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Quantum Communication
GAP Optique Geneva University

Nicolas Gisin
Quantum cryptography:
•
•
•
•
•
•
•
Q crypto:
BB84 and uncertainty relations
RMP 74, 145-195, 2002
Ekert and entanglement
no cloning theorem
Q cloning:
BB84  Ekert
RMP 77, 1225-1256, 2005
Implementations
Eve: optimal individual attack
Error correction, privacy amplification, advantage distillation

Quantum Teleportation

Optimal and generalized quantum measurements
• principle, connection to optimal state estimation and cloning
• experiments
• quantum relays and quantum repeaters
• optimal quantum cloning
• POVMs (tetrahedron, unambiguous state discrimination)
• weak measurements
1
Quantum cryptography: a beautiful idea
GAP Optique Geneva University
• Basic Quantum Mechanics:
• A quantum measurement perturbs the system
QM  limitations
• However, QM gave us the laser, microelectronics, superconductivity, etc.
• New Idea:
Let's exploit QM for secure communications
2
 If Eve tries to eavesdrop a "quantum
communication channel", she has to perform
some measurements on individual quanta (single
photon pulses).
GAP Optique Geneva University
 But, quantum mechanics tells us: every
measurement perturbs the quantum system.
 Hence the "reading" of the "quantum signal" by a
third party reduces the correlation between
Alice's and Bob's data.
 Alice and Bob can thus detect any undesired third
party by comparing (on a public channel) part of
their "quantum signal".
3
 The "quantum communication channel" is
not used to transmit a message
(information), only a "key" is transmitted
(no information).
GAP Optique Geneva University
 If it turns out that the key is corrupted,
they simply disregard this key (no
information is lost).
 If the key passes successfully the control,
Alice and Bob can use it safely.
 Confidentiality of the key is checked before
the message is send.
 The safety of Quantum Cryptography is
based on the root of Quantum Physics.
4
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Modern Cryptology
Secrecy is based on:
Complexity theory
The key is public
Information theory
The key is secrete
The public key contains the
decoding key, but it is very
difficult to find (one way
functions)
The key contains the decoding
key: Only the two partners have
a copy !
The security is not proven (no
one knows whether one way
functions exist)
Example:
127 x 229 = 29083
The security is proven (Shannon
theorem)
Example:
Message:
011001001
Key:
110100110
Coded message: 101101111
5
Eve
 25% errors
Bob
BB84 protocol:
H/V Basis
Alice
Polarizers
45 Basis
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Horizontal - Vertical
Diagonal (-45, +45)
Alice's Bit Sequence
Bob's Bases
Bob's Results
Key
0 1 0 - 0 1 1 1 1 -
1 0
-
1 0
1 - -
0 1
-
- 1 -
6
Security from Heisenberg uncertainty
relations
Alice
Bob
GAP Optique Geneva University
Eve
P(X, Y, Z)
Theorem 1: (I. Csiszàr and J. Körner 1978, U. Maurer 1993)
If I(A:B) > min{I(A:E),I(B:E)},
then Alice & Bob can distil a secret key using 1-way
communication over an error free authenticated public channel.
where I(A:B) = Shannon mutual information
= H(A)-H(A|B)
= # bits one can save when writing A
knowing B
7
GAP Optique Geneva University
Finite-coherent attacks
Theorem 2 (Hall, PRL 74,3307,1995)
Heisenberg uncertainty relation in Shannon-information terms:
I(A:B) + I(A:E) < 2.log(d.c)
where c=maximum overlap of eigenvectors and d is the
dimension of the Hilbert space.
For BB84 with n qubits, d=2n and c=2^(-n/2). Hence,
Theorem 2 reads: I(A:B) + I(A:E) < n
It follows from Csiszàr and Körner theorem that the security
is guaranteed whenever I(A:B) < 1/2 (per qubit)
This corresponds exactly to the bound of the Mayers et al.
proofs, i.e. QBER<11%
Note: same reasoning valid for 6-state protocols, and for higher
dimensions (M. Bourennane et al.).
8
Eve: optimal individual attack
1.0
I AB  1  H (QBER)
Shannon Inform ation
GAP Optique Geneva University
0.8
IAE1-IAB
0.6
0.4
0.2
0.0
0.0
0.1
0.2
QBER
9
Quantum Communication

Quantum Communication is the art of transferring a Q state from
one place to another. Example:
GAP Optique Geneva University
• Q cryptography
• Q teleportation

Quantum Information is the art of turning a Q paradox into a
potentially useful task. Example:
• Q communication: from no-cloning to Q crypto
• Q computing: from superpositions to Q parallelism

Note that entanglement and Q nonlocality are always present, at
least implicitely. Though their exact power is not yet fully
understood
10
Ekert protocol (E91)
a=x,z
b=x,z
source
GAP Optique Geneva University
a
b
  0,0  1,1
0,1
0,1
 x  x

  z  z

1
Theorem: let ABE  H A  H B  H E
If AB is pure, then ABE  AB  E
11
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Quantum cryptography on noisy channels
No cloning theorem:
0  0,0
1  1,1

0  1  0,0  1,1
 ( 0  1 )( 0  1 )
12
No cloning theorem and the
compatibility with relativity
GAP Optique Geneva University
No cloning theorem: It is impossible to copy an unknown
quantum state,  
/  
Proof #1:
0  0,0
1  1,1

0  1  0,0  1,1
 ( 0  1 )( 0  1 )
Proof #2: (by contradiction)
Alice
M
Source of
entangled
particules
*
Arbitrary fast signaling !
Bob
}
clones
13
Optimal Universal non-signaling Quantum
Cloning
A

m  UQCM  
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B
symmetric and universal   A   B 
  AB

