quantum cryptography december, 3 rd 2007 philippe labouchere annika behrens
Post on 22-Dec-2015
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How to measure information (1)
• Claude E. Shannon 1948
• Information entropy
• Mutual information
ii
i
i x
n
i
xx
n
i
x ppp
pXH 2
1
2
1
log1
log)(
Xx Yy ypxp
yxpyxpYXI
)()(
),(log),(, 2
[bits
]
How to measure information (2)
• Relation between H and I
• Mutual information between 2 parties
•
•
)|()(),( YXHXHYXI
posterioriaprioriaKL HHI
)(log2 NH apriori
XxYy
iaposterior yxpyxpypH )|(log)|()( 2
The BB84 protocol: principle
• 2 conjugate basis
• Information encoded in photon’s polarization→ ’0’ ≡ — & /→ ’1’ ≡ | & \
• Quantum & classical channels used for key exchange
Charles H. Bennett
Gilles Brassard
From random bits to a sifted key
Alice’s random bits 0 1 1 O O 1
Random sending bases D D R R D RPhoton Alice
sends / \ — — / —Random
receiving bases R D R D D RBits as received
by Bob 1 1 1 0 0 1Bob reports
basis of received bits
R D R D D RAlice says which
were correct no OK OK no OK OKPresumably
shared information
. 1 1 . 0 1Bob reveals
some key bits at random
. . 1 . 0 .Alice confirms
them . . OK . OK .Remaining shared bits . 1 . . . 1
Quantu
m
transm
issi
on
Public
dis
cuss
ion
The no-cloning theorem
• Dieks, Wootters, Žurek 1982
”It is forbidden to create identical copies of an arbitrary
unknown quantum state.”
• Quantum operations : unitary & linear transformations on the state of a quantum system
Sources of photons
• Thermal light
• Coherent light
• Squeezed light
11)(
m
m
th n
nmp
n
m
em
nmp
!)( 2nn
1nAverage photon number of photons in a mode
Number of photons
nm
Faint-laser pulses
• <n> = μ ~ 0.1 photon / pulse
• Photon-number splitting attack!
• Dark counts of detectors vs high pulse rate & weaker pulses
darkAB pT detdetdetdet
2
2
darkdark
AB
ppT
2)0(1
)1()0(1
1
2 n
p
pp
p
pp
n
nmulti
nnp 1)0(!
Detection yield
Transmission efficiency
detABT
Tradeoff
Entangled photon pairs
• SpontaneousParametric Down Conversion
• Idler photon acts as trigger for signal photon
• Very inefficient
Single-photon sources
• Intercept/resend attack=> error rate < dark count rate !
• Condition for security:
• Drawback : dark counts & afterpulses
detdark
AB
pT
Transmission efficiency
Detection yielddet
ABT
Practical limits of QC
• Realization of signal
• Stability under the influence of the environment (transportation)
- Birefringence- Polarization dispersion- Scattering
• Need of efficient sources & detectors (measurements)
Bite rate as function of distance after error correction
and privacy amplificationPulse rate = 10 MHz
μ = 0.1 (faint laser pulses)
Losses: @ 800nm : 2dB / km @ 1300 nm: 0.35dB / km @ 1550 nm: 0.25 dB /km
QSS (2)
• N-qubit GHZ source
• Define
z
N
z
NNGHZ 10
2
1
z
N
z
N
xN
N10
2
11
0
z
N
z
N
yN
Ni 10
2
11
0
z
N
z
Nz
z
N
z
Nz
11
00
Sequentially polarized single photon protocol
Original BB84 Modified BB84
Diagonal and Rectilinear bases Classes X and Y
/ and — ≡ ‘0’| and \ ≡ ‘1’
φj = {0, π/2} ≡ ’0’
φj = {π, 3π/2} ≡ ’1’
Correlated results if same bases used
Correlated results if 1cos1
N
jj