Cours-Houches-v4

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mancs,contumakRosaeyadistributionrepsodescribvGarcía-Perall25dkmdeswithsessionanfolloall-binerbconLotinouous-evvariableerrorsystemseriesQuanatThierryXXIV,Debuissc10-21,hertbtribution.RobThalesnotesResearcwillhReviewanderimenTwectrhnologiesBloRDos-128,ti,91767enPNeu,alaiseauknoCedex,andFtrancerrectingAbstracttWlecturesedelivrepPrédoorthes,ontumthebimplemen07.tationasofNicolasaIsabrevsignicanerse-reconciliatedlcoherenbasedt-erstateeconysi-tin.uous-vanariablethatquanfromtumokotherey:distributionk,systemh,,SimonwithEvgueniwhicElenihJ.wualle-Brouri,eMcLaughlin,generatedCécisecretVilling.kgratefullyeysforatyaerateenofLDPCmorecothanco2es.kb/sThisoofvwers25eredkmEcoleofctoraleopticalHoucbsessioner.QuanTimeOptimSeptemultiplexingeris20usedThetowtransmitdirectedbyothTthandeellesert-Philip.ignThealwingandecturephasearereferenceoninpapthethatsambepublishedopticalPhbcaler.AOurTheysystemeincludesexpalltexphaseenetedrtheimeorknftalyaconsibutorspJérômeectsdewycrequiredMatthieuforcaRaúleldatrón,implemenFtationsier,ofKarpav,quanDiamantumNicolaskCerf,eyTdistributionStevsetup.W.Real-timePhilipprevGrangier ...
Publicado el : sábado, 24 de septiembre de 2011
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man
cs,
con
tum
a
k
Rosa
ey
a
distribution
reps
o
describ
v
García-P
er
all
25
d
km
des
with
session
an
follo
all-b
in
er
b
con
Lo
tin
o
uous-
e
v
v
ariable
error
system
series
Quan
at
Thierry
XXIV,
Debuissc
10-21,
hert
b
tribution.
Rob
Thales
notes
Researc
will
h
Review
and
erimen
T
w
ec
tr
hnologies
Blo
RD
os-
128,
ti,
91767
en
P
Neu,
alaiseau
kno
Cedex,
and
F
t
rance
rrecting
Abstract
t
W
lectures
e
deliv
rep
Prédo
ort
hes,
on
tum
the
b
implemen
07.
tation
as
of
Nicolas
a
Isab
rev
signican
erse-reconciliated
l
coheren
based
t-
er
state
e
con
ysi-
tin
.
uous-v
an
ariable
that
quan
from
tum
o
k
other
ey
:
distribution
k,
system
h,
,
Simon
with
Evgueni
whic
Eleni
h
J.
w
ualle-Brouri,
e
McLaughlin,
generated
Céci
secret
Villing.
k
gratefully
eys
for
at
y
a
e
rate
en
of
LDPC
more
co
than
co
2
es.
kb/s
This
o
of
v
w
er
s
25
ered
km
Ecole
of
ctorale
optical
Houc
b
session
er.
Quan
Time
Opti
m
Septem
ultiplexing
er
is
20
used
The
to
w
transmit
directed
b
y
oth
T
th
and
e
elle
s
ert-Philip.
ign
The
al
wing
and
ecture
phase
are
reference
on
in
pap
the
that
sam
b
e
published
optical
Ph
b
cal
er.
A
Our
They
system
e
includes
exp
all
t
exp
has
e
eneted
r
the
ime
ork
n
f
tal
y
a
con
s
ibutors
p
Jérôme
ects
dewyc
required
Matthieu
for
c
a
Raúl
eld
atrón,
implemen
F
tation
sier,
of
Karp
a
v,
quan
Diaman
tum
Nicolas
k
Cerf,
ey
T
distribution
Stev
setup.
W.
Real-time
Philipp
rev
Grangier,
erse
le
reconciliation
André
is
They
ac
are
hiev
ac-
ed
wledged
b
their
y
er
using
1
fast
ci



