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to face this problem. This is clearly true for polarization photon level by Hariharan et al. (1993), studied in connection
to photon pairs by Brendel et al. (1995).
based systems; but it is equally a concern for phase based
20
PM ¬bers might be of use for phase based QC systems.
systems, since the interference visibility depends on the
However, this requires the whole setup “ transmission lines
polarization states. Hence, although polarization e¬ects
as well as interferometers at Alice™s and Bob™s “ to be made
are not the only source of di¬culties, we shall describe
of PM ¬bers. While this is principally possible, the need of
them in some detail, distinguishing between 4 e¬ects: the
installing a completely new ¬ber network makes this solution
geometrical one, birefringence, polarization mode disper-
not very practical.


15
Alice and Bob can compensate for it. The e¬ect of bire- ni¬cant in components like phase modulators. In par-
fringence is thus similar to the geometrical e¬ect, though, ticular, some integrated optics waveguides actually guide
in addition to a rotation, it may also a¬ect the elliptic- only one mode and thus behave almost like polarizers
ity. Stability of birefringence requires slow thermal and (e.g. proton exchange waveguides in LiNbO3 ). PDL
mechanical variations. is usually stable, but if connected to a ¬ber with some
Polarization Mode Dispersion (PMD) is the pres- birefringence, the relation between the polarization state
ence of two di¬erent group velocities for two orthogonal and the PDL may ¬‚uctuate, producing random outcomes
polarization modes. It is due to a delicate combination (Elamari et al. 1998). PDL cannot be described by a uni-
of two causes. First, birefringence produces locally two tary operator acting in the polarization state space (but
group velocities. For optical ¬bers, this local modal dis- it is of course unitary in a larger space (Huttner et al.
persion is in good approximation equal to the phase dis- 1996a). It does thus not preserve the scalar product. In
persion, of the order of a few ps/km. Hence, locally an particular, it can turn non-orthogonal states into orthog-
optical pulse tends to split into a fast mode and a slow onal ones which can then be distinguished unambiguously
mode. But because the birefringence is small, the two (at the cost of some loss) (Huttner et al. 1996a, Clarke et
modes couple easily. Hence any small imperfection along al. 2000). Note that this could be used by Eve, specially
the ¬ber produces polarization mode coupling: some en- to eavesdrop on the 2-state protocol (paragraph II D 1).
ergy of the fast mode couples into the slow mode and Let us conclude this paragraph on polarization e¬ects
vice-versa. PMD is thus similar to a random walk21 and in ¬bers by mentioning that they can be passively com-
grows only with the square root of the ¬ber length. It pensated, provided one uses a go-&-return con¬guration,
is expressed in √ps , with values as low as 0.1 √ps for using Faraday mirrors, as described in section IV C 2.
km km
modern ¬bers and possibly as high as 0.5 or even 1 √ps km
for older ones.
3. Chromatic dispersion e¬ects in singlemode ¬bers
Typical lengths for the polarization mode coupling
vary from a few meters up to hundreds of meters. The
In addition to polarization e¬ects, chromatic disper-
stronger the coupling, the weaker the PMD (the two
sion (CD) can cause problems for quantum cryptography
modes do not have time to move away between the cou-
as well. For instance, as explained in sections IV C and
plings). In modern ¬bers, the couplings are even arti¬-
V B, schemes implementing phase- or phase-and-time-
cially increased during the drawing process of the ¬bers
coding rely on photons arriving at well de¬ned times,
(Hart et al. 1994, Li and Nolan 1998). Since the cou-
that is on photons well localized in space. However, in
plings are exceedingly sensitive, the only reasonable de-
dispersive media like optical ¬bers, di¬erent group ve-
scription is a statistical one, hence PMD is described as
locities act as a noisy environment on the localization of
a statistical distribution of delays δ„ . For long enough
the photon as well as on the phase acquired in an inter-
¬bers, the statistics is Maxwellian and PMD is related to
ferometer. Hence, the broadening of photons featuring
the ¬ber length „“, the mean coupling length h, the mean
non-zero bandwidth, or, in other words, the coupling be-
modal birefringence B and to the RMS delay as follows
tween frequency and position must be circumvented or
2
(Gisin et al. 1995): PMD≡ << δ„ >> = Bh „“/h.
