Download Quantum Cryptography, Quantum Teleportation

Quantum Information
Quantum Cryptography
Quantum Teleportation
Quantum Cryptography
What hath God wrought. - first message
sent by telegraph, by Samuel F. B. Morse
Classical Cryptography
Cryptography – The art of encoding a message in such a way
that only the person to whom it is addressed
can read it.


The ENIGMA code
The key – one-time-pad

Θ = Subtraction mod 2:
Θ

110111111000001
111011000111001
001100111111000
Solutions to key transfer problem:


Multiple use of the same key
Public key encryption. EX: RSA
Basic Principles Of Quantum Cryptography
The purpose: To provide a reliable method for
transmitting a secret key and knowing that
no one has intercepted it along the way
The process of sharing a secret key is called –
Quantum Key Distribution. The methods:




quantum measurement on single particles
entagled states
Quantum cryptography does not protect against eavesdropper
attacks, but provides a failsafe way of knowing if there has been
intercepting
Basic Principles Of Quantum Cryptography
BS+Amplifier
Strong pulse=1 Weak pulse=0

Classically – It is possible to make
an exact duplicate of the signal.
Quantum mechanics – No-Cloning Theorem. Proof:


Suppose a state in system A and B: 
We can:




A
e
B

A
e
B
Perform an observation
Control the hamiltonian:
We choose:
U
A
e
B

   e B  A U †U 
A
A

e
U
B
B
 
B

A
A
e

B
A


A
B

B
 
2
Basic Principles Of Quantum Cryptography
Example:
Measure the polarization state of a single photon



PBS - Polarizing Beam Splitter
On a general case:   cos
 sin  
2

P

cos
 



2

 P  sin  

Eve can reproduce the same state only for the
special case of   0,90 !!
Quantum Key Distribution According To The
BB84 Protocol
A protocol for sharing the private key in a secure
way.
Basis Binary 1 Binary 0
2 possible bases:


The procedure:

1.
2.
3.
4.
5.

 0 
  90  

  45 
  135 
The sender determines the basis.
The receiver determines his basis.
The bases are compared in an unsecured communication.
The receiver resend a subset of the correct bits.
If the error is less then 25% - the transmission was safe
and the rest of the correct bits are the private key.
Quantum Key Distribution According To
The BB84 Protocol
Perror  PEve  has  wrong basis  PBob  gets  wrong  result  25%
System Errors

Errors might occur even without
eavesdropper:



Photon deletion
Birefringence
Detector dark counts
System Errors
1. Photon Deletion

Can occur for a number of reasons:





Absorption
Scattering
Detector inefficiency
Does not affect information security
When comparing selected bases (stage #3)
times comparison must be added
System Errors
2. Birefringence


Such a medium would change the
polarization angle
Shannon’s noisy channel coding theorem:
Ncorrection  N   log2   (1   )log2 (1   )
N =number of correction bits
 = error probablility


N grows with 
It can be reduced on the account of the
transmission rate
System Errors
3. Detector dark counts


The original photon never reaches the detector
and the detector randomly registers due to
thermal noise in the photocathode
Error correction can be established as in
Birefringence
These errors cause a reduction in the length of
the private key. It reduces the efficiency of the
system, but do not affect it’s security
Error correcting

The procedure:
1.
2.
3.
4.
5.
6.
Permuting a random number of bits
Divide the string into blocks of size b
Compare their parity discarding the last bit
Repeat the procedure for the blocks containing errors with
smaller size blocks
Correct the error giving only the chosen basis
Repeat the total process to eliminate the even error number
blocks
After 20 parity checks, the probability of an
undiscovered eavesdropper is 1 in a million!
Identity verification


It is impossible to know whether an
eavesdropper impersonates the receiver.
Only by using the first authenticate private
key, it is possible to transmit a newer one.
Single Photon Source





