Dalle disuguaglianze di Bell alla crittografia quantistica

Dalle disuguaglianze di Bell alla crittografia quantistica:
applicazioni della fisica fondamentale e dell’Informazione Quantistica
Giuseppe Vallone
email: [email protected]
Dip. di Fisica, Università di Torino - 5 Giugno 2015
What is Quantum Information?
Information Theory
Quantum Mechanics
⇔
Merging two big XXth century revolutions:
information theory (Shannon, Turing) and Quantum Mechanics.
Pag. 2
Examples of applications
Quantum computer
Quantum cryptography
Quantum metrology
Pag. 3
Esempi di applicazioni
Pag. 4
Quantum sensing
Quantum imaging
Quantum simulation
Quantum random number
generation
But...
...be aware of fake!
Pag. 5
Summary
Pag. 6
1
Quantum Mechanics
2
Quantum Key Distribution
3
Quantum Random Number Generators
4
Entanglement and Bell inequalities
5
Protocols exploiting entanglement
Teleportation
“Device Independent” protocols
6
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Summary
Pag. 7
1
Quantum Mechanics
2
Quantum Key Distribution
3
Quantum Random Number Generators
4
Entanglement and Bell inequalities
5
Protocols exploiting entanglement
Teleportation
“Device Independent” protocols
6
Conclusions
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Superposition principle
I
Pag. 8
Physical states are representes as vectors |ψi
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Superposition principle
I
Physical states are representes as vectors |ψi
I
Superposition principle: if |ψ1 i and |ψ2 i are physical states,
any linear combination is a physical state:
|Ψi = a|ψ1 i + b|ψ2 i
Pag. 8
a, b ∈ C
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Superposition principle
I
Physical states are representes as vectors |ψi
I
Superposition principle: if |ψ1 i and |ψ2 i are physical states,
any linear combination is a physical state:
|Ψi = a|ψ1 i + b|ψ2 i
I
From classical bit (two orthogonal states |0i and |1i) to
quantum-bit , or qubit:
|ψi = α|0i + β|1i
Pag. 8
a, b ∈ C
α, β ∈ C , |α|2 + |β|2 = 1
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Superposition principle
I
Physical states are representes as vectors |ψi
I
Superposition principle: if |ψ1 i and |ψ2 i are physical states,
any linear combination is a physical state:
|Ψi = a|ψ1 i + b|ψ2 i
I
From classical bit (two orthogonal states |0i and |1i) to
quantum-bit , or qubit:
|ψi = α|0i + β|1i
I
Pag. 8
a, b ∈ C
α, β ∈ C , |α|2 + |β|2 = 1
indistinguishability ⇒ INTERFERENCE!
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
State superposition
Example 1: photons on a semi-reflective mirror (beam splitter)
Pag. 9
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
State superposition
Example 1: photons on a semi-reflective mirror (beam splitter)
|1i
|0i
Pag. 9
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
State superposition
Example 1: photons on a semi-reflective mirror (beam splitter)
√1 (|0i
2
Pag. 9
+ |1i)
Quantum Mechanics
QKD
QRNG
State superposition
Example 2: two-slit experiment
Pag. 10
Bell
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
State superposition
Example 3: Schrödinger cat
1
|ψi = √ (|livei + |deadi)
2
Pag. 11
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
State superposition
up to 6910 AMU, 430 atoms
Pag. 12
Quantum interference of large organic molecules, Nature Communication 2, 263 (2011)
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Misurement and no-cloning
I
Pag. 13
The measurement (in general) perturbs quantum states
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Misurement and no-cloning
Pag. 13
I
The measurement (in general) perturbs quantum states
I
The output of a measurement is probabilistic (if the state is
not an eigenstate of the observable)
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Misurement and no-cloning
I
The measurement (in general) perturbs quantum states
I
The output of a measurement is probabilistic (if the state is
not an eigenstate of the observable)
I
Impossibility of perfect cloning: quantum copy-machine is
not physical
@ U | U |ψiA → |ψiA |ψiB
Pag. 13
∀|ψi
Quantum Mechanics
QKD
QRNG
Uncertainty principle
I
Bound on the precision of
non-commuting observables:
Heisenberg uncertainty
principle
∆x∆p ≥
Pag. 14
~
2
Bell
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Uncertainty principle
I
Bound on the precision of
non-commuting observables:
Heisenberg uncertainty
principle
∆x∆p ≥
I
Pag. 14
~
2
The lower is the uncertainty on the position, the larger is
the uncertainty on the momentum (and viceversa)
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Summary
Pag. 15
1
Quantum Mechanics
2
Quantum Key Distribution
3
Quantum Random Number Generators
4
Entanglement and Bell inequalities
5
Protocols exploiting entanglement
Teleportation
“Device Independent” protocols
6
Conclusions
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
QKD principles
The best method to encrypt a message is the One-Time-Pad
(OTP) protocol: for a n-bit message, a n-bit secure key is
needed
Quantum key distribution (QKD) allows two users to exchange
random and secret keys
Pag. 16
Quantum Mechanics
QKD
QRNG
Bell
QKD in a nutshell
BB84 protocol
Pag. 17
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Secret key rate
Basic tools:
Pag. 18
I
two non-commuting basis
I
no-cloning theorem
I
any measurement (generally)
perturbs the systems