controlled. This implies working with photons of small
PMD could cause depolarization which would be devas-
bandwidth, or, as long as the bandwidth is not too large,
tating for quantum communication, similar to any deco-
operating close to the wavelength »0 where chromatic
herence in quantum information processing. But fortu-
dispersion is zero, i.e. for standard ¬bers around 1310
nately, for quantum communication the remedy is easy, it
nm. Fortunately, ¬ber losses are relatively small at this
su¬ces to use a source with a coherence time larger than
wavelength and amount to ≈0.35 dB/km. This region
the largest delay δ„ . Hence, when laser pulses are used
is called the second telecommunication window22 . There
(with typical spectral widths ∆» ¤ 1 nm, corresponding
are also special ¬bers, called dispersion-shifted, with a
to a coherence time ≥ 3 ps, see paragraph III A 1), PMD
refractive index pro¬le such that the chromatic disper-
is no real problem. For photons created by parametric
sion goes to zero around 1550 nm, where the attenuation
down conversion, however, PMD can impose severe lim-
is minimal (Neumann 1988)23 .
itations since ∆» ≥ 10 nm (coherence time ¤ 300 fs) is
not unusual.
Polarization Dependent Losses (PDL) is a di¬er-
ential attenuation between two orthogonal polarization 22
The ¬rst one, around 800 nm, is almost no longer used. It
modes. This e¬ect is negligible in ¬bers, but can be sig-
was motivated by the early existence of sources and detectors
at this wavelength. The third window is around 1550 nm
where the attenuation reaches an absolute minimum (Thomas
et al. 2000) and where erbium doped ¬bers provide convenient
21
In contrast to Brownian motion describing particles di¬u- ampli¬ers (Desurvire 1994).
sion in space as time passes, here photons di¬use in time as 23
Chromatic dispersion in ¬bers is mainly due to the mate-
they propagate along the ¬ber. rial, essentially silicon, but also to the refractive index pro¬le.


16
CD does not constitute a problem in case of faint laser Transmission over free space features some advan-
pulses where the bandwidth is small. However, it be- tages compared to the use of optical ¬bers. The atmo-
comes a serious issue when utilizing photon pairs cre- sphere has a high transmission window at a wavelength
ated by parametric downconversion. For instance, send- of around 770 nm (see Fig. 8) where photons can eas-
ing photons of 70 nm bandwidth (as used in our long- ily be detected using commercial, high e¬ciency photon
distance Bell inequality tests, Tittel et al. 1998) down counting modules (see chapter III C 1). Furthermore, the
10 km of optical ¬bers leads to a temporal spread of atmosphere is only weakly dispersive and essentially non-
birefringent25 at these wavelengths. It will thus not alter
around 500 ps (assuming photons centered at »0 and a
ps
typical dispersion slope of 0.086 nm2 km ). However, this the polarization state of a photon.
can be compensated for when using energy-time entan- However, there are some drawbacks concerning free-
gled photons (Franson 1992, Steinberg et al. 1992a and space links as well. In contrast to transmitting a signal
1992b, Larchuk et al. 1995). In contrast to polariza- in a guiding medium where the energy is “protected” and
tion coding where frequency and the physical property remains localized in a small region in space, the energy
used to implement the qubit are not conjugate variables, transmitted via a free-space link spreads out, leading to
frequency and time (thus position) constitute a Fourier higher and varying transmission losses. In addition to
pair. The strict energy anti-correlation of signal and idler loss of energy, ambient daylight, or even light from the
photon enables one to achieve a dispersion for one pho- moon at night, might couple into the receiver, leading
ton which is equal in magnitude but opposite in sign to to a higher error rate. However, the latter errors can be
that of the sister photon, corresponding thus to the same maintained at a reasonable level by using a combination
delay24 (see Fig. 7). The e¬ect of broadening of the two of spectral ¬ltering (¤ 1 nm interference ¬lters), spatial
wave packets then cancels out and two simultaneously ¬ltering at the receiver and timing discrimination using
emitted photons stay coincident. However, note that the a coincidence window of typically a few ns. Finally, it
arrival time of the pair varies with respect to its emission is clear that the performance of free-space systems de-
time. The frequency anticorrelation provides also the pends dramatically on atmospheric conditions and is
basis for avoiding decrease of visibility due to di¬erent possible only with clear weather.