1
1    m  
2


1
 14    m  12   12  m  i , j  x , y , z tij i   j
4



universal   AB (U m )  U  U   AB ( m ) U   U 
 t xx  t yy , t xy  t yx , t xz  t yz  t zx  t zy  0
no.signaling   AB (  x )   AB (  x )   AB (  z )   AB (  z )
 t xx  t zz
 AB  0    2 3 
achievable by the Hillery-Buzek UQCM
N.Gisin, Phys. Lett.A 242, 1-3, 1998 14
BB84  E91
a=x,z
b=x,z
source
GAP Optique Geneva University
a
0,1
source
b
0,1
Alice
Indistinguishable from a
single photon source. The
qubit is coded in the a-basis
And holds the bit value given
by Alice results.
15
16
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Experimental Realization
 Single photon source
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• laser pulses strongly attenuated ( 0.1 photon/pulse)
• photon pair source (parametric downconversion)
• true single-photon source
 Polarization or phase control during the
single photon propagation
• parallel transport of the polarization state (Berry topological phase)
 no vibrations
• fluctuations of the birefringence  thermal and mechanical stability
• depolarization  polarization mode dispersion smaller than the
source coherence
• Stability of the interferometers coding for the phase
 Single photon detection
• avalanche photodiode (Germanium or InGaAs) in Geiger mode
 dark counts
• based on supraconductors  requires cryostats
17
Telecommunication wavelengths
 Attenuation
(  transparency)
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l [mm]
a [dB/km]
T10km
0.8
2
1%
1.3
0.35
44%
1.55
0.2
63%
 Chromatic dispersion
 Components available
 Two windows
18
Single Photon Generation (1)
• Attenuated Laser Pulse
Poissonian Distribution
100%
Probability
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Attenuating Medium
80%
Mean = 1
Mean = 0.1
60%
40%
20%
0%
" 0 or 1 or 2 or..." rather than 1
0
1
2
3
4
5
Number of photons per pulse
• Simple, handy, uses reliable technology
 today’s best solution
19
Avalanche photodiodes
 Single-photon detection with avalanches in
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Geiger mode
 macroscopic avalanche triggered
by single-photon
E
x
c
E Gap  h
l
1 photon  absorption avalanche
Silicon:
1000 nm
Germanium: 1450 nm
InGaAs/InP: 1600 nm
20
Noise sources
 Charge tunneling across the junction
 Band to band thermal excitation
 reduce temperature
 Afterpulses  release of charges
trapped during a previous avalanche
 increase temperature
Optimization !!!
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 not significant
21
Efficiency and Dark Counts
1.0E+00
30%
Detection efficiency (1.55 mm)
T = - 40 C
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1.0E-01
20%
1.0E-02
1.0E-03
10%
1.0E-04
Dark count probability per gate (2.4 ns)
1.0E-05
35.5
0%
36
36.5
37
37.5
38
38.5
39
Bias voltage [V]
22
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experimental Q communication
for theorists
tomorrow: Bell inequalities and
nonlocal boxes
23
Polarization Encoding (1)
GAP Optique Geneva University
Alice
Bob
PBS
Laser
Det
Laser
Det
Laser
Det
Pol. Control
Det
Laser
PBS
24
GAP Optique Geneva University
Polarization effects in optical fibers:
Cause
Consequence
Limitations for QC
Geometrical phase
(Berry phase)
Polarization rotation
The fiber must be motionless
Birefringence
Polarization
transformation
(unitary)
The fiber should not undergo
fast thermal or stress
variations
PMD
(Polarization Mode
Dispersion)
Depolarization
(decoherence)
The fiber should have a PMD
delay smaller than the
coherence time of the source
PDL
Non-unitary
(Polarization
polarization
Dependent Loss)
transformation
Can be compensated only with
additional losses
 Polarization encoding is a bad choice !
25
Phase Coding
 Single-photon interference
Alice
A
1
D1
Bob
B
Basis 1: A = 0; p
Basis 2: A = p/2; 3 p/2
Bases
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0.5
D1
D2
0
0
D2
Basis: B = 0; p/2
Compatible:
Alice A  Di
(A-B = np)
Bob Di  A
2
4
6
Phase [radians]
Incompatible: Alice and Bob ??
(A-B = p/2)
26
Difficulties with Phase Coding
 Stability of a 20 km long interferometer?
Time Window
Bob
A
B
Coincidences
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Alice
long -long
short -short
Time (ns)
0

short - long +
long - short
-3
-2
-1
0
1
2
3
Problems:
• stabilization of the path difference  active feedback control
• stability of the interfering polarization states
27
The Plug-&-Play configuration
J.Mod.Opt. 47, 517, 2000
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 Simplicity, self-stabilization
28
Faraday mirrors
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l
• 4 Faraday rotator
• standard mirror ( incidence)
• l 4 Faraday rotator
  R( )m  ( R( )m )
m


1
 m  R ( )( R( )m ) 
FM
Independent of 
29
GAP Optique Geneva University
QC over 67 km, QBER  5%
RMP 74, 145-195, 2002,
Quant-ph/0101098
+ aerial cable (in Ste Croix, Jura) !
D. Stucki et al., New Journal of Physics
4, 41.1-41.8, 2002. Quant-ph/0203118 30
 Company established in 2001
• Spin-off from the University of Geneva
GAP Optique Geneva University
 Products
• Quantum Cryptography
(optical fiber system)
• Quantum Random Number Generator
• Single-photon detector module (1.3 mm and 1.55
mm)
 Contact information
email: [email protected]
web: http://www.idquantique.com
31
Quantum Random Number Generator
to be announced next week at CEBIT
GAP Optique Geneva University
 Physical randomness source
 Commercially available
 Applications
• Cryptography
• Numerical simulations
• Statistics
32
Photon pairs source
lp
ls,i
GAP Optique Geneva University
laser
nonlinear
birefringent
crystal



filtre
Parametric fluorescence
Energy and momentum conservation
p  s  i

 
k p  ks  ki
Phase matching determines the wavelengths and propagation
directions of the down-converted photons
33
2-photon Q cryptography:
Franson interferometer
2
1
Two unbalanced interferometers  no first order interferences
photon pairs  possibility to measure coincidences
2.0
One can not distinguish between
"long-long" and "short-short"
Coinc. window
1.5
coinc. (a.u.)
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SOURCE
short-short +
long-long
1.0
long-short
short-long
Hence, according to QM, one should
add the probability amplitudes
0.5
0.0
-3
-2
-1
0
1
 t betw een Start and Stop (ns )
2
3
 interferences (of second order)
34
GAP Optique Geneva University
2- source of Aspect’s 1982 experiment
35
Photon pairs source (Geneva 1997)
F L
P
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KNbO3
output 1


output 2
crystal
Energy-time entanglement
lens


l p  655 nm; ls,i  1310 nm
diode laser
simple, compact, handy
40 x 45 x 15 cm3
filter
laser


Ipump = 8 mW
with waveguide in LiNbO3
with quasi phase matching,
Ipump  8 mW
36
GAP Optique Geneva University
1 j1 _ 2 j2
single counts
single counts
Quantum non locality
b
1
j1
2
j2
analyzer
b
analyzer
a-b

the statistics of the correlations can‘t be described by
local variables
Quantum non locality
37
The qubit sphere and the time-bin qubit


i

qubit : 

a
0


e
1
0
different properties : spin, polarization,
0 i1
time-bins
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
2
0 1
any qubit state can be created and
0 1
2
2
0 i1
measured in any basis
2
1
Alice

  a s  e i l
j
1
hn
1
0
variable coupler
Bob
D0
0
D1
switch
switch
variable coupler
38
The interferometers
C
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1
FM
2
3
FM





single mode fibers
Michelson configuration
circulator C : second output port
Faraday mirrors FM: compensation of birefringence
temperature tuning enables phase change
39
entangled time-bin qubit

l
s
A
A
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variable coupler
i
  a 0 A 0 B  e 1 A 1 A

non-linear
crystal
l
B
s
B
depending on coupling ratio and phase , maximally and non-maximally
entangled states can be created


extension to entanglement in higher dimensions is possible
robustness (bit-flip and phase errors) depends on separation of time-bins
40
test of Bell inequalities over 10 km
GAP Optique Geneva University
Bellevue
APD 1 +
Genève
P
F L
KNbO 3
APD1R++
R-+
R+R--
&
classical channels
APD 2APD2+
Bernex
41
results