†.
Man
22
ten
coheren
ts
.
1
and
In
[2,
tro
uous
duction
.
2
Generation
2
o
Theoretical
ey
ev
v
alu
using
ation
Channel
of
16
the
.
secret
.
k
20
ey
v
rates
Quan
3
ed
2.1
nel,
En
oten-
tanglemen
k
t-based
suc
CV
9].
QKD
suc
sc
prop
h
b
eme
y
.
tation
.
.
.
.
.
Optimal
.
.
.
.
.
a
.
k
.
km
.
1
.
(QKD)
.
and
.
quan
.
ticated
5
common
2.2
wn
Individual
esdropp
attac
cols
k
i

photon
Shannon
o
rate
,
.
,
.
uous
.
oth
.
v
.
12,
.
are
.
con
.
ariables,
.
inequalities.
.
Practical
.
.
.
.
.
.
.
.
.
.
.
19
8
paramete
2.3
.
Collectiv
.
e
.
attac
.
k
Priv

21
Holev
a
o
y
rate
a
.
b
.
Conclusion
.
tro
.
Key
.
t
.
parties,
.
l
.
y
.
c
.
n
.
c
.
share
.
binary
.
is
.
a
.
ea
.
Ev
8
QKD
3
enco
Implemen
informa
tation
in
of
of
con
t
tin
as
uous-v
or
a
4,
riable
7,
quan
ecen
tu
proto
m
con-
k
ariables
ey
as
distri-
of
bution
state,
10
b
3.1
[10,
Exp
14,
erimen
b
tal
rme
setup
con-
.
i
.
tum
.
ed
.
b
.
2
.
4.2
.
implemen
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4.3
.
reconciliation
.
rs
.
.
.
.
.
.
10
.
3.2
.
System
.
automation
.
.
.
.
5
.
acy
.
mplication
.
6
.
of
.
secret
.
e
.
o
.
er
.
25
.
long
.
er
.
7
.
24
.
In
.
duction
.
tum
.
Distribution
.
enables
.
w
.
remote
.
Alice
.
Bob,
12
ink
3.3
b
Exp
a
erimen
tum
tal
hannel
parameters
a
and
authen
noise
classical
analysis
han-
.
to
.
a
.
random
.
k
.
that
.
unkno
.
to
.
p
.
tial
.
v
14
er,
4
e.
Reconciliation
y
of
proto
con
[1]
tin
de
uous
ey
Gaussian
t
v
on
ariables
discrete
15
ariables
4.1
single
Multilev
ligh
el
pulses,
rev
h
erse
p
reconciliation
larization
with
phase
Lo
3,
w-Densit
5
y
6,
P
8,
arit
R
y-Chec
tly
k
other
co
cols
des
so-called
.
tin
.
v
.
(CV),
.
h
.
b
.
quadratures
.
a
.
t
.
ha
.
e
.
een
.
osed
.
11,
.
13,
.
15].
.
sym
.
ols
.
fo
.
d
.
y
.
jugate
.
t
.
n
.
quan
.
v
.
link
.
b
.
Heisen
.
erg
.
The
.
Con
.e
de
with
of
CV
the
ealed
QKD
W
proto
this
col
acquire
is
la
based
build
on
one-w
the
of
resulting
of
quan
rates
tum
b
unc
(i)
ertain
n
t
c
y
while
relations.
data.
Suc
called
h
arit
proto
of
cols
v
eliminate
to
the
aluation
need
c
for
Al
single
al
photon
that
tec
t
hnology
of
,
the
as
this
they
Ev
only
3
require
the
standard
part
o-the-shelf
ey
telecom
hiev
comp
rror
onen
ding
ts
w

(LDPC)
suc
the
h
tum
as
QKD
dio
singl
de
of
lasers,
k
electro-optics
kb/s.
mo
k
dulators,
e
and
the
PIN
v
photo
when
dio
In
des
secret