wavepacket broadening in the two arms of an interferom- Finally, let us brie¬‚y comment on the di¬erent sources
eter. And since the CD properties of optical ¬bers do leading to coupling losses. A ¬rst concern is the trans-
not change with time “ in contrast to birefringence “ no mission of the signals through a turbulent medium, lead-
on-line tracking and compensation is required. It thus ing to arrival-time jitter and beam wander (hence prob-
turns out that phase and phase-time coding is particu- lems with beam pointing). However, as the time-scales for
larly suited to transmission over long distances in optical atmospheric turbulences involved are rather small “
¬bers: nonlinear e¬ects decohering the qubit “energy” around 0.1 to 0.01 s “, the time jitter due to a varia-
are completely negligible, and CD e¬ects acting on the tion of the e¬ective refractive index can be compensated
localization can be avoided or compensated for in many for by sending a reference pulse at a di¬erent wavelength
cases. at short time (around 100 ns) before each signal pulse.
Since this reference pulse experiences the same atmo-
spheric conditions as the subsequent one, the signal will
4. Free-space links arrive essentially without jitter in the time-window de-
¬ned by the arrival of the reference pulse. In addition,
the reference pulse can be re¬‚ected back to the transmit-
Although telecommunication based on optical ¬bers is
ter and used to correct the direction of the laser beam by
very advanced nowadays, such channels may not always
means of adaptive optics, hence to compensate for beam
be available. Hence, there is also some e¬ort in devel-
wander and to ensure good beam pointing
oping free space line-of-sight communication systems -
Another issue is the beam divergence, hence increase of
not only for classical data transmission but for quantum
spot size at the receiver end caused by di¬raction at the
cryptography as well (see Hughes et al. 2000a and Gor-
transmitter aperture. Using for example 20 cm diameter
man et al. 2000).
optics, the di¬raction limited spot size after 300 km is
of ≈ 1 m. This e¬ect can in principle be kept small
taking advantage of larger optics. However, it can also
be of advantage to have a spot size large compared to the
Indeed, longer wavelengths feel regions further away from the
receiver™s aperture in order to ensure constant coupling
core where the refractive index is lower. Dispersion-shifted
in case of remaining beam wander. In their 2000 paper,
¬bers have, however, been abandoned by today™s industry, be-
cause it turned out to be simpler to compensate for the global
chromatic dispersion by adding an extra ¬ber with high neg-
ative dispersion. The additional loss is then compensated by
25
an erbium doped ¬ber ampli¬er. In contrast to an optical ¬ber, air is not subject to stress,
24
Assuming a predominantly linear dependence of CD in hence isotropic.
function of the optical frequency, a realistic assumption.


17
Gilbert and Hamrick provide a comprehensive discussion • In active quenching circuits, the bias voltage is
of free-space channels in the context of QC. actively lowered below the breakdown voltage as
soon as the leading edge of the avalanche current
is detected (see e.g. Brown et al. 1987). This
mode enables higher count rates compared to pas-
C. Single-photon detection
sive quenching (up to tens of MHz), since the dead-
time can be as short as some tens of ns. How-
With the availability of pseudo single-photon and
ever, the fast electronic feedback system renders
photon-pair sources, the success of quantum cryptogra-
active quenching circuits much more complicated
phy is essentially dependent on the possibility to detect
than passive ones.
single photons. In principle, this can be achieved using
a variety of techniques, for instance photo-multipliers,
• Finally, in gated mode operation, the bias volt-
avalanche-photodiodes, multichannel plates, supercon-
age is kept below the breakdown voltage and is
ducting Josephson junctions. The ideal detector should
raised above only for a short time when a photon
ful¬ll the following requirements:
is expected to arrive, typically a few ns. Maxi-
mum count-rates similar to active quenching cir-
cuits can be obtained using less complicated elec-
• it should feature a high quantum detection e¬- tronics. Gated mode operation is commonly used in
ciency over a large spectral range, quantum cryptography based on faint laser pulses
where the arrival-times of the photons are well
• the probability of generating noise, that is a signal
known. However, it only applies if prior timing
without a photon arriving, should be small,
information is available. For 2-photon schemes, it
• to ensure a good timing resolution, the time be- is most often combined with one passive quenched
tween detection of a photon and generation of an detector, generating the trigger signal for the gated
electrical signal should be as constant as possible, detector.