1.0

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0.5

0.0
-0.5
V = (85.3 ± 0.9)%
raw

15 Hz coincidences
Sraw = 2.41
Snet = 2.7
violation of Bell
inequalities by 16
(25) standarddeviations
close to quantummechanical
predictions
same result in the
lab
V = (95.5 ± 1) %
net.
0
1000
4000
7000
10000
13000
time [sec]
42
GAP Optique Geneva University
le labo
43
Bell test over 50 km

S  E (a  0 ,   45 )  E (a  90 ,   45 )  E (a  0 ,   45 )  E (a  90 ,   45 )
E (a  0 ,   45 )  0.533  0.006
E (a  90 ,   45 )  0.581  0.007
S  2.185  0.012
Violation of Bell inequalities
by more than 15 
Correlation Function
E (a  90 ,   45 )  0.554  0.005
1.0
1.0
1.0
1.0 1.0
0.8
0.8
0.8
0.8 0.8
0.6
0.6
0.6
0.6 0.6
0.4
0.4
0.4
0.4 0.4
0.2
0.2
0.2
0.2 0.2
0.0
0.0
0.0 0.0 0.0
-0.2
-0.2
-0.2 -0.2-0.2
-0.4
-0.4
-0.4 -0.4-0.4
-0.6
-0.6 -0.6-0.6
-0.6
-0.8
-0.8
-0.8 -0.8
-0.8
-1.0
-1.0
-1.0 -1.0
0.0 0.0 0.2 0.2 0.4
0.8
1.0
-1.0
0.40.6
0.6
0.8
0.0 0.0 0.2
0.2 0.5 0.4
0.4 1.0
0.6
0.0
0.61.5[h] 0.8
0.8
2.0
Time
Correlation Function
E (a  0 ,   45 )  0.518  0.006
Correlation
Function
Correlation
Function
Correlation
Function
GAP Optique Geneva University

With phase control we can choose four different
settings a = 0° or 90° and  = -45° or 45°
Violation of Bell inequalities:
Time [h]
1.2
1.0
1.0
1.0
2.5
1.4
1.2
1.2
1.2
3.0
3.5
Time
[h]
TimeTime
[h] [h]
44
GAP Optique Geneva University
Qutrit Entanglement
  c0 ss  c1e
i (a m   m )
mm  c2e
i (al  l )
ll
45
PRL 93, 010503, 2004
Bell Violation
GAP Optique Geneva University
I(lhv) = 2
<
I(2) = 2.829
< I(3) = 2.872
I = 2.784 +/- 0.023
46
Two-photon Fabry-Perot interferometer
GAP Optique Geneva University
Aim :
direct detection of
high dimensional
entanglement
NLC : non linear crystal
0  1  2    0,0  1,1  2,2  
Coincidences DaDb (red) and DaDb’ (blue) as
function of time
while varying the
phase a
D. Stucki et al., quant-ph/0502169
47
Plasmon assisted entanglement transfer
Corresponding inteference fringes
Vp = 97±2%
Vref = 97±1%
140
polarization direction
20nm
BCB
15 m
BCB
15 m
c o in c id e n c e s / 5 s e c .
fiber
120
100
Si-waffer
60
40
20
0
0
2
4
6
8
10
phase [arbitrary units]
12
14
SS+LL
1 cm
events
GAP Optique Geneva University
phase
80
LS
SL
TAC
difference of detection time
 a short lived phenomenon like a plasmon can be coherently
excited at two times that differ by much more than its lifetime.
At a macroscopic level this would lead to a “Schrödinger cat” in
superposition of living at two epochs that differ by much more
than a cat’s lifetime.
48
Experimental QKD with entanglement
cw source
GAP Optique Geneva University
Alice
*
Bob
NL crystal
J. Franson, PRL 62, 2205, 1989
W. Tittel et al., PRL 81, 3563-3566, 1998
49
QKD
GAP Optique Geneva University
Alice
*
Bob
* *
U 1 00  11  1 U 00  11
*
G. Ribordy et al., Phys. Rev. A 63, 012309, 2001
S. Fasel et al., European Physical Journal D, 30, 143-148, 2004
P.D. Townsend et al., Electr. Lett. 30, 809, 1994
R. Hughes et al., J. Modern Opt. 47, 533-547 , 2000
A. Shields et al., Optics Express 13, 660, 2005
N. Gisin & N. Brunner, quant-ph//0312011
50
Quantum cryptography below lake Geneva
GAP Optique Geneva University
Alice
*
A tt.
Bob
PBS
F.M.
Applied Phys. Lett. 70, 793-795, 1997.
Electron. Letters 33, 586-588, 1997; 34, 2116-2117, 1998.
J. Modern optics 48, 2009-2021, 2001.
51
52
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Limits of Q crypto
Secret bit per pulse
- distance
- bit rate
I AB  I Eve (optical noise )
10-6
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10-2
n    0.1
Detector noise
 100 km
distance
53
PNS Attack: the idea
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90,5% 9% 0.5%
Alice 0 ph 1 ph 2 ph
QND
measurement of
photon number
0 ph 0 ph 1 ph
Losses
Eve!!!
Bob
Lossless channel
(e.g.
teleportation)
Quantum memory
PNS (photon-number splitting):
 The photons that reach Bob are unperturbed
 Constraint for Eve: do not introduce more losses
than expected
 PNS is important for long-distance QKD
54
Limits of Q crypto
Secret bit per pulse
I AB  I Eve ( multi  photon pulse )
I AB  I Eve (optical noise )
10-6
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10-2
- distance
- bit rate
Detector noise
 50 km
 100 km
distance
55
1-photon Q crypto
Alice
Bob
GAP Optique Geneva University
2-n
*
31 km CDC
IF
 single-photon source : P(1) = 0.5 … 0.7, P(2)  0.015 & g2  0.1
Results: (PRA 63,012309, 2001 and S. Fasel et al., quant-ph/0403xxx)
Compensation
Interf. Filter
Sifted
key rate
23 Hz
11 Hz
Optical
QBER
5.5 %
4%
Accident
.QBER
1%
1%
Detector
QBER
4%
1.7 %
Dispers.
QBER
0%
0.5 %
Total
QBER
10.5 %
7.2 %
56
Generalized measurements: POVM
GAP Optique Geneva University
A set {Pm} defines a POVM iff
1. Pm  0
2.  m Pm=1
Pm 
 
