Quantum Cryptography
December, 3rd 2007
Philippe LABOUCHEREAnnika BEHRENS
1. Introduction
2. Photon sources
3. Quantum Secret Sharing
1. Introduction
2. Photon sources
3. Quantum Secret Sharing
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
Venn diagrams
)|()|():,(),:( ABHCBHCBAICBAI
The BB84 protocol
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
Mutual information vs quantum bit error rate
bitsreceived
bitswrong
N
NQBER
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
1. Introduction
2. Photon sources
3. Quantum Secret Sharing
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
1. Introduction
2. Photon sources
3. Quantum Secret Sharing
Quantum Secret Sharing (1)
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
Goodbye GHZ, welcome single qubit
}2/3,2/{
},0{
Y
X
j
j
11)(
00)(
jijj
jj
eU
U
102
1 jji
N e
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
Questions ?
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