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(at ωs 1, ωs 2) and idler photon (at ωi 1, ωi 2) is the same for
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

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