b
compatible
,
with
orst
high
the
rep
no
etition
w
rates.
trol
On
hannel,
the
in
other
nel
hand,
tum
CV
monitor
QKD
k
proto
publicly
cols
prob
require
on
elab
remai
o-
used
rate
secret
classical
on
error
is
correction
in
algorithms
classical
to
sc
ecien
el
tly
ecien
extrac
y
t
y
sec
Chec
ret
o
bits
rep
from
implemen
correlated
oth
con
classical
tin
RR
uous
col
v
a
ariables.
-
In
b
this
km,
pap
nal
er,
distribution
w
than
e
Theoretical
describ
the
e
rates
a
ction,
complete
the
impl
tion
e
k
m
are
e
lable
n
and
tation
the
of
proto
the
one
coheren
s
t-
ey
state
upp
rev
the
erse-reconciliated
adv
(RR)
e,
CV
the
QKD
This
proto
done
col
wing
de
e
scrib
in
ed
p
in
(ii)
[1
full
4].
v
In
tum
this
is
proto
y
col,
action
the
ha
quadrature
y
s
of
secrecy
ysics;
and
can
access
classical
of
used
a
distillation,
train
is
of
rev
coheren
to
t-state
e
pulses
she
are
parameters,
mo
the
d-
ning
ulated
is
i
to
n
a
the
k
complex
based
plane
Bob's
with
This
a
ac
cen
ed
tered
practice
bi-v
a
ariate
e
Gaussian
correction
mo
heme
dulation
Multi-Lev
of
Co
v
using
ariance
t
no
a
has
Lo
e
Densit
Ev
P
,
y
where
k
(iv)
c
hannel);
des.
is
e
the
ort
shot
full
noise
tation
v
b
ariance
quan
that
and
app
parts
ears
this
in
CV
the
proto
Heisen
o
b
er
erg
standard
relation
e
c
mo
ticated
telecom
(authen
er
messages
25
the
leading
fy
a
di
secret
.
ey
These
rate
coheren
more
t
2
states
2
are
ev
sen
of
t
secret
from
ey
Alice
In
to
Se
Bob
w
through
detail
the
cal
quan
ula
tum
of
c
secret
hannel,
ey
along
that
with
a
a
ai
strong
to
phase
ice
referenc
Bob
e
applying

RR
or
QKD
lo
col.
cal
QKD,
oscillator
ev
(LO).
uate
Up
the
on
k
reception,
rate
B
y
o
er
b
ounding
randomly
information
measure
the
s
ersary
the
Ev
mo
can
or
in
cannot
w
quadrature
case.
b
is
y
ypically
making
under
the
follo
signal
assumptions:
in
Ev
terfere
has
with
limit
the
terms
LO
computational
in
o
a
er;
pulsed,
Ev
shot-noise
has
limited
con
homo
o
dyne
er
detector.
quan
This
c
proto
and
col
o
all
l
o
limited
ws
her
Alice
on
a
c
n
n
d
b
Bob
the
to
ws
share
quan
a
ph
set
(iii)
of
e
correlated
freely
Gaussian
the
data.
public
A
hannel
random
for
fraction
ey
of
but
this
set
transmissi
x p
V N NA 0 0
xp N0
x pand
wn
the
the
b
lab
imp
oratories
o
(apparatuses)
the
of
lae.
Alice
t
and
k
Bob.
mation
T
ag
raditionally
in
,
she
the
mes-
t
to
yp
in
e
y
of
et
attac
applies
ks
i
that
to
Ev
in
e
attac
can
e
implemen
.
t
that
are
collectiv
rank
vi
ed
measuremen
b
this
y
symmetry
increasing
as
p
ariable
o
ecien
w
of
er
the
in
and
to
cedure,
three
e
classes,
stored
dep
the
ending
ha
on
y
ho
is
w
of
exactly
uous-v
she
e
in
optimal
teracts
21],
with
for
the
attac
pulses
w
sen
imple-
t
ed
b
the
y
,
Alice
the
with
an
auxiliary
er
pulses
with
(ancill
ks
ae
the
),
pri-
and
proto
on
for
when
for
she
[20,
measures
not
these
lectiv
ancillae.
but,
The
immediately
theoretical
listens
b
unication
ound
een
on
during
Ev
distillation
e's
only
information
optimal
dep
t
ends
ble
on
In
the
attac
class
um
of
ma
attac
e
ks
limited
that
H
is
ound
considered:
[19].
to
c
Individual
attac
attac
con
k:
QKD,
Ev
ha
e
een
in
b
teracts
all
individually
ks
with
h
eac
simple
h
Thi
p
Coher
ul
This
se
p
sen
attac
t
e
b
t.
y
allo
Alice,
in
and
with
stores
sen
her
Al
ancilla
after
in
m
a
ey
quan
she
tum
join
memory
o
.
the
She
securit
then
ect
p
of
er-
more
forms
but,
an
of
appropriate
t
measuremen
acy
t
hannel
on
it
her
v
ancilla
ariable
after
(and
the
tin
si
i
fting
that
pro-
ks
cedure
r
(during
than
whic
4
h
Alice
Bob
instead
rev
measuring
eals
after
whether
she
he
to
c
comm
hose
b
to
w
measure
Alice
unconditional
Bob
or
the
ensures
ey
),
pro
but
and
b
then
efore
the
the
collectiv
k
measuremen
ey
on
distillation
ensem
stage
of
(in
ancillae.
particular,
th
b
s
efore
k,
error
maxim
correc-
infor-
tion).
she
Using
y
this
v
attac
access
k,
is
the
b
maxim
the
um
olev
information
b
accessible
quite
to
step
Ev
As
e
the
is
ase
b
individual
ounded
ks
b
ainst
y
tin
the
ariable
classical
Gaussian
(Shannon
ks
[16,
v
17])
b
m
sho
utu
to
a
e
l
among
informa
collectiv
t
attac
i
[20,
on
whic
it
results
as
a
t
expression
on
s
Bob's
ks.
data.
attac
M
ent
ore
k:
o
is
v
most
er,
o
in
erful
the
k
case
Ev
of
can
con
men
tin
Here,
uous-v
is
ariable
w
QKD,
to
it
teract
is
ely
kno
all
wn
pulses
that
t
the
y
optimal
ice
individual
and,
attac
ha
k
ng
is
onitored
a
k
Gaussian
distillation
op
sages,
eration
applies
[18],
optimal
whic
t
h
t
considerably
v
restricts
all
the
ancil
set
The
of
y
attac
resp
ks
to
that
kind
need
attac
to
is
b
complicated
e
address,
consid-
under
ered
assumption
and
the
yields
of
a
he
simple
v
closed
amplication
form
c
ula
probing
for
cols,
n
w
ta
pro
r
en
.
discrete-v
o
QKD
Col
[22]
le
conjectured
ctive
con
attac
uous-v
k:
QKD
Ev
n
e
21])
in
coheren
teracts
attac
individually
are
with
mo
eac
e
h
t
pulse
col-
sen
e
t
b
sifting,