More then one photon at time would risk giving up
information to the eavesdropper
The laser must have attenuated beam at a certain
frequency
The average number of photons per pulse, n, has
Poisson distribution
A rate of 5% of n>1 to n=1 is alarmingly high!
Genuine single-photon source is the solution but still on
development
Single Photon Source
HBT - Hanbury–Brown and Twiss
Practical Demonstrations Of Quantum
Cryptography

Two broad categories:


Free-space quantum cryptography
Quantum cryptography in optical fibres
Practical Demonstrations Of Quantum
Cryptography
Free-space quantum cryptography





Bennett & Brassard ’92:
 Light Emitting Diode in 550nm
 Air gap – 0.32m (today – 23km).
The goal: communicating with satellites in low earth orbitals.
At long range 600-900nm are used in which the atmospheric
losses are small, and low-noise detectors with high quantum
efficiency are available.
These conditions provide two main sources of errors:
 Air turbulence
 Stray light
Most of these problems are occurred at low heights.
Practical Demonstrations Of Quantum
Cryptography
Quantum cryptography in optical fibers


Advantage: The beam does not diverges
Disadvantages:
 signal decay:
 850nm introduces greater decay but is used with silicon
low-noise SPAD
 1300/1550nm energy is below silicon’s band gap and
have higher dark count rate, and suffer from afterpulsing.
 Birefringence of the fiber:
 In long distances thermal and mechanical induction
changes the fiber birefringence.
 A solution – optical phase encoding – Mach-Zender.
Practical Demonstrations Of Quantum
Cryptography

Mach-Zender

The achievements:


850nm – 100kbps over 4.2 km
1550nm – 122km.
Gentlemen do not read each other's mail Henry Stimson, U.S. Secretary of State
Quantum Teleportation
"God does not play dice with the universe" -- Albert Einstein
Entangled States
Entangled state – the wave function can not be factorized into a
product of the wave functions of the of the individual particles.

EPR –
 Correlated photon pairs:
 The polarization of either photon measured independently
of the other is random
 The polarization of the pair of photons is perfectly
correlated:
1

 
 01 , 02  11 ,12     1  01,12  11, 02 
2
2
 The measurement of one photon can determine in 100% the
state of the other photon
Entangled States

Shroedinger’s Cat Paradox
1
 
 live,1  dead , 2
2

Generation Of Entangled Photon
Pairs
1967 - Kocher and Commins


The initial and final state are both J  0 states → the photons emitted
are with J  0
Both have even parity


The photon pair have the polarization correlation properties required
for EPRB


1

 01 , 02  11 ,12
2

Generation Of Entangled Photon
Pairs
Down-conversion





Using non-linear optics
Phase matching condition: 1. 0  1  2
2. k0  k1  k2
Entangled state occur only on intersection

If 1  2  20 the process is called- degenerate
The dispersion relation Vs. the birefringence produces the
one of the matching:
 Type 1 – Same polarization
 Type 2 – Different polarization
Generation Of Entangled Photon
Pairs



Type 2 example:
UV from a pump laser is focused
into a  -barium borate crystal
The photon is converted into 2
red degenerate photons:
1
 
1 ,

2


2
 ei
1
, 2

By gauge of  one can achieve either state.
This method is widely used today
Single-Photon Interference
Experiment
1987 Hong-Ou-Mandel Interferometer

Coherence length - The distance over which interference will occur. Coherence length of an
optical source is affected by the size of the source, spatial coherence, the phase purity of the
source, temporal coherence, and the spectral bandwidth of the light.
Single-Photon Interference
Experiment
1991- Zou-Wang-Mandel

The mere possibility of which-path information is
sufficient to destroy the single photon interference
Bell’s Ineqaulity

The conditions:
1
2
P11 (1 ,2 )  P10 (1 ,2 )  0.5
P11 ( , )  0.5
P01 (1 ,2 )  P00 (1 ,2 )  0.5
P10 ( , )  0
P11 (1 ,2 )  P01 (1 ,2 )  0.5
P01 ( , )  0