i.e. the time jitter should be small,
Apart from Geiger mode, Brown et al. also investi-
• the recovery time (i.e. the deadtime) should be gated the performance of Silicon APDs operated in sub-
small to allow high data rates. Geiger mode (Brown et al. 1989). In this mode, the bias
voltage is kept slightly smaller than the breakdown volt-
In addition, it is important to keep the detectors age such that the multiplication factor “ around 100 “
handy. For instance, a detector which needs liquid he- already enables to detect an avalanche, however, is still
lium or even nitrogen cooling would certainly render a small enough to prevent real breakdowns. Unfortunately,
commercial development di¬cult. the single-photon counting performance in this mode is
Unfortunately, it turns out that it is impossible to meet rather bad and initial e¬orts have not been continued,
all mentioned points at the same time. Today, the best the major problem being the need for extremely low-noise
choice is avalanche photodiodes (APD). Three di¬erent ampli¬ers.
semiconductor materials are used: either Silicon, Ger-
manium or Indium Gallium Arsenide, depending on the
An avalanche engendered by carriers created in the
wavelengths.
conduction band of the diode can not only be caused
APDs are usually operated in so-called Geiger mode.
by an impinging photon, but also by unwanted causes.
In this mode, the applied voltage exceeds the breakdown
These might be thermal or band-to-band tunneling pro-
voltage, leading an absorbed photon to trigger an elec-
cesses, or emissions from trapping levels populated while
tron avalanche consisting of thousands of carriers. To re-
a current transits through the diode. The ¬rst two causes
set the diode, this macroscopic current must be quenched
produce avalanches not due to photons and are referred
“ the emission of charges stopped and the diode recharged
to as darkcounts. The third process depends on previous
(Cova et al. 1996). Three main possibilities exist:
avalanches and its e¬ect is called afterpulses. Since the
number of trapped charges decreases exponentially with
• In passive-quenching circuits, a large (50-500 k„¦)
time, these afterpulses can be limited by applying large
resistor is connected in series with the APD (see
deadtimes. Thus, there is a trade-o¬ between high count
e.g. Brown et al. 1986). This causes a decrease of
rates and low afterpulses. The time-constant of the ex-
the voltage across the APD as soon as an avalanche
ponential decrease of afterpulses shortens for higher tem-
starts. When it drops below breakdown voltage,
peratures of the diode. Unfortunately, operating APDs
the avalanche stops and the diode recharges. The
at higher temperature leads to a higher fraction of ther-
recovery time of the diode is given by its capaci-
mal noise, that is higher dark counts. There is thus again
tance and by the value of the quench resistor. The
a tradeo¬ to be optimized. Finally, increasing the bias
maximum count rate varies from some hundred kHz
voltage leads to a larger quantum e¬ciency and a smaller
to a few MHz.
time jitter, at the cost of an increase in the noise.


18
We thus see that the optimal operating parameters, from Germanium or InGaAs/InP semiconductor materi-
voltage, temperature and dead time (i.e. maximum count als. In the third window (1.55 µm), the only option is
rate) depend on the very application. Besides, since the InGaAs/InP APDs.
relative magnitude of e¬ciency, thermal noise and af- Photon counting with Germanium APDs, although
ter pulses varies with the type of semiconductor material known for 30 years (Haecker, Groezinger and Pilkuhn
used, no general solution exists. In the two next para- 1971), started to be used in the domain of quantum com-
graphs we brie¬‚y present the di¬erent types of APDs. munication with the need of transmitting single photons
The ¬rst paragraph focuses on Silicon APDs which en- over long distances using optical ¬bers, hence with the
able the detection of photons at wavelengths below 1µm, necessity to work at telecommunications wavelength. In
the second one comments on Germanium and on Indium 1993, Townsend, Rarity and Tapster (Townsend et al.
Gallium Arsenide APDs for photon counting at telecom- 1993a) implemented a single photon interference scheme
munication wavelength. The di¬erent behaviour of the for quantum cryptography over a distance of 10 km, and
three types is shown in Fig. 9. Although the best ¬g- in 1994, Tapster, Rarity and Owens (1994) demonstrated
ure of merit for quantum cryptography is the ratio of a violation of Bell inequalities over 4 km. These experi-
dark count rate R per time unit to detection e¬ciency ·, ments where the ¬rst ones to take advantage of Ge APDs
we depict here the better-known noise equivalent power operated in passively quenched Geiger mode. At a tem-
NEP which shows similar behaviour. The NEP is de- perature of 77K which can be achieved using either liquid
¬ned as the optical power required to measure a unity nitrogen or Stirling engine cooling, typical quantum ef-
signal-to-noise ratio, and is given by ¬ciencies of about 15 % at dark count rates of 25 kHz
can be found (Owens et al. 1994), and time jitter down
hν √
N EP = 2R. (25) to 100 ps have been observed (Lacaita et al. 1994) “ a
·
normal value being 200-300 ps.