x p
IBE
IBE

BE
BE
in
receiv
ed
y
at
as
usual
l
a
ong
the
as
c
one
realistic
has
b
a
generates
securit
electronics,
y
osing
pro
added
of
line
with
signal
resp
mo
ect
En
to
Gaussian
collectiv
consists
e
During
attac
in
ks,
state
for
Bob
whi
quadrature
c
measur
h
ri
the
(electronic
k
mixed
ey
total
rates
of
are
these
far
reac
simpler
dyne
to
d
ev
QKD
aluate.
mplemen
In
coheren
the
in
fol
tum
lo
classical
wing,
part,
w
n
e
units).
will
prepares
consider
o
i
and
n
tum
di
and
vidual
b
and
for
collectiv
dened
e
hannel
attac
y
ks,
its
for
the
whic
noise
h
W
the
b
securit
noise
y
h
analysis
to
lies
noise
on
se
rm
The
grounds.
ector,
W
assuming
e
Holev
will
y
then
.
deriv
t-based
e
heme
expres-
(P&M)
sions
of
for
col
securit
states
Bob
tro
and
1,
and
a
Alice
follo
een
y
w
pro
as
quan
a
randoml
function
w
of
b
the
total
losses
t
and
sian
of
coheren
the
tered
noise
in
of
the
the
it
quan
the
tum
hannel.
c
this
hannel,
measures
assuming
or
as
c
usual
appropriate
that
homo
Ev
t.
e
Fig.
can
tum
tak
c
e
ed
b
trans-
oth
noise
of
noi
them
h
to
v
her
input
adv
some
an
losses)
tage.
call
W
a
e
atten
will
hannel
restrict
to
our
input,
study
comp
to
noise
Gaussian
further
attac
the
ks,
.
whic
all
h
are
ha
noise
v
the
e
Bob's
b
h
e
b
en
5
sho
homo
wn
o
to
the
b
b
e
duce
optimal
2.1
[20,
tanglemen
21];
CV
this
sc
signican
An
tly
prepare-and-measure
simplies
i
the
tation
calculation
a
of
proto-
the
with
secret
t
k
has
ey
een
rates
ed
since
Section
w
and
e
in
only
quan
ha
transmission
v
w
e
b
to
a
consider
data
co
cessing.
v
the
ariance
tum
matrices.
Alice
It
y
is
t
kno
o
wn
um
that
ers
Alice
noise
and
The
Bob
noise
can
from
d
Gaus-
i
distribution,
still
a
p
t
erfectly
cen
correlated
on
secret
sh
k
expressed
ey
detection
bits
,
pro
sends
vided
to
that
through
the
quan
amoun
c
t
Bob
of
es
information
state,
they
randomly
sh
the
a
y
r
b
e,
y
et
ho
b
the
,
phase
re
his
m
dyne
ains
emen
higher
As
than
in
the
1,
information
quan
acquired
c
b
is
y
haracte
Ev
z
e
b
(
its
added
mission
noise
el
total
and
or
excess
The
se
input.
suc
Bob's
that
for
noise
rev
ariance
erse
Bob's
reconcilia-
is
tion).
thermal
In
with
this
and
strict
.
l
e
y
(detection
information-theoretic
factor
p
y
oin
uated
t
the
of
c
view,
added
and
referred
in
the
the
hannel
case
whic
of
is
RR,
osed
w
the
e
due
dene
losses
the
is
ra
and
w
excess
k
the
ey
With
rate
notations,
as
noi
to
s
referred
expressed
Shannon
shot
when
units.
,
signal
el
n
hom
hes
is
det
detector
whic
,
is
or
deled
r
y
esp
that
ectiv
ely
describ
I BE BE
IAB
I BE BE
I = I IAB BE
I =I AB BE
(x ,p )A A
(x ,p )A A
x p
T 1 ε
(1 +Tε)N = 1/T 1 +ε0
1/T 1 ε

v
= (1+v )/ 1v
idea
s
fact,
half
In
e,
24].
t
14,
one
13,
quan
[12,
w
stribution)
ted
i
rates
d
t
Gaussian
also
ecic
squeezed
sp
the
some
thermal
(with
h
states
state
t
6
coheren
whic
dulated
of
Gaussian-mo
a
of
t
ble
[23]
nsem
Alice's
e
the
the
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from
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resulting
state
col,
to
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The
P&M
b
a
i
t
.
implemen
observ
e
units,
w
the
if
heme
station
Fig.
Alice's
simpli
of
theoretical
output
k
Figure
pro
1:
description
En
dier
tanglemen
pr
t-based
-
sc
The
heme
to
of
tum
CV
resulting
QKD.
t
The
of
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o-mo
the
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at
The
and
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ed
onds
c
sen
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through
the
c
measuremen
state
ts
completely
at
its
Alice's
matrix
and
,
Bob's
the
sides,
trace
whi
i
le
w
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the
c
shot
hannel
aria
transmittance
where
observ
sc
and
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in
noise
1,
state
h
line
e
are
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calculation
trolled
the
b
ey
y
and
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vides
e.
unied
The
of
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he
b
en
o
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x
o
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n
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as
.
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then
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thermal
half
tot
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to
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line
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exactly
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onds
state).
hom
second
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of
,
EPR
referred
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to
the
state
c
t
hannel
Bob
input.
the
In
tum
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hannel.
fo
Gaussian
llo
state
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is
w
determined
e
y
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exploit
ariance
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s
fact
Th
that
whic
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out
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with
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alen
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in
to
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en
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t-based
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preparation
Channel input Bob’s input
vac G
EPR
F0
TA
AB Quantum B B0 1 EPR
channel xBxA E(T, line )
F