⇒ Eve detection!
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Secret key rate
Basic tools:
I
two non-commuting basis
I
no-cloning theorem
I
any measurement (generally)
perturbs the systems






⇒ Eve detection!





Secret key rate:
r = 1 − 2h2 (Q)
with
Q = QBER
Pag. 18
h2 (Q) = −Q log2 (Q) − (1 − Q) log2 (1 − Q)
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Secret key rate
Basic tools:
I
two non-commuting basis
I
no-cloning theorem
I
any measurement (generally)
perturbs the systems






⇒ Eve detection!





Secret key rate:
r = 1 − 2h2 (Q)
with
Q = QBER
h2 (Q) = −Q log2 (Q) − (1 − Q) log2 (1 − Q)
If Eve is gaining information on the key, the key is discarded.
Eve has no information on the secret message
Pag. 18
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
QKD in the lab
QKD system for
BB84 protocol
Free-space QKD
prototype
Pag. 19
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Aligment-free OAM QKD: 210m free space link
Hybrid qubit: α|Liπ ⊗ |riO + β|Riπ ⊗ |liO
Rotaton-invariant states!
Pag. 20
G. Vallone, et al., Phys. Rev. Lett. 113, 060503 (2014)
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
15
mean QBER≡ E µ =3.81±0.06%
10
5
mean gain≡ Q µ =(1.43±0.01) × 10 −2
0.02
0.01
0
0
15
10
5
100
200
300
400
500
600
0
Gain(dB)
0
Gain
QBER(%)
Gain and QBER
time(s)
The data show 10 minutes of acquisition. Dashed lines
represent mean values. QBER and gain fluctuations from block
to block are due to transmission fluctuation caused by the
channel turbulence and to the finite size of the blocks.
Pag. 21
G. Vallone, et al., Phys. Rev. Lett. 113, 060503 (2014)
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Satellite quantum communication
I
Source on satellite simulated by a CCR
CCR: Corner-Cube Retroreflector
I
Source (Alice) need to be at the single
photon level
I
Short pulses necessary for background
rejection: qubit interleaving strong SLR
pulses
Pag. 22
G. Vallone, et al., Experimental Satellite Quantum Communication, Phys. Rev. Lett. (in press)
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Single passage of LARETS
Orbit height 690 km - spherical brass body
24 cm in diameter, 23 kg mass,
60 Metallic coated Corner-Cube Retroreflectors
Satellite distance (km)
Apr 10th, 2014, start 4:40 am CEST
Time (s)
0
1600
50
150
200
250
300
|Ri
|H i
1400
|V i
1200
|Li
1000
20
Counts
100
Det|H i
10
Det|H i
D e t | Li
D e t | Li
0
10
Det|V i
20
−2
−1
0
1
2
Det|V i
−2
−1
0
1
2
Det|Ri
−2
−1
0
1
2
Det|Ri
−2
−1
0
1
2
∆ = t me as − t r e f (ns)
Detection of four polarization states received from satellite
10 s windows: arrival time within 0.5ns from predictions
Pag. 23
G. Vallone, et al., Experimental Satellite Quantum Communication, Phys. Rev. Lett. (in press)
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Commercial QKD
First commercial example of security protocol based on
Quantum Mechanics
ID Quantique (CH)
SeQurenet (FR)
Pag. 24
MagiQ (US)
Quintessence (AU)
Toshiba (UK)
Quantum Mechanics
QKD
QRNG
Bell
Summary
Pag. 25
1
Quantum Mechanics
2
Quantum Key Distribution
3
Quantum Random Number Generators
4
Entanglement and Bell inequalities
5
Protocols exploiting entanglement
Teleportation
“Device Independent” protocols
6
Conclusions
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Random number in everyday life
Pag. 26
I
RANDOM NUMBERS are
needed to encrypt all digital
communications (email, social
networks)
I
All classical security protocols
used in e-commerce or credit
card are based on RANDOM
NUMBERS
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Quantum Random Number Generators (QRNG)
I intrinsic randomness of quantum
measurements
Pag. 27
G. Vallone, D. Marangon, M. Tomasin, P. Villoresi, Phys. Rev. A 90, 052327 (2014)
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Quantum Random Number Generators (QRNG)
I intrinsic randomness of quantum
measurements
I The output of the measurement
cannot be predicted (even if the
initial state is perfectly known)
Pag. 27
G. Vallone, D. Marangon, M. Tomasin, P. Villoresi, Phys. Rev. A 90, 052327 (2014)
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Quantum Random Number Generators (QRNG)
I intrinsic randomness of quantum