Here, h is Planck™s constant and ν is the frequency of the Traditionally, Germanium APDs have been imple-
impinging photons. mented in the domain of long-distance quantum com-
munication. However, this type of diode is currently get-
ting replaced by InGaAs APDs and it is more and more
1. Photon counting at wavelengths below 1.1 µm di¬cult to ¬nd Germanium APDs on the market. Mo-
tivated by pioneering research reported already in 1985
Since the beginning of the 80™s, a lot of work has (Levine, Bethea and Campbell 1985), latest research fo-
been done to characterize Silicon APDs for single pho- cusses on InGaAs APDs, allowing single photon detection
ton counting (Ingerson 1983, Brown 1986, Brown 1987, in both telecommunication windows. Starting with work
Brown 1989, Spinelli 1996), and the performance of Si- by Zappa et al. (1994), InGaAs APDs as single photon
APDs has continuously been improved. Since the ¬rst counters have meanwhile been characterized thoroughly
test of Bell inequality using Si-APDs by Shih and Al- (Lacaita et al. 1996, Ribordy et al. 1998, Hiskett et al.
ley in 1988, they have completely replaced the photo- 2000, Karlsson et al. 1999, and Rarity et al. 2000, Stucki
multipliers used until then in the domain of fundamental et al. 2001), and ¬rst implementations for quantum cryp-
quantum optics, known now as quantum communication. tography have been reported (Ribordy 1998, Bourennane
Today, quantum e¬ciencies of up to 76% (Kwiat et al. et al. 1999, Bethune and Risk 2000, Hughes et al. 2000b,
1993) and time jitter down to 28 ps (Cova et al. 1989) Ribordy et al. 2000). However, if operating Ge APDs
have been reported. Commercial single photon counting is already inconvenient compared to Silicon APDs, the
modules are available (EG&G SPCM-AQ-151), featuring handiness of InGaAs APDs is even worse, the problem
quantum e¬ciencies of 70 % at a wavelength of 700 nm, a being a extremely high afterpulse fraction. Therefore,
time jitter of around 300 psec and maximum count rates operation in passive quenching mode is impossible for
larger than 5 MHz. Temperatures of -20oC “ su¬cient to applications where noise is crucial. In gated mode, In-
keep thermally generated dark counts as low as 50 Hz “ GaAs APDs feature a better performance for single pho-
can easily be achieved using Peltier cooling. Single pho- ton counting at 1.3 µm compared to Ge APDs. For in-
ton counters based on Silicon APDs thus o¬er an almost stance, at a temperature of 77 K and a dark count prob-
ability of 10’5 per 2.6 ns gate, quantum e¬ciencies of
perfect solution for all applications where photons of a
wavelength below 1 µm can be used. Apart from funda- around 30% and of 17% have been reported for InGaAs
mental quantum optics, this includes quantum cryptog- and Ge APDs, respectively (Ribordy et al. 1998), while
raphy in free space and in optical ¬bers, however, due to the time jitter of both devices is comparable. If working
high losses, the latter one only over short distances. at a wavelength of 1.55 µm, the temperature has to be
increased for single photon detection. At 173 K and a
dark count rate of now 10’4 , a quantum e¬ciency of 6%
2. Photon counting at telecommunication wavelengths can still be observed using InGaAs/InP devices while the
same ¬gure for Germanium APDs is close to zero.
When working in the second telecommunication win- To date, no industrial e¬ort has been done to opti-
dow (1.3µm), one has to take advantage of APDs made mize APDs operating at telecommunication wavelength

19
for photon counting, and their performance is still far In the BB84 protocol Alice has to choose randomly
behind the one of Silicon APDs26 . However, there is between four di¬erent states and Bob between two bases.