1  mm
4
The result m happens with probability Tr( Pm)
Example: unambiguous discrimination between 2 non-orthogonal
Q states
 POVM with 3 outcomes:
1. the state was definitively the first one
2. the state was definitively the second one
3. inconclusive result
 minimal probability of an inconclusive result = (1-sin(a))/2
where cos(a) is the overlap
PRA 54, 3783, 1996
57
A new protocol: SARG
GAP Optique Geneva University
The quantum protol is
identical to the BB84
During the public discussion
phase of the new protocol
Alice doesn’t announce bases
but sets of non-orthogonal
states
 even if Eve hold a copy, she
can’t find out the bit with
certainty
 More robust against PNS attacks !
Joint patent UniGE + id Quantique pending
PRL 92, 057901, 2004; Phys. Rev. A 69, 012309, 2004
58
SARG vs BB84
Secret key rate, log10 [bits/pulse]
GAP Optique Geneva University
PNS, optimal m, detector efficiency , dark
counts D
Perfect detectors
=1, D=0
Typical detector
=0.1, D=10-5
m = 0.335
mexp = 0.2
m = 0.014
SARG
BB84
Distance [km] 67km = Geneva-Lausanne
59
Protocols for high secret bit rate
Bob
GAP Optique Geneva University
Alice
bit rate at emission
goal: > 1 Gbit/s
channel
loss
« no » loss in
detector
Bob’s
optics
+ noise  secret bit rate
goal: > 1 Mbit/s
60
protocols for high secret bit rate:
an example (patent pending)
GAP Optique Geneva University
Wish list:
•
•
•
•
•
•
quant-ph/0411022
APL 87, 194105, 2005
low loss at Bob’s side
use one of the 2 bases more frequently
make that basis simple
telecom compatible
resistant to PNS attacks
does not work with single photons
tB
Laser IM
bit 0
bit 1
decoy
sequence
DB
DM1
DM2
61
Pulse rate
434 Mhz
Link loss (25km) 5 dB
QBERoptical
1%
QBERtot
<4%
GAP Optique Geneva University
First results :
quant-ph/0411022 62
APL 87, 194105, 2005
GAP Optique Geneva University
GHz Telecom QKD
1.27 GHz
up-conversion detector:
R(secre
t[es tim
)
1550
nated]
+ 980
pump = 600 nm
L
Raw Ra te (a ve)
QBER
25k m
2.62 M H z
1.2 %
500 kH z
L
Raw Ra te (a ve)
QBER
R( secre t[es tim ated] )
50k m
530 k H z
7.3 %
75 kH z
Rob Thew et al., 2005
63
64
GAP Optique Geneva University
GAP Optique Geneva University
Bell measurement
0 11 2  i 0 A 1 A i 0 B 1 B  0 A 1 B  1 A 0 B 1 1 0 2  i 0 A 1 A i 0 B 1 B  0 A 1 B  1 A 0 B
0 1 1 2  1 1 0 2  i( 0 A 1 A  0 B 1 B ) 0 1 1 2  1 1 0 2  0 A 1 B  1 A 0 B
65
Bell measurement
GAP Optique Geneva University
1.   
D1
00
1/16
2.   
D1
p
1/16
1/16
1
2
D1
22
0
1
2
D2
1/16
1/8
1/2
1/8
22
00
1/16
0
p
1
 00  11 
2
00
D2
3.   
22
00
D2
p
1
 00  11 
2
1/16
11
1/16
01
12
01
1/
8
1/8
4.   
22
1/16
1
 01  10
2
1/4
Psuccess = ½
0
2
D1
11
0
2
D2
1/4
1/8
1/8
p
1/8
1
0
2
1
12
0
1
1
2
1/8
1/8
1/8
1/8
1/8
1
 01  10
2
11
02
11
1/
4

1/4
1/8

2
0
02
0
2
1/8
1/8
1/8
3 Bell states are detected!
66
67
GAP Optique Geneva University
Q repeaters & relays
*
.
.
*
entanglement
entanglement
entanglement
J. D. Franson et al, PRA 66,052307,2002; D. Collins et al., quant-ph/0311101
Bell
measurement
REPEATER
GAP Optique Geneva University
RELAY
Bell
measurement
*
.
??
*
entanglement
entanglement
QND measurement
+ Q memory
H. Briegel, W. Dür, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 81, 5932 (1998)
68
3-photon: Q teleportation & Q relays

2 bits
U
Bell

EPR
2 km

Classical channel
Charlie
Alice
BSM
Bob
2 km
EPR source
2 km

1.00
0.98
0.96
0.94
Fidelity
GAP Optique Geneva University

0.92
n=1
0.90
n=2
n=3
n=4
0.88
0.86
0.84
0.82
0.80
0
50
100
150
200
Distance [km]
250
300
350
69
The Geneva Teleportation
experiment over 3x2 km
GAP Optique Geneva University
Photon = particle (atom) of light
Polarized photon
( structured photon)
Unpolarized photon
( unstructured  dust)
70
GAP Optique Geneva University
55 metres
2 km of
optical fibre
2 km of
optical fibre
Two entangled
photons
71
GAP Optique Geneva University
55 metres
2 km of
optical fibre
72
GAP Optique Geneva University
55 metres
Bell measurement
(partial)
the 2 photons
interact
4 possible results:
0, 90, 180, 270 degrees
73
GAP Optique Geneva University
55 metres
Bell measurement
(partial)
the 2 photons
interact
4 possible results:
0, 90, 180, 270 degrees
The correlation is
independent of the quantum
state which may be unknown
or even entangled with a
fourth photon
74
Quantum teleportation