P10 (1 ,2 )  P00 (1 ,2 )  0.5
P00 ( , )  0.5

1
cos 2 
2
1
P10 (1 , 0)  sin 2 
2
1
P01 (1 , 0)  sin 2 
2
1
P00 (1 , 0)  cos 2 
2
P11 (1 , 0) 
Bell’s Ineqaulity
Quantum Machanics
LHV
1
1
2
2
2
2
P11 (1 , 2 )  cos 2 (1   2 )
P
(

,

)

(sin

sin


cos

cos
2 )
11 1
2
1
2
1
2
2
1 2
1
2
2
2
2
P10 (1 ,  2 )  sin (1   2 )
P
(

,

)

(sin

cos


cos

sin
2 )
10
1
2
1
2
1
2
2
1
1
P01 (1 , 2 )  sin 2 (1   2 )
P01 (1 , 2 )  (cos 2 1 sin 2  2  sin 2 1 cos 2  2 )
2
2
1
1
2
P
(

,

)

(cos 2 1 cos 2  2  sin 2 1 sin 2  2 )
P00 (1 , 2 )  cos (1   2 )
00
1
2
2
2
E (1 ,2 )  P11 (1 ,2 )  P00 (1 ,2 )  P10 (1 ,2 )  P01 (1 , 2 )
S  E (1, 2 )  E (1, 2 )  E (1, 2 )  E (1, 2 )
Bell’s Inequality:
Inequality violation:
1  0
 2  22.5
2  S  2
1  45
 2  67.5

S 2 2
Experimental Confirmation Of Bell's
Theory

1981-82 Alain Aspect

3 experiments:




Different angels
Limits of S
Non-local correlation
The violation of Bell’s inequality has been
confirmed
Principles Of Teleportation

The basic idea: To transfer the quantum state of one
photon to another that is physically separated from it. To
exchange quantum information without direct
transformation of Qubits.

General principles:
1.
2.
3.
4.
The input photon must either be destroy or lose its initial state in an
irretrievable way (according to the no-cloning theorem.
The more info gleaned about  - the less fidelity gained.
No matter is teleported between the input and output, only quantum
information.
It is impossible transmit information faster then the speed of light.
Principles Of Teleportation
1993 – Bennett –
Teleporting of photon polarization
 1  C0 0 1  C1 1 1
C0  C1  1
2
2

23

1
0

2
2
1312 0
3

Principles Of Teleportation
The process:

The full wave function of the 3 particles:

1
C0 0 1  C1 1 1   0 2 1 3  1 2 0 3 
123
2
1

C0 0 1 0 2 1 3  C0 0 1 1 2 0 3  C1 1 1 0
2



With Bell’s notation:


123
2
1 3  C1 1 1 1 2 0
    C0 1  C1 0  
3
3
12





1    12  C0 1 3  C1 0 3  

2      C0 0  C1 1  
3
3 
12

  
C0 0 3  C1 1 3  


12

3

2 remarks:
 The protocol can only work after transmitting the result classically.
 Photons 1 and 2 are entangled and there is no information about
the coefficients, hence there was no duplication.
Experimental Demonstration of
Teleportation
1997-8 Bouwmeester
Bibliography







Mark Fox, “Quantum Optics”, Oxford University Press, 2006
http://www.ai.sri.com/~goldwate/quantum.html#PrivAmp
http://www.csa.com/discoveryguides/crypt/overview.php
Bennett, C. H., Bessette, F., Brassard, G., Salvail, L., and Smolin,
J., "Experimental Quantum Cryptography", Journal of Cryptology,
vol. 5, no.1, 1992, pp. 3-28.
A. Muller, T. Herzog, B. Huttner,a) W. Tittel, H. Zbinden, and N.
Gisin ‘‘Plug and play’’ systems for quantum cryptography, 1996
www.wikipedia.org
A. Poppe, A. Fedrizzi, H. Hübel, R. Ursin, A. Zeilinger “Entangled
State Quantum Key Distribution and Teleportation”