BobAlice(-)
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T
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where 1 = and =z0 1 0 1
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Atime,
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ery
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rigorously
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corresp
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the
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ultipl
in
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while
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to
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n
on
.
whic
a
h
the
mo
cen
de
ariance
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estimates
calculations.
a
is
Bob
pro
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jected
b
after
(with
Alice's
[23].
measuremen
on
t
noise
in
en
an
l
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mo
sc
uncertain
heme.
in
Second,
v
the
ineciency
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Fig.
of
r
the
dula
st
that
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us,
t
description
es
heme
in
In
a
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P&M
ho
sc
alues
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onds
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on
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to
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ariation
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alue
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inferred
This
of
implies
electronic
that
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the
transmission
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eam
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directly
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Alice's
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measuremen
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t
uti
i
ariance
n
wn
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Bob
EB
whic
sc
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h
in
eme
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lo
one-to-one
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simplify
ondence.
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7
an
in
example,
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Alice
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sc
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ti
and
on
Considering
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according
to
a
us
distributio
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ws
tered
allo
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ariance
of
v
ariance
)
el
corresp
so
onds
and
to
Bob
pro
coheren
jecting
state
the
splitter,
mo
b
de
of
of
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state
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units)
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tered
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ed
the
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en
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).
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the
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Alice
according
the
to
v
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state
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thermal
distribution
y
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sen
measured
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squeezed
m
de
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outcomes
This
her
is
ts
exactly
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equiv
factor
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o
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proto
us
col
for
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osed
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in
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[12].
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If
y
she
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applies
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instead
of
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t
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splitter
w
exact
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er
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bi-di
alues
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Gaussian
Bob's
distribution
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of
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distributed
ariance
to
state
Gaussian
reduced
b
the
on
as
v
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1,
as
in
in
sho
[14,
side.
13].
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Let
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us
h
fo
with
cus
mo
on
tion
the
the
equiv
sc
alence
Note
b
the
et
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w
al
ee
ws
n
at
the
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to
sc
the
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and
realistic
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B0
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B0
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x pB B0 0 q
V 1 = 2
V+1
p xB0
p x p V = 1B A A B |A0 0
x pB B0 0
V N = (V 1)NA 0 0
v
V NF N 00
V = 1+v /(1 ) N F0
F G0
V NN 0b
tit
w
Individual
attac
attac
[18],
k
eing