2 bits
U
GAP Optique Geneva University
Bell


EPR
(c0 0  c1 1 )  ( 0,0  1,1 ) /





2
1
1 ( 0,0  1,1 )
( 0,0  1,1 )  ( c0 0  c1 1 )
2

2 2
1
1 ( 0,0  1,1 )
( 0,0  1,1 )  ( c0 0  c1 1 )
2

2 2
1
1 ( 0,1  1,0 )
( 0,1  1,0 )  ( c1 0  c0 1 )
2
2 2

1
1 ( 0,1  1,0 )
( 0,1  1,0 )  ( c1 0  c0 1 )
2
2 2


z 
x
y
75
What is teleported ?
 According to Aristotle, objects are constituted by matter
and form, ie by elementary particles and quantum states.
GAP Optique Geneva University
 Matter and energy can not be teleported from one place to
another: they can not be transferred from one place to
another without passing through intermediate locations.
 However, quantum states, the ultimate structure of
objects, can be teleported. Accordingly, objects can be
transferred from one place to another without ever
existing anywhere in between! But only the structure is
teleported, the matter stays at the source and has to be
already present at the final location.
76
Implications of entanglement
 The world can’t be understood in terms of
GAP Optique Geneva University
“little billiard balls”.
 The world is nonlocal (but the nonlocality can’t
be used to signal faster than light).
 Quantum physics offers new ways of processing
information.
77
Experimental setup
Bob
&
InGaAs
Charlie
fs laser @ 710 nm
Ge
InGaAs
55 m
Alice:creation of qubits to
be teleported
BS
RG
creation of entangled qubits
Charlie:the Bell measurement
WDM
WDM
RG
LBO
LBO
Bob:analysis of the
teleported qubit, 55 m from
Charlie
2 km of optical fiber
fs laser
GAP Optique Geneva University
Alice
coincidence electronics
sync out
78
results
1
F 
Equatorial states
0
Mean Fidelity
40
F  0,1  = 78 ± 3%
8000
7000
6000
25
5000
20
4000
15
3000
10
2000
5
1000
Three-fold coincidence [/500s]
2
1
F  1,0  = 77 ± 3%
Fmean  Feq  Fp
3
3
 77.5 ±2.5 % mean fidelity: Fpoles=77.5 ± 3 %
30
North & south poles
» 67 % (no entanglement)
0
0
0
2
4
6
8
10
12
14
16
18
1.0
9000
35
30
7000
25
6000
5000
20
4000
15
3000
10
2000
5
Feq1Vraw
2
0
0
2
4
6
8
10
12
14
16
1000
Three-fold coincidence [/500s]
Raw visibility : Vraw= 55 ± 5 %
8000
0
18
Phase [arb. units]
= 77.5 ± 2.5 %
coincidence[arb unit]
Phase [arb. units]
40
four-fold coincidences [1/500s]
four-fold coincidences [1/500s]
GAP Optique Geneva University
9000
35
Ccorrect
Ccorrect  C wrong
0.8
0.6
0.4
0.2
0.0
0
1
2
3
4
time between start and stop [ns]
5
6
79
Size of the classical communication
One proton in one cm3 at a temperature of 300 K:
p2
h
kT 
 l   1.7 1010 [m]
2m
p
GAP Optique Geneva University
3
 L
d  dim( H )     1.9 10  23
l
 bits  ln 2 (d 2 )  155 bits
1020 protons in one cm3 at a temperature of 300 K  1020 x 155  1022 bits
To be compared to today’s optical fiber communication in labs:
1 Tbyte x 1024 WDW channels x 1000 fibers  1019 bits/sec.
  1 hour !!
80
GAP Optique Geneva University
1: EPR
2: Distribute
3: Create Qubit
4: Prepare BSM
5: BSM
6: Send result
7: Store photon
8: Wait for BSM
9: Analysis
81
GAP Optique Geneva University
&
LBO
n
n+1
PC
LBO
n
n+1
1: EPR
2: Distribute
3: Create Qubit
4: Prepare BSM
5: BSM
6: Send result
7: Store photon
8: Wait for BSM
9: Analysis
Laser fs
82
Entanglement swapping
GAP Optique Geneva University
Entangled photons that never interacted
3-Bell-state analyzer
N.Brunner et al., quant-ph/0510034
Bell state
measurement
A
B
C
D
2 independent sources
EPR source
(  )
AB
i
( )  0 A ,0 B  e 1A ,1B
EPR source
( )
CD
i
( )  0C ,0 D  e 1C ,1D
83
Superposition basis: results
Deriedmatten, Marcikic et al., PRA 71, 05302, 2005
5000
90
80
4000
70
60
3000
50
40
2000
30
20
1000
10
0
3-photon coincidences [/6h]
without BSM
Four-photon coincidences [/6h]
GAP Optique Geneva University
100
0
100
200
300
400
500
600
Phase [degrees]
V = (80 ± 4) %
F  90 %
78 hours of measurement !
84
85
GAP Optique Geneva University
Coin tossing at a distance
GAP Optique Geneva University
correlated
Each side the
results are
random
correlated
Non correlated
the statistics of the correlations can‘t be described by
local variables
Quantum non locality
86
Bell’s inequality:
Bob
Left
Alice
same
different
GAP Optique Geneva University
Left
Middle
Right
Middle
same
different
Right
same
different
1/4
0%
1/4
1/4
3/4
3/4
3/4
100 %
0%
100 %
0%
1/4
1/4
3/4 100 %
3/4 1/4
LMR
GGG
GGR
GRG
RGG
GRR
RGR
RRG
RRR
Arbitr. mixture
Quantum
Mechanics
(D. Mermin, Am. J. Phys. 49, 940-943, 1981)
Si param 
Prob(resultats =)
100 %
1/3
1/3
1/3
1/3
1/3
1/3
100 %
 1/3
=1/4
3/4
Bell Inequality
Quantum non-locality
87
Bell inequality
GAP Optique Geneva University


P (1 | a ,  ,)  a(a ,  , l )
Locality 

 

(ab)(a , b , , l )  a(a , , l )  b(b , , l )



In particular: a (a, , l )  a (a, b, , l )
a.b+a.b’+a’.b-a’.b’=a.(b+b’)+a’.(b-b’)2
E(a,b)=

a(l).b(l) (l) dl
S=E(a,b)+E(a,b’)+E(a’,b)-E(a’,b’)2
Bell inequality
88
89
GAP Optique Geneva University
GAP Optique Geneva University
Generalized measurements: POVM
A set {Pm} defines a POVM iff
1. Pm  0
2.  m Pm=1
The result m happens with probability Tr( Pm)
Example: Pm 
 
1  mm
4
where the m m are the 4 vectors of the
thetrahedron
90
50%
GAP Optique Geneva University
D1
input
PBS
D2
l/2
33.3%
 4  4 D2 
  j 1 j

