory CVQKD XP Next
Key-Rates
SECOQC Performance
100 kb/s
(2008)
L
Exce osses o
ss n n
95% oise ly
effi
90
cie
%
nt c
eff
ode
ici
10 kb/s Slo
en
t
w
co
co
de
de
1 kb/s
0k
10
20
30
40
50
km
km
km
km
km
m

52. Intro. Cont. Var. Information Theory CVQKD XP Next
Key-Rates

53. Intro. Cont. Var. Information Theory CVQKD XP Next
Key-Rates
SECOQC Performance
100 kb/s
(2008)
L
Exce osses o
ss n n
95% oise ly
Increases modulation rate :
effi
90
cie
×10 easy, ×100 doable
%
nt c
eff
ode
ici
10 kb/s Slo
en
t
w
co
co
de
de use modern codes :
ocotonion based protocol
us
+multi-edge LDPC codes
eG
+ repetition codes
PU
s
1 kb/s
0k
10
20
30
40
50
km
km
km
km
km
m

54. Intro. Cont. Var. Information Theory CVQKD XP Next
Integration with classical cryptography

55. Intro. Cont. Var. Information Theory CVQKD XP Next
Integration with classical cryptography

56. Intro. Cont. Var. Information Theory CVQKD XP Next
1 Introduction
Prefect Secrecy and Quantum Cryptography
Various Secure Systems
2 Continuous variables
Field quadratures
Homodyne Detection : Theory
3 Information Theory
XXth century CVQKD
Where are the bits ?
4 Continuous Variable Quantum Key Distribution
Spying
Protocols
5 Experimental systems
1st and 2nd generation demonstrators
Key-Rates
Integration with classical cryptography
6 Open problems

57. Intro. Cont. Var. Information Theory CVQKD XP Next
Open Problems
Finite size effects
Link with post-selection based protocols (.de, .au)
Side-channels and quantum hacking
Other cryptographic applications