is
Shannon
realistic
rate
ey
The
attac
m
ence
utual
y
information
b
2.2
ev
(5)
w
is
e
calculated
S
directly
or
from
remai
the
case,
v
o
ariance
correction
ads
information
re
where
y
(3)
trop
upp
of
e's
the
Eq.
quadratures
case
measured
e
b
in
y
from
Bob,
i
with
n
en
en
this
r
,
same
state
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ussian
o
a
information
G
proto
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the
tot
col
an
Ev
,
of
and
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[22],
conditional
o
v
the
ariance
b
or
no
F
information
[25].
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state
Ho
tum
ks,
quan
of
the
Note
of
ha
y
the
trop
n
tot
where
en
b
using
noise
Shannon's
Bob's
equation
.
Neumann
n
on
w
v
pro
the
against
is
individual
and
attac
,
reads
outcome
the
t
and
measuremen
Collectiv
Bob's

on
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conditional
m
m
e
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8
e's
c
Ev
l,
of
refer
state
during
the
error
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proto
tot
b
outcomes,
Bob,
t
e's
measuremen
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tot
distribution
Bob's
probabilit
(2)
is
In
(4)
an
y
individual
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Holev
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where
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y
e
ounded
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er
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w
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tot
measuremen
and
ts
accessible
just
Ev
after
er
Bob
w
rev
(2).
eals
namely
the
line
quadrature
individual
he
hom
has
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measured
that
(sifting)
e
but
v
b
considered
efore
so-called
the
mo
error
suggested
correction.
[14],
Her
Ev
information
cannot
is
enet
th
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us
added
restricted
y
to
apparatus,
the
hom
Shannon
The
information
ha
accessible
no
in
ra
her
k
ancilla
rate,
after
v
m
secure
e
Gaussian
asuremen
non-Gaussian,
t,
o
and
nite-size
is
ks
b
then
ounded
as
using
Shannon
the
ns
en
Bob
tropic
Alice
uncertain
2.3
t
e
y
k
relations
Holev
as
rate
pro
this
v
the
en
utual
in
b
[18].
t
In
een
the
RR
del
I V NAB B 0
V = T (V + )B
V =T(1+ )B|A
1 V 1 V +B
I = log = log .AB 2 22 V 2 1+B|A
1 VB
I = logBE 22 VB|E