DOP  3
 1
2
4
  Dj

j

1



D3
D4

91
92
GAP Optique Geneva University
Non-locality according to Newton
GAP Optique Geneva University
 Newton was very conscious of an unpleasant
characteristics of his theory of universal
gravitation :
 A stone moved on the moon would
immediately affect the gravitational field on
earth.
 Newton didn’t like this non-local aspect of his
theory at all, but, due to a lack of alternatives,
physics had to live with it until 1915.
93
GAP Optique Geneva University
Let’s read Newton’s words:
That Gravity should be innate, inherent and essential to
Matter, so that one Body may act upon another at a
Distance thro’ a Vacuum, without the mediation of any
thing else, by and through which their Action and Force
may be conveyed from one to another, is to me so great
an Absurdity, that I believe no Man who has in philosophical
Matters a competent Faculty of thinking, can ever fall into it.
Gravity must be caused by an Agent acting constantly
according to certain Laws, but whether this Agent be
material or immaterial, I have left to the Consideration
of my Readers.
Isaac Newton
Papers & Letters on Natural Philosophy and related documents
Edited by Bernard Cohen, assisted by Robert E. Schofield
Harvard University Press, Cambridge, Massachusetts, 1958
94
Einstein, the greatest mechanical engineer
GAP Optique Geneva University
 Today, thanks to Einstein, gravitation is no longer
considered as a kind of action at a distance. A
moon-quake triggers a bunch of gravitons that
propagate through space and « informs » Earth.
The propagation is very fast, but at finite speed,
the speed of light, i.e. about 1 second from the
moon to our Earth.
95
Einstein, the greatest mechanical engineer
 In 1905, Einstein also gave a description of
GAP Optique Geneva University
Brownian motion: the statistics of collisions
between invisible atoms and molecules support
the atomic hypothesis:
 Still in 1905, Einstein gave a mechanical
explanation of the photo-electric effect:
96
Classical
physics:
GAP Optique Geneva University
Nature is made out of many
little “billiard balls” that
mechanically bang into
each other
Quantum physics:
Named by historical accident quantum mechanics,
the new physics is precisely characterized by the fact that
it does not provide a mechanical description of Nature
97
Non-locality according to Einstein
GAP Optique Geneva University
 Einstein was very conscious of an "unpleasant"
characteristic of quantum physics :
 Spatially separated systems behave as a single
entity: they are not logically separated.
 Acting “here” has an apparent, immediate, effect
“there”.
 Einstein-Podolski-Rosen argued that this being
obviously impossible, quantum physics is
incomplete.
 Most physicists didn’t like this non-local aspect of
quantum theory, but again, due to a lack of
alternatives, … it remained in the curiosity-lab. 98
GAP Optique Geneva University
Non-locality for non-physicists
99
Quantum exams
x
GAP Optique Geneva University
Alice
y
Bob
a
b
P ( a , b | x, y )
Joint
conditional
probability
Events at
2 separated
locations.
Not under the
professor’s control
Settings
(experimental
conditions).
Under the
professor’s control
100
Quantum exam #1
GAP Optique Geneva University
 Suppose Alice is asked to output the question
received by Bob, and vice-versa.
 Can they succeed?
 Clearly, not!
Why? Because it would imply signaling
(arbitrarily fast communication) and every
physicists knows – since Einstein – that this is
impossible.
And even long before Einstein, Newton and
others had the strong intuition that signaling is
impossible.
 The relativistic no-signaling condition implies that some
conditional probabilities (i.e. some exams) are
impossible !
101
Quantum exam #2
GAP Optique Geneva University
 Suppose that Alice and Bob are asked to always
output the same answer, whenever they receive
the same question.
 Can they succeed?
 Clearly yes!
It suffice that Alice and Bob prepare a common
strategy before being spatially separated; i.e.
they should prepare one precise answer for each
question.
 Is there an alternative strategy? No, as all
students preparing exams know.
 Some conditional probabilities can be explained in the
frame of classical physics only with common causes.
102
Quantum exam #3: binary case
 But now, assume that A&B should always output the
GAP Optique Geneva University





same value, except when both receive the input 1
Formally a+b=x•y modulo 2
Can they succeed? Note that the exam doesn’t require
signaling.
If A’s output is predetermined by some strategy, then
this would allow signaling.
Consequently, A’s output has to be random. Similarly,
B’s output has to be random.
A and B’s randomness should be the same whenever
x.y=0, but should be opposite whenever x=y=1.
 This is impossible, although there is no signaling.
 How close to a+b=x y can they come?
•
Can they achieve a probability larger than 50%?
103
GAP Optique Geneva University
Prob(a+b=x•y)=?
optimal for classical
Alice and Bob
CHSH-Bell inequality:
P(a+b=x•y|x=0,y=0) + P(a+b=x•y|x=0,y=1) +
P(a+b=x•y|x=1,y=0) + P(a+b=x•y|x=1,y=1)  3

 2+2  3.41
optimal for Alice and Bob
sharing quantum entanglement
Quantum correlations (entanglement) allows one to
perform some tasks, including some useful tasks, that
are classically impossible !
104
GAP Optique Geneva University
Entanglement is everywhere!
old wisedom:
entanglement is like a dream, as soon as one tries to
tell it to a friend, it evaporates!
Entanglement is fragile !
recent experiments:
Entanglement is not that fragile !
Entanglement is everywhere, but hard to detect.
This new wisedom raises new questions:
Can entanglement be derived from a more primitive concept?
Can Q physics be studied from the outside ?
105
Theoretical Physics
Q concepts without Hilbert space
GAP Optique Geneva University
 Can entanglement,
non-locality,
no-cloning,
uncertainty relations,
cryptography,
etc
be derived from one primitive concept ?
 Can all these be studied « from the outside »,
i.e. without all the Hilbert space artillery?
106
binary local correlations
x
y
GAP Optique Geneva University
Alice
a
Bob
p(a,b|x,y)
b
QM
all facets correspond
to the CHSH-Bell :
P3
polytope of local
correlations p(a,b|x,y)
107
CHSH-Bell inequality
GAP Optique Geneva University
P(a+b=x•y|x=0,y=0) + P(a+b=x•y|x=0,y=1) +
P(a+b=x•y|x=1,y=0) + P(a+b=x•y|x=1,y=1)  3
use non-signaling to remove the output 1:
P(0,1|x,y)=P(a=0|x)-P(0,0|x,y)
P(1,1|x,y)=1-P(a=0|x)-P(b=0|y)+P(0,0|x,y)
 P(00|00)+P(00|01)+P(00|10)-P(00|11)
 P(a=0|0)+P(b=0|0)
x -1
y
0
-1
+1
+1
0
+1
-1
0
No better inequality is known to detect non-locality of Werner
2 qubit states !!!
108
detection loophole
GAP Optique Geneva University
P(00|00)+P(00|01)+P(00|10)-P(00|11)  P(a=0|0)+P(b=0|0)
detection efficiency  
2 P(00|00)+ 2 P(00|01)+ 2 P(00|10)- 2 P(00|11)
  P(a=0|0)+  P(b=0|0)
P(00 | 00)  P(00 | 01)  P(00 | 10) - P(00 | 11)
a violation requires:  
P(a  0 | 0)  P(b  0 | 0)
 threshold for max entangled qubit pair  82%
 threshold decreases for partially entangled qubit pairs towards 2/3 !
(P. H. Eberhard, Phys. Rev. A 47, R747,1993)
 find better inequalities
109
The new inequality for qubits with
3 settings
GAP Optique Geneva University
This is the only new inequality for 3 inputs and binary outputs.
I3322 =
x -1
y
0
0
-2
+1
+1
+1
-1
+1
+1
-1
0
+1
-1
0
0
Collins & Gisin, J.Phys.A 37, 1775, 2004
110
For each q, let lCHSH be the critical weight such that
(q)= lCHSH Pcos(q)|00>+sin(q)|11> + (1- lCHSH) P|01>
is at the limit of violating the CHSH inequality
1.03
1.01
1.00
trace(B)
GAP Optique Geneva University
1.02
0.99
0.98
0.97
0.96
0.95
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
q
111
Non-locality without signaling
GAP Optique Geneva University
J.Barrett
x et al,
quant-ph/0404097
y
set of correlations p(a,b|x,y) s.t.
1. p(a|x,y)=bp(a,b|x,y) = p(a|x)
2. p(b|x,y)=ap(a,b|x,y) = p(b|y)
Alice
Bob
facet corresponding
to the no-signaling :
binary case:
unique extremal point!
a+b=xy
one above each CHSHa
p(a,b|x,y)
b
Bell inequality
QM
facet corresponding
to the CHSH-Bell :
P3
polytope of local
correlations p(a,b|x,y)
112
A unit of non-locality, or non-locality
without the Hilbert space artillery
GAP Optique Geneva University
x
Alice
Bob
y
Non Local Machine
a
a + b= x.y
b
A single bit of communication suffice to simulate
the NL Machine (assuming shared randomness).
But the NL Machine does not allow any communication.
Hence, the NL Machine is a strickly weaker ressource
than communication.
113
no-cloning theorem without quantum
GAP Optique Geneva University
L.Masanes, A.Acin, NG
quant-ph/0508016
x
z
a
y
b
If
c
a+b=x.y
then b+c=x(y+z), and Alice can signal to B-C
a+c=x.z
Non-signaling no-cloning theorem
114
From Bell inequality to cryptography
CHSH Q-crypto protocol
Alice
Bob
sifting:
• 1-way
• all bits are kept
• noisy even
without Eve
A.Acin, L.Masanes, NG
quant-ph/0510094
GAP Optique Geneva University
facet corresponding
to the no-signaling :
a+b=xy
intrinsic
info > 0
1-way
distillation
2-way
???
1
secure QKD against individual
2-1
QM attacks by any post-quantum
non-signaling Eve !
0
isotropic
correlations
polytope of
local correlations
facet corresponding
to the CHSH-Bell :
P3
115
From Bell inequality to cryptography
A.Acin, L.Masanes, NG
quant-ph/0510094
Uncertainty relations, i.e.
information / disturbance trade-off:
I(E:B|x=0) = fct(QBERx=1)
I(E:B|x=1 ) = fct(QBERx=0)
GAP Optique Geneva University
facet corresponding
to the no-signaling :
V. Scarani
a+b=xy
intrinsic
info > 0
1-way
distillation
2-way
???
1
secure QKD against individual
2-1
QM attacks by any post-quantum
non-signaling Eve !
0
isotropic
correlations
polytope of
local correlations
facet corresponding
to the CHSH-Bell :
P3
116
Simulating entanglement with a few bits
of communication (+ shared
randomnes)
 

