all ωs , ωi . Note that this cancellation scheme is not restricted

to signal and idler photons at nearly equal wavelengths. It

applies also to asymmetrical setups where the signal photon

(generating the trigger to indicate the presence of the idler

FIG. 5. Photo of our entangled photon-pair source as used photon) is at a short wavelength of around 800 nm and travels

in the ¬rst long-distance test of Bell inequalities (Tittel et only a short distance. Using a ¬ber with appropriate zero

al. 1998). Note that the whole source ¬ts in a box of only dispersion wavelength »0 , it is still possible to achieve equal

40 — 45 — 15cm3 size, and that neither special power supply DGD with respect to the energy-correlated idler photon at

nor water cooling is necessary. telecommunication wavelength, sent through a long ¬ber.

52

FIG. 10. Normalized net key creation rate ρnet as a func-

tion of the distance in optical ¬bers. For n = 1, Alice uses

a perfect single photon source. For n > 1, the link is di-

vided into n equal length sections and n/2 2-photon sources

are distributed between Alice and Bob. Parameters: detec-

tion e¬ciency · = 10%, dark count probability pdark = 10’4 ,

¬ber attenuation ± = 0.25 dB/km.

1'000'000

100'000

10'000

Rnet [bit/s]

1550 nm "single"

1'000

100

FIG. 8. Transmission losses in free space as calculated us- 800 nm 1300 nm 1550 nm

10

ing the LOWTRAN code for earth to space transmission at

the elevation and location of Los Alamos, USA. Note that 1

there is a low loss window at around 770 nm “ a wavelength 0 20 40 60 80 100 120

Distance [km]

where high e¬ciency Silicon APD™s can be used for single

photon detection (see also Fig. 9 and compare to Fig. 6).

FIG. 11. Bit rate after error correction and privacy ampli-

¬cation vs. ¬ber length. The chosen parameters are: pulse

rates 10 Mhz for faint laser pulses (µ = 0.1) and 1 MHz for the

case of ideal single photons (1550 nm “single”); losses 2, 0.35

1E-13

InGaAs APD

and 0.25 dB/km, detector e¬ciencies 50%, 20% and 10%, and

150 K

dark count probabilities 10’7 , 10’5 , 10’5 for 800nm, 1300nm

1E-14

NEP [W/Hz1/2]

and 1550 nm respectively. Losses at Bob and QBERopt are

neglected.

1E-15

Ge APD

77 K

1E-16 Si APD

1E-17

400 600 800 1000 1200 1400 1600 1800

Wavelength [nm]

FIG. 9. Noise equivalent power as a function of wavelength

for Silicon, Germanium, and InGaAs/InP APD™s.

FIG. 12. Typical system for quantum cryptography using

polarization coding (LD: laser diode, BS: beamsplitter, F:

0.0

neutral density ¬lter, PBS: polarizing beam splitter, »/2: half

-10.0

waveplate, APD: avalanche photodiode).

n=1

-20.0

-30.0

10 Log (ρnet)

ρ

-40.0

n=2

-50.0

n=4

-60.0

-70.0

-80.0

-90.0

0 25 50 75 100 125 150 175 200

Distance [km]

53

FIG. 15. Poincar´ sphere representation of two-levels quan-

e

tum states generated by two-paths interferometers. The

states generated by an interferometer where the ¬rst coupler

is replaced by a switch correspond to the poles. Those gener-

ated with a symetrical beamsplitter are on the equator. The

azimuth indicates the phase between the two paths.

FIG. 13. Geneva and Lake Geneva. The Swisscom optical

¬ber cable used for quantum cryptography experiments runs

under the lake between the town of Nyon, about 23 km north

FIG. 16. Double Mach-Zehnder implementation of an in-

of Geneva, and the centre of the city.

terferometric system for quantum cryptography (LD: laser

diode, PM: phase modulator, APD: avalanche photodiode).

The inset represents the temporal count distribution recorded

as a function of the time passed since the emission of the pulse

by Alice. Interference is observed in the central peak.

FIG. 14. Conceptual interferometric set-up for quantum

cryptography using an optical ¬ber Mach-Zehnder interferom-

eter (LD: laser diode, PM: phase modulator, APD: avalanche

photodiode).

FIG. 17. Evolution of the polarization state of a light pulse

represented on the Poincar´ sphere over a round trip propa-

e

gation along an optical ¬ber terminated by a Faraday mirror.

FIG. 18. Self-aligned “Plug & Play” system (LD: laser

diode, APD: avalanche photodiode, Ci : ¬ber coupler, PMj :

phase modulator, PBS: polarizing beamsplitter, DL: optical

delay line, FM: Faraday mirror, DA : classical detector).

54

FIG. 23. System for phase-coding entanglement based

quantum cryptography (APD: avalanche photodiode). The

FIG. 19. Implementation of sideband modulation (LD: photons choose their bases randomly at Alice and Bob™s cou-

laser diode, A: attenuator, PMi : optical phase modulator, plers.

¦j : electronic phase controller, RFOk : radio frequency oscil-

lator, FP: Fabry-Perot ¬lter, APD: avalanche photodiode).