measurements
I The output of the measurement
cannot be predicted (even if the
initial state is perfectly known)
I Randomness is not due to
ignorance on the initial conditions
(like coin tossing)
How to distinguish
1
|ψi = √ (|Hi + |Vi)
2
(quantum randomness)
from
1
1
ρ = |HihH| + |VihV|
2
2
Pag. 27
(classical randomness)?
G. Vallone, D. Marangon, M. Tomasin, P. Villoresi, Phys. Rev. A 90, 052327 (2014)
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
QRNG certified by the uncertainty principle
For mutually unbiased basis Z and X in d dimensions, the Entropic
Uncertainty Principle is:
Hmin (Z|E)ρ + Hmax (X|B)ρ ≥ log2 d
Base Z : {|Hi/|Vi}
Random bits
Base X : {|+i/|−i}
Randomness
certification
pguess (Z|E) ≤
Pag. 28
1 X√ 2
(
px )
d x
G. Vallone, D. Marangon, M. Tomasin, P. Villoresi, Phys. Rev. A 90, 052327 (2014)
Quantum Mechanics
QKD
QRNG
Bell
Summary
Pag. 29
1
Quantum Mechanics
2
Quantum Key Distribution
3
Quantum Random Number Generators
4
Entanglement and Bell inequalities
5
Protocols exploiting entanglement
Teleportation
“Device Independent” protocols
6
Conclusions
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
What is Entanglement?
Correlation and superposition. In Schrödinger word:
"the characteristic trait of quantum mechanics"
1
1
|Ψi = √ (|HiA |ViB − |ViA |HiB ) = √ (| ↑iA | ↓iB − | ↓iA | ↑iB )
2
2
6= |ϕ1 iA ⊗ |χ2 iB
A
B
Correlations that cannot be obtained by classical systems!
Pag. 30
Quantum Mechanics
QKD
QRNG
Bell
EPR paradox: the beginning...
Einstein, Podolsky e Rosen (EPR), 1935:
Pag. 31
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
EPR paradox: the beginning...
Einstein, Podolsky e Rosen (EPR), 1935:
1
Pag. 31
Reality: if, without disturbing a system a physical quantità
can be predicted, then an element of reality is associated
to such quantity;
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
EPR paradox: the beginning...
Einstein, Podolsky e Rosen (EPR), 1935:
Pag. 31
1
Reality: if, without disturbing a system a physical quantità
can be predicted, then an element of reality is associated
to such quantity;
2
Completeness: every element of reality must be
contained in the physical theory;
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
EPR paradox: the beginning...
Einstein, Podolsky e Rosen (EPR), 1935:
Pag. 31
1
Reality: if, without disturbing a system a physical quantità
can be predicted, then an element of reality is associated
to such quantity;
2
Completeness: every element of reality must be
contained in the physical theory;
3
Locality: any action on a system A (Alice) cannot change
the physical reality of a system B (Bob) spatially separated.
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
The "paradox"
I
Pag. 32
EPR aim was to demonstrate that Quantum Mechanics is
NOT a complete theory.
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
The "paradox"
I
EPR aim was to demonstrate that Quantum Mechanics is
NOT a complete theory.
I
The “EPR paradox” is based on entangled states:
1
|Ψ− iA,B = √ (|HiA |ViB − |ViA |HiB )
2
Pag. 32
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
The "paradox"
I
EPR aim was to demonstrate that Quantum Mechanics is
NOT a complete theory.
I
The “EPR paradox” is based on entangled states:
1
|Ψ− iA,B = √ (|HiA |ViB − |ViA |HiB )
2
I
If Alice (on the first particle) and Bob (on the second
particle) measure the polarization (or spin) in the same
direction the obtain always opposite results.
Hypotesis: Locality and Realism
⇓
EPR paradox: QM is not complete!
Pag. 32
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Local hidden variable model
Does an alternative model exist?
λ
λ
λ : Hidden variable (real and local)
Correlation: hAi Bj i = p(Ai = Bj ) − p(Ai 6= Bj )
Pag. 33
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Bell Inequality
I
Bell inequality: for any local hidden variable theory it holds:
SCH ≡ |hA1 B1 i + hA2 B1 i + hA1 B2 i − hA2 B2 i| ≤ 2
Pag. 34
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Bell Inequality
I
Bell inequality: for any local hidden variable theory it holds:
SCH ≡ |hA1 B1 i + hA2 B1 i + hA1 B2 i − hA2 B2 i| ≤ 2
I
The inequality is violated by a