b
GAP Optique Geneva University

a
Alice
a
Bob
a & b define measurement
bases.
The output a &  should
reproduce the Q statistics:



 1  a a 1   b  


P(a ,  | a , b )  Tr  


2
2



Case of singlet: 8 bits, Brassard,Cleve,Tapp, PRL 83, 1874 1999
2 bits, Steiner, Phys.Lett. A270, 239 2000,
Gisins Phys.Lett. A260, 323, 1999
1 bit! Toner & Bacon, PRL 91, 187904, 2003
0 bit: impossible (Bell inequality) … but …
117
Simulating singlets with the NL Machine


a


sg (al1 )  sg (al2 )
GAP Optique Geneva University
Alice
 
l1 , l2 


sg (b l )  sg (b l )
Bob

l1 , l2

Non local Machine
a
b

a  a  sg (al1 )

b


  b  sg (b l )

where l1 and l2 are distributed uniformly on S ( 2) ,

 
1 if x  0
sg ( x)  
and l  l1  l2
0 if x  0
 
Given a & b, the statistics of a &  is that of the singlet state:
 


1 a b
E (a ,  | a , b ) 
2
118
GAP Optique Geneva University
hint for the proof:
l1
l2
119
GAP Optique Geneva University

2
l
x=0
l

1
x=1
x=1
x=0
120
(x,y)
l
l
GAP Optique Geneva University
l

2
(0,0)
(1,0)
l

1
(0,1)
(1,1)
(1,1)
(0,1)
(1,0)
(0,0)
121
a= a +1
(x,y)
=b
l
l
GAP Optique Geneva University
l

2
(0,0)
(1,0)
l

1
(0,1)
(1,1)
(1,1)
(0,1)
 =b+1
(1,0)
(0,0)
a=a
« cqfd »
122
Simulating partial entanglement
GAP Optique Geneva University
Partially entangled states seem more nonlocal than the max entangled ones !
Partially entangled states are more robust against the
detection loophole (P. H. Eberhard, Phys. Rev. A 47, R747,1993)
Bell inequalities are more violated by partially entangled states
than by max entangled ones (for dim > 2 & all known cases).
When testing Bell inequality, the use of a partially entangled
state provides more information per experimental run
than the use of max entangled states.
(T. Acin, R. Gilles & N. Gisin, PRL 95, 210402, 2005 )
123
How to prove that some correlation can’t be simulated
with a single use of the nonlocal machine ?
GAP Optique Geneva University
 same idea as Bell inequality, i.e.
1. List all possible strategies
2. Notice that they constitute a convex set
3. Notice that this convex set has a finite number of
extremal points (vertices), i.e. it’s a polytope
4. Find the polytope’s facets
5. Express the facets as inequalities
124
Ai
Bj
l x
y
l
a+b=xy
GAP Optique Geneva University
a
rA
b
rB
For given Ai and l, there are 6 extremal local strategies:
1. rA=0
3. X=0 and rA=a
5. X=0 and rA=a+1
2. rA=1
4. X=1 and rA=a
6. X=1 and rA=a+1
For 2 settings per side, there are 64 strategies defining
264 different vertices.
The polytope is the same as the “no-signaling polytope”
studied by J. Barrett et al in quant-ph/0404097
Consequently, no quantum state can violate such a
2-settings inequality
125
The
1
nl-bit
inequality
For 3 settings per side:
x -2
y
0
0
-2
+1
+1
+1
-1
+1
+1
-1
0
+1
-1
0
y
0
P(rA =0|x)
x
P(rB =0|y)
GAP Optique Geneva University
• there are 66 strategies defining 3880 different vertices.
• There is a unique new inequality:
P(rA = rB =0|x,y)
Recall: for standard Bell inequalities (i.e with no nonlocal
machines) and 3 settings per side, there is also a unique
x -1
new inequality:
0
0
y
I3322 =
Collins & Gisin, J.Phys.A 37, 1775, 2004
-2
+1
+1
+1
-1
+1
+1
-1
0
+1
-1
0
0
126
GAP Optique Geneva University
Geometric intuition
NLM
NLM
D
D
CHSH
NLM
D
D
I3322
127
GAP Optique Geneva University
Very partially entangled states do violate the
1-nl bit-Bell inequality :
partial ent.
max ent.
 Very partially entangled states can’t be simulated with
only 1 nl-bit
 Partially entangled states are more nonlocal
than the singlet !
128
129
GAP Optique Geneva University
130
GAP Optique Geneva University
The I3322-Bell inequality is
not monogamous
There exists a 3-qubit state ABC, such that
A-B violates the I3322-Bell inequality and
A-C violates it also.
GAP Optique Geneva University
A
ABC
B
C
(see D. Collins et al., J.Phys. A 37, 1775-1787, 2004)
131
132
GAP Optique Geneva University
GAP Optique Geneva University
Quantum Cryptography guaranties confidentiality