no fundamental reasons why photon counting at wave- The limited random number generation rate may force
lengths above 1 µm should be more delicate than below, Alice to produce her numbers in advance and store them,
except that the photons are less energetic. The real rea- opening a security weakness. On Bob™s side the random
sons for the lack of commercial products are, ¬rst, that bit creation rate can be lower since, in principle, the basis
Silicon, the most common semiconductor, is not sensitive must be changed only after a photon has been detected,
(the band gap is too large), and secondly that the mar- which normally happens at rates below 1 MHz. However,
ket for photon counting is not yet mature. But, without one has to make sure that this doesn™t give the spy an
great risk, one can forecast that good commercial pho- opportunity for a Trojan horse attack (see section VI K)!
ton counters will become available in the near future, and An elegant con¬guration integrating the random num-
that this will have a major impact on quantum cryptog- ber generator into the QC system consists in using a pas-
raphy. sive choice of bases, as discussed in chapter V (Muller et
al. 1993). However, the problem of detector induced
correlation remains.
D. Quantum random number generators

E. Quantum repeaters
The key used in the one-time-pad must be secret and
used only once. Consequently, it must be as long as the
message and must be perfectly random. The later point Todays ¬ber based QC systems are limited to tens of
proves to be a delicate and interesting one. Computers kilometers. This is due to the combination of ¬ber losses
are deterministic systems that cannot create truly ran- and detectors™ noise. The losses by themselves do only
dom numbers. But all secure cryptosystems, both classi- reduce the bit rate (exponentially with the distance), but
cal and quantum ones, require truly random numbers27 ! with perfect detectors the distance would not be limited.
Hence, the random numbers must be created by a ran- However, because of the dark counts, each time a pho-
dom physical process. Moreover, to make sure that the ton is lost there is a chance that a dark count produces
random process is not merely looking random with some an error. Hence, when the probability of a dark count
hidden deterministic pattern, it is necessary that it is becomes comparable to the probability that a photon
completely understood. It is thus of interest to imple- is correctly detected, the signal to noise ratio tends to
ment a simple process in order to gain con¬dence in its 0 (more precisely the mutual information I(±, β) tends
to a lower bound29 ). In this section we brie¬‚y explain
proper operation.
A natural solution is to rely on the random choice of how the use of entangled photons and of entanglement

a single photon at a beamsplitter28 (Rarity et al. 1994). swapping (Zukowski et al. 1993) could open ways to
In this case the randomness is in principle guaranteed by extend the achievable distances in a foreseeable future
the laws of quantum mechanics, though, one still has to (some prior knowledge of entanglement swapping is as-
be very careful not to introduce any experimental arte- sumed). Let us denote tlink the transmission coe¬cient
fact that could correlate adjacent bits. Di¬erent experi- (i.e. tlink =probability that a photon sent by Alice gets
mental realizations have been demonstrated (Hildebrand to one of Bob™s detectors), · the detectors™ e¬ciency and
2001, Stefanov et al. 2000, Jennewein et al. 2000a) pdark the dark count probability per time bin. With a
and prototypes are commercially available (www.gap- perfect single photon source, the probability Praw of a
optique.unige.ch). One particular problem is the dead- correct qubit detection reads: Praw = tlink ·, while the
time of the detectors, that may introduce a strong an- probability Pdet of an error is: Pdet = (1 ’ tlink ·)pdark .
P
ticorrelation between neighboring bits. Similarly, after- Accordingly, the QBER= Prawdet det and the normalized
+P
pulses may provoke a correlation. These detector-related net rate reads: ρnet = (Praw + Pdet ) · f ct(QBER) where
e¬ects increase with higher pulse rates, limiting the bit the function f ct denotes the fraction of bits remaining
rate of quantum number generator to some MHz. after error correction and privacy ampli¬cation. For the
sake of illustration we simply assume a linear dependence
dropping to zero for QBER≥ 15% (This simpli¬cation
does not a¬ect the qualitative results of this section.
26
For a more precise calculation, see L¨ tkenhaus 2000.):
u
The ¬rst commercial photon counter at telecommunication
wavelengths came out only this year (Hamamatsu photomul-
tiplier R5509-72). However, the e¬ciency does not yet allow
an implementation for quantum cryptography.
27
The pin number that the bank attributes to your credit 29
The absolute lower bound is 0, but dependening on the
card must be random. If not, someone knows it! assumed eavesdropping strategy, Eve could take advantage of
28
Strictly speaking, the choice is made only once the photons the losses. In the latter case, the lower bound is given by her
are detected at one of the outports. mutual information I(±, «).



20
f ct(QBER) = 1 ’ QBER . The corresponding net rate IV. EXPERIMENTAL QUANTUM
15%

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