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 ?