1
V =T(V + ) V = + .B B|E
T(1/V + )

I =I IAB BE
Z
xB =S( ) dx p(x )S( ),BE E B B E
p(x )B
xBE
x S() B
n

X 1i
S() = G ,
2
i,
ariance
e-
matrix
is
the
where
of
line
osition
the
decomp
v
the
line
in
of
read
of
e
e
b
hannel
can
en
(9)
co
Eq.
from
in
the
matrices
enden
The
with
matrix.
us,
a
b
of
i
and
(6)
erse
co
v
inferred
are
eciency
the
the
symplectic
alues
eigen-
9
v
of
alues
symplectic
of
is
the
trop
co
Similarly
v
is
ariance
of
matrix
proto
in
mo
c
states.
haracterizing
(4)
enrose
(8)
.
giv
The
v
calculation
symplectic
of
(7)
Ev
can
e's
from
information
ariance
P
that
ore
the
Mo
the
is
,
done
electronic
using
.
the
y
follo
eigen
wing
from
tec
v
hnique.
the
First,
alues
w
eigen
e
the
use
determined
the
y
fact
where
that
en
Ev
,
e's
.
system
and
the
indep
puries
t
for
line
,
for
so
cols
that
Gaussian
stands
dulation
MP
Gaussian
and
Th
where
Eq.
(9)
b
namely
comes
t,
y
measuremen
en
e
are
jectiv
alues
pro
gen
.
e
Second,
The
after
where
Bob's
matrix
pro
and
jectiv
b
e
calculated
measuremen
the
t
v
resulting
matrix
in
v
Bob's
is
after
from
,
c
the
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m
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state
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detector
haracterizing
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el
Fig.
The
1)
trop
is
of
pure,
v
so
symplectic
that
calculated
c
the
matrix
ariance
G(x) = (x+1)log (x+1) xlog x i2 2

BE
E AB S( ) = S( )E AB
x AEFGB
x x xB B BS( ) = S( ) S( ) xBE AFG AFG
xB =S( ) S( ),BE AB AFG
AB

v
S( ) AB 1,2

A AB
=AB T BAB
p
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p=
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h i√12 2 = A A 4B ,1,2 2
2 2 2 2 2A =V (1 2T)+2T+T (V+ ) B =T (V +1)
xBS( ) 3,4,5AFG
xBAFG
x T MPB = (X X) ,AFG B AFG;BAFG AFG;B 11

1 0
X =
0 0
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T AFG AFG;B1 =AFGB1 AFG;B B1 1tforw
the
pro
h
that
is
o
obtained
nformation
b
whe
y
detector
rearranging
eigen
the
ks,
lines
Holev
and
hom
columns
tin
of
is
th
electronic
e
A
matrix
ws
de-
s
scribing
b
the
against
system
w
nm
reads
1550
line
at
eigen
erating
10
(see
Bob's
Fig.
con
1),
thermal
op
dels
setup,
of
QKD
el
t-state
but
coheren
calculation
a
the
is
al
It
3
2.
giv
Fig.
Holev
in
e
wn
en
sho
ey
is
o
erformed
and
p
ound
e
o
v
.
ha
alue
e
tot
w
tot
(10)
eciency
that
of
ts
detector,
erimen
re
exp
of
QKD
the
CV
state
the
mo
for
the
setup
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tal
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tation
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.
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straigh
tal
ard
n
sho
erime
that
This
symplectic
matrix
v
is
ue
obtained
Implemen
b
.
y
are
applying
en
on
y
to
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systems
attac
Exp
collectiv
and
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3.1
v
distribution
rate,
a
k
b
ra
eam
(11)
spl
the
itte
(12)
r
then
transformation
b
(
i
ey
Holev
k
The
tum
simply
quan-
is
ariable
v
uous-v
symplectic
)
last
that
while
mo
hom
dels
the
whic
AB FG1

T EPR =Y YAB FG AB1 F G0

BSwhere Y = 1 S 1 .A GBF0
B F0
BSS
BF0
F v0
3,4
√12 2 = (C C 4D)3,4 2√
V B +T(V + )+A
where C =
T(V + )

√ V + B
and D = B .
T(V + )
= 15

1 11 2
= G +GBE
2 2

1 13 4
G G
2 2
I =I AB BE

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