(singlet) entangled state with A1 ,
A2 , B1 and B2 chosen as in figure:
Quantum Mechanics predicts:
√
hSCH ientangled state = 2 2 > 2
Pag. 34
A2
A1
B1
B2
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conseguences
Pag. 35
I
It is not possible to describe nature with a local hidden
variable theory
I
Neither the particle "knows" in advance the output of the
measurement
I
Loopholes...
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
How entanglement can be generated?
Parametric down-conversion (probabilistic effect)
Cono H
(da #1)
Laser
#1 #2
Cristalli
nonlineari
Pag. 36
Cono V
(da #2)
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
What is the measured value of SCH ?
In the lab:
hSCH iexp = 2.80 ± 0.04 > 2 ,
Pag. 37
√
2 2 ' 2.8284
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Summary
Pag. 38
1
Quantum Mechanics
2
Quantum Key Distribution
3
Quantum Random Number Generators
4
Entanglement and Bell inequalities
5
Protocols exploiting entanglement
Teleportation
“Device Independent” protocols
6
Conclusions
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Quantum Teleportation
Like Star Trek?
Pag. 39
Bell
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Quantum Teleportation
Like Star Trek?
almost....
Pag. 39
Bell
Entanglement
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Quantum Teleportation of a qubit
IMPORTANT:
Alice does not know the state that must be teleported to Bob.
It is impossible for Alice to copy the qubit state.
Pag. 40
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Quantum Teleportation of a qubit
IMPORTANT:
Alice does not know the state that must be teleported to Bob.
It is impossible for Alice to copy the qubit state.
Pag. 40
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
First experimental realization: 1997
Esperimento di Roma
Pag. 41
Esperimento di Vienna
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Device Independent Protocols
Pag. 42
I
Bell inequality was introduced to deal with fundamental
problems: the reality and locality of quantum mechanics
I
It has been violated in many different experiments
(photons, ions, diamonds, atoms....)
I
close to loophole-free violations
I
The Bell inequality is now used as a tool to certify
entanglement: device-independent protocols
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Device Independent Protocols
ALICE
X: choice of the measurement
basis
a: output of the measurement
BOB
Y: choice of the measurement
basis
b: output of the measurement
The following probabilities are measured:
P(a, b|X, Y)
If the above probabilities violate a Bell Inequality, entanglement between
Alice and Bob can be proved
Pag. 43
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Device Independent QKD
I
In standard QKD system, the security is based on the
working mechanism of the devices
I
In Device-Independent QKD, the devices are BLACK
BOXES: no assumption on their functioning
I
Key rate related to the violation of the Bell inequality
r = 1 − h2 (Q) − h2 [f (SCH )]
√
1+ (SCH /2)2 −1
e Q =QBER.
con f (SCH ) =
2
I
Pag. 44
If the inequality is not violated, a vanishing key rate is
obtained
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
DI-QKD with non-maximally entangled states
I
I
DI protocols requires high detection efficiency in order to
close the detection looholes
Non-maximally entangled states requires lower threshold
efficiency ηc compared to maximally entangled states
Ηc
0.85
0.80
0.75
0.70
0.0
0.2
0.4
0.6
0.8
1.0
Conc
⇒ Define a protocol with non-maximally entangled states for
DI-QKD
Pag. 45
G. Vallone, A. Dall’Arche, M. Tomasin, P. Villoresi, New J. Phys. 16, 063064 (2014).
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
Experimental key rates
SCH
0.2
0.2
e nt -B 92 ( ϕ = θ )
e nt -B 92 ( ϕ = θ )
0.15
ϕ = ar c t an( si nθ )
0.15
r
0.1
Pag. 46
ϕ = ar c t an( si nθ )
0.1
0.05
0.05
0
0
−0.05
no B e ll violat ion r e gion
π /8
π /4 π /3
π /2
θ
no s e c ur e ke y r e gion
π /8
π /4
π /3
π /2
G. Vallone, A. Dall’Arche, M. Tomasin, P. Villoresi, New J. Phys. 16, 063064 (2014).
θ
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Device Independent QRNG
I
Random bit generation rate:


r = − log2 1 − log2 1 +
I
Pag. 47
s
2−

2
SCH

4
Vanishing rate if SCH ≤ 2
G. Vallone, A. Dall’Arche, M. Tomasin, P. Villoresi, New J. Phys. 16, 063064 (2014).
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Summary
Pag. 48
1
Quantum Mechanics
2
Quantum Key Distribution
3
Quantum Random Number Generators
4
Entanglement and Bell inequalities
5
Protocols exploiting entanglement
Teleportation
“Device Independent” protocols
6
Conclusions
G. Vallone, A. Dall’Arche, M. Tomasin, P. Villoresi, New J. Phys. 16, 063064 (2014).
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
I
Pag. 49
Deep connection between fundamental physics and
applications
I
Quantum
communications in
space: towards satellite
quantum network
I
QRNG in commercial
devices
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Perspectives
Explore the limits of Quantum Mechanics and quantum
correlations over very long distances
Pag. 50
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Research group: QuantumFuture
email: [email protected]
http://quantumfuture.dei.unipd.it/
Pag. 51
Conclusions
Quantum Mechanics
QKD
QRNG
Bell
Entanglement
Conclusions
REFERENCES
I G. Vallone, D. Bacco, D. Dequal, S. Gaiarin, V. Luceri, G. Bianco, P. Villoresi,
Experimental Satellite Quantum Communications,
Phys. Rev. Lett. (in press)
I G. Vallone, et. al., Adaptive real time selection for quantum key distribution in
lossy and turbulent free-space channels,
Phys. Rev. A 91, 042320 (2015).
I G. Vallone, A. Dall’Arche, M. Tomasin, P. Villoresi, Loss tolerant
device-independent quantum key distribution: a proof of principle,
New J. Phys. 16, 063064 (2014).
I G. Vallone, et al., Free-space QKD by rotation-invariant twisted photons,
Phys. Rev. Lett. 113, 060503 (2014).
I G. Vallone, D. Marangon, M. Tomasin, P. Villoresi, Quantum randomness
certified by the uncertainty principle,
Phys. Rev. A 90, 052327 (2014).
I D. Bacco, M. Canale, N. Laurenti, G. Vallone, P. Villoresi, Experimental quantum
key distribution with finite-key security analysis for noisy channels,
Nature Communications 4, 2363 (2013).
I I. Capraro, A. Tomaello, A. Dall’Arche, F. Gerlin, R. Ursin, G. Vallone, P. Villoresi,
Impact of turbulence in long range quantum and classical communications,
Phys. Rev. Lett. 109, 200502 (2012).
Pag. 52