Bell’s inequalities are violated

Quantum correlation can’t be explained by local variables
133
Alice

A
Eavedropping
(cloning)
machine
white
paper E
GAP Optique Geneva University

B
Bob
clone
U
E
int ern . st .
Q machine E '
int ern. st .
Q machine E '
U :C C C  C C C
 A   am ,nU m ,n  B Bm ,  n E , E '
d
d
d
d
d
d
m , n  0..d 1
Error operator:
U m ,n 
e
2pi dn
k  0.. d 1
d
km k
Bell states
134
  B
GAP Optique Geneva University
 E
Where:
 am ,n
2
 bm ,n
2
m ,n 0.. d 1
m ,n 0.. d 1
bm ,n
U m ,n  

U m ,n
U m ,n  

U m ,n
1
2pi ( nm '  mn') / d

e
am ',n'

d m ',n'0..d 1
1
1 
   D0   1 

2
d
N. Cerf et al., PRL 84,4497,2000 & 88,127902,2002135
Case d=2 (qubits):
GAP Optique Geneva University
U m ,n
 1 x

 
 z  y 
am ,n
2
F
FD 

 
2 
 FD D 
2
Classical random variables:
Alice Bob
Eve
X=0,1 Y=0,1 Z=[Z1,Z2]
Z1 =X+Y
1
Z2=X with prob.    FD
2
Conditional mutual information:
I( X :Y | Z )  H( X | Z )  H( X | Y , Z )
 H ( )  0 F  1
2
136
Optimal individual attack on BB84
GAP Optique Geneva University
Page 182 à 185 de Rev.Mod.Phys. 74, 145, 2002
137
Eve: optimal individual attack
1.0
I AB  1  H (QBER)
IAE1-IAB
Shannon Inform ation
GAP Optique Geneva University
0.8
0.6
IAE
0.4
0.2
Bell inequ. violated
Bell inequ. not violated
0.0
0.0
0.1
0.2
QBER
138
GAP Optique Geneva University
Advantage distillation
Alice
X0=1
X1=1
X2=0
X3=1
Bob
Y0
Y1
Y2
Y3
….
….
Xj
Yj
Alice announces {0,1,3},
Bob accepts iff Y0= Y1= Y3
Eve can’t do better than a majority vote!
Alice and Bob take advantage of their
public authenticated channel
Theorem: if the intrinsic information vanishes,
then advantage distillation does not produce a secert key.
Theorem: In arbitrary dimensions d and
either the case of 2 bases or of d+1 bases:
Advantage distillation produces a secret key
iff Alice and Bob are not separated.
N. Gisin & S. Wolf, PRL 83, 4200-4203, 1999
139
1.0
I AB  1  H (QBER)
Shannon Inform ation
GAP Optique Geneva University
0.8
IAE
0.6
0.4
0.2
0.0
0.0
2-way
quantum. Inf. Proc.
suffice
1-way
class. Inf. Proc.
suffice
Bell inequality:
can be
never
violated violated
0.1
D0
0.2
Alice
and
Bob
separated
or
classical
0.3
QBER
0.4
140
Quantum Cryptography
GAP Optique Geneva University
Entanglement
Q nonlocality
AB measurement
Entanglement
distillation
AB
Where is Eve ?
P(A,B,E)
I(A:B), I(A:E)
I(A:B|E)
I(A:BE) intrinsic info.
Secret key
distillation
measurement
shared secret bit
In the Q scenario one assumes that Eve holds the entire
universe except the Q systems under Alice and Bob’s
direct control. Ie Eve holds the purification of .
141
Intrinsic information
GAP Optique Geneva University
Eve
Alice Bob
0
1
0
0
¼
0
1
¼
1
0
¼
1
1
¼
I(A:B|E) = 1
0
EE
e
1
142
Intrinsic information
GAP Optique Geneva University
Eve
Alice Bob
0
1
¼
e
0
0
¼
0
1
¼
¼
1
0
¼
¼
1
1
¼
I(A:B|E) = 1
¼
0
EE
I(A:B|E) = 0
e
1
Intrinsic information: I(A:BE) = Min I(A:B|E)
EE
143
Intrinsic info  entanglement
Theorem:
Let P(A,B,E) be a probability distribution shared between Alice, Bob
and Eve after measuring a quantum state ABE.
GAP Optique Geneva University
I(A:BE) > 0 iff AB is entangled
N. Gisin and S. Wolf, PRL 83, 4200-4203, 1999.
S. Wolf and N. Gisin, Proceedings of Crypto 2000, pp 482-500
Theorem:
If moreover Alice and Bob hold qubits,
then
AB is entangled iff P(A,B,E) is such that Alice and Bob
can distil a secret key
A. Acin, L. Masanes and N. Gisin, PRL 91, 167901, 2004.
144
Quantum Cryptography
GAP Optique Geneva University
Entanglement
Q nonlocality
AB measurement
Entanglement
distillation
AB
P(A,B,E)
I(A:B), I(A:E)
I(A:B|E)
I(A:BE) intrinsic info.
Secret key
distillation
measurement
shared secret bit
In the binary case, the diagram commutes.
A counter example in dimension 3 is known.
The existence of bound information is conjectured.
145
What is secure ?
GAP Optique Geneva University
Quantum cryptography is technically ready to provide absolute
secure key distribution between two end-points:
Where are Alice’s and
Bob’s boundaries ??
At the quantum/classical split:
and old question in a modern setting!
Alice
Secure QKD channel
Bob
146
How to improve Q crypto ?
GAP Optique Geneva University
Effect on
distance
Effect on
bit rate
Feasibility
Detectors
1- source
Q channel
Protocols
Q relays
Q repeater
147
horizontal
vertical
pol.
-45° pol.
PBS@45°
GAP Optique Geneva University
port 2
Faraday
effect
PBS@0°
port 1
horizontal
vertical
pol.
pol.
+45° pol.
port 3
148
Rayleighback-scaterrings delay line
Bob
Alice
Laser
GAP Optique Geneva University
FR
PM
APD
D
A
PBS
PM
APD
Drawback 1:
Drawback 2:
Perfect
interference
(V99%) withoutTrojan
any adjustments
, since:
Rayleigh
backscattering
horse attacks
•
•
both pulses travel the same path in inverse order
both pulses have exactly the same polarisation thanks to FM
149