FIG. 24. Quantum cryptography system exploiting pho-

tons entangled in energy-time and active basis choice. Note

the similarity with the faint laser double Mach-Zehnder im-

plementation depicted in Fig. 16.

FIG. 20. Multi-users implementation of quantum cryptog-

raphy with one Alice connected to three Bobs by optical

¬bers. The photons sent by Alice randomly choose to go to

one or the other Bob at a coupler.

FIG. 25. Schematic diagram of the ¬rst system designed

and optimized for long distance quantum cryptography and

exploiting phase coding of entangled photons.

FIG. 21. Typical system for quantum cryptography ex-

ploiting photon pairs entangled in polarization (PR: active

polarization rotator, PBS: polarizing beamsplitter, APD:

Laser

avalanche photodiode).

t0

s P, l A ; l P, s s P , l B; l P , s

A B

Alice Bob l P, l

s P, s s P, s l P, l

single count rate

single count rate

A

A B

B

β

±

φ

source

β

±

tA - t0 tB - t 0

nonlinear

crystal .

beam-splitter

stop

start

± β

perfect correlation

80

+

long/long+

coincidence

+

count rate

short/short

60

short/long

long/short

’

40

’

20

±+β

anticorrelation

0

0

-3 -2 -1 1 2 3

Alice Bob

time difference [ns]

FIG. 22. Principle of phase coding quantum cryptography FIG. 26. Schematics of quantum cryptography using en-

using energy-time entangled photons pairs. tangled photons phase-time coding.

55

1.0

one w ay com m uni- tw o w ay com m unication

is necessary

-cation suffices

0.8

secret-key rate

E ve's inform ation

Inform ation [bit]

0.6

error correction and quantum privacy am pl. or

0.4

classical privacy am pl. classical advantage distillation

0.2

B ell-C H S H B ell-C H S H ineq.

B ob's inform ation

ineq. is violated is not violated

0.0

QBER0

IR 6

IR 4

0.0 0.1 0.2 0.3 0.4 0.5

Q uantum bit error rate (Q B E R )

FIG. 27. Poincar´ representation of the BB84 states and

e

the intermediate basis, also known as the Breidbart basis,

FIG. 30. Eve and Bob information versus the QBER, here

that can be used by Eve.

plotted for incoherent eavesdropping on the 4-state protocol.

For QBERs below QBER0 , Bob has more information than

Eve and secret-key agreement can be achieved using classical

Eve error correction and privacy ampli¬cation. These can, in prin-

ciple, be implemented using only 1-way communication. The

Alice Bob secret-key rate can be as large as the information di¬erences.

For QBERs above QBER0 (≡ D0 ), Bob has a disadvantage

A B

U with respect to Eve. Nevertheless, Alice and Bob can apply

quantum privacy ampli¬cation up to the QBER correspond-

ing to the intercept-resend eavesdropping strategies, IR4 and

IR6 for the 4-state and 6-state protocols, respectively. Alter-

natively, they can apply a classical protocol called advantage

distillation which is e¬ective precisely up to the same maxi-

mal QBER IR4 and IR6 . Both the quantum and the classical

perturbation information

protocols require then 2-way communication. Note that for

the eavesdropping strategy optimal from Eve™ Shannon point

FIG. 28. Eavesdropping on a quantum channel. Eve ex-

of view on the 4-state protocol, QBER0 correspond precisely

tracts information out of the quantum channel between Alice

to the noise threshold above which a Bell inequality can no

and Bob at the cost of introducing noise into that channel.

longer be violated.

FIG. 29. Poincar´ representation of the BB84 states in the

e

event of a symmetrical attack. The state received by Bob after

the interaction of Eve™s probe is related to the one sent by

Alice by a simple shrinking factor. When the unitary operator

U entangles the qubit and Eve™s probe, Bob™s state (eq. 46)

is mixed and is represented by a point inside the Poincar´ e

sphere.

56

FIG. 31. Intuitive illustration of theorem 1. The initial

situation is depicted in a). During the 1-way public discussion

phase of the protocol Eve receives as much information as

Bob, the initial information di¬erence δ thus remains. After

error correction, Bob™s information equals 1, as illustrated on

b). After privacy ampli¬cation Eve™s information is zero. In

c) Bob has replaced all bits to be disregarded by random bits.

Hence the key has still the original length, but his information

has decreased. Finally, removing the random bits, the key is

shortened to the initial information di¬erence, see d). Bob

has full information on this ¬nal key, while Eve has none.

FIG. 32. Realistic beamsplitter attack. Eve stops all

pulses. The two photon pulses have a 50% probability to

be analyzed by the same analyzer. If this analyzer is compat-

ible with the state prepared by Alice, then both photon are

detected at the same outcome; if not there is a 50% chance

that they are detected at the same outcome. Hence, there

is a probability of 3/8 that Eve detects both photons at the

same outcome. In such a case, and only in such a case, she

resends a photon to Bob. In 2/3 of these cases she introduces

no errors since she identi¬ed the correct state and gets full

information; in the remaining cases she has a probability 1/2

to introduce an error and gains no information. The total

QBER is thus 1/6 and Eve™s information gain 2/3.

57