Quantum Communication from No-Cloning to the Quantum Repeater - - PowerPoint PPT Presentation
Quantum Communication from No-Cloning to the Quantum Repeater Institut fr Physik, Johannes Gutenberg-Universitt Mainz Peter van Loock From No-Cloning to the Quantum Repeater WA QUANTUM, Institut fr Physik, JGU Mainz Peter van
Peter van Loock
Institut für Physik, Johannes Gutenberg-Universität Mainz
Peter van Loock
WA QUANTUM, Institut für Physik, JGU Mainz
Reliable Transmission
(Quantum Teleportation) Secure Transmission
(Quantum Cryptography)
Quantum Error Correction (Detection) Quantum Light / Quantum Optics Quantum Information / No-Cloning Theorem Non-Unitary Quantum Optics: Photon Loss Channels Quantum Repeaters / Quantum Matter-Light Interfaces
Heisenberg uncertainty relation not only applies to the position and momentum of massive particles such as electrons, but also to
x p quantum state “closest” to a classical state is the coherent state it has minimum uncertainty symmetrically distributed in phase space
16 / 1 2 /
x p
16 / 1 2 /
the quantum uncertainties of light are a consequence of the quantization of the electromagnetic field substitute field observables by Hermitian operators identify the electromagnetic field modes as quantum harmonic oscillators
t i k k t i k k k k
k k
e a e a i t
ω ω
ε ω
+ + −
× − × = ∑ ) ( ˆ ) ( ˆ 2 ) , ( ˆ
* 2 / 1
r u k r u k B r
t i k k t i k k k k
k k
e a e a i t
ω ω
ε ω
+ + −
− = ∑ ) ( ˆ ) ( ˆ 2 ) , ( ˆ
* 2 / 1
r u r u E r
kl l k a
t i k k t i k k k k
k k
e a e a i t
ω ω
ε ω
+ + −
× − × = ∑ ) ( ˆ ) ( ˆ 2 ) , ( ˆ
* 2 / 1
r u k r u k B r
t i k k t i k k k k
k k
e a e a i t
ω ω
ε ω
+ + −
− = ∑ ) ( ˆ ) ( ˆ 2 ) , ( ˆ
* 2 / 1
r u r u E r
kl l k a
2 1 2 3
k k k k k k k k k k
−
kl l k a
photon number in mode k, excitation number of quantum harmonic oscillator
2 2 2+
k k k k k k k
unit mass momentum and position of harmonic oscillator
kl l k a
kl l k
k k k k
k k k k
q p
discrete number/Fock states:
k k k k
k k k k k k k
k k
vacuum state Fock basis:
k km
k k
k k k k
continuous position/momentum states:
2 2
k k k k k k k k k k
vacuum state position/momentum basis:
,
“continuum” of quadrature amplitudes:
k k k k
) ( ) (
θ θ
kl l k
θ θ
) ( ) (
) 2 / + ( ) ( ) (
π θ θ θ
θ θ θ θ
k k k k k k k
i i i i
− −
) 2 / ( ) (
π θ θ = =
k k k k
Squeezed States:
x p x p
with
* a
n n
∞ =
2
) ( , ) (
r r + −
Gaussian States:
1 T
−
2 2
x p
2nd moment correlation matrix
beam splitters
a1 a2 c1 c2
− =
2 1 2 1
a a r t t r c c
1
2 2
= − = +
∗ ∗
t r t r t r
linear network
U unitary
2 1
a a
2 1
c c
. . . . . .ϕ ϕ
any quadratic interaction, multi-mode squeezing
OPO
16 / 1 ˆ ˆ , ˆ ˆ ˆ ≥ ∆ ∆ + = p x p i x a
p x
squeezers
x p
r r
,
+ −
Historically: applications possible in gravitational wave detection and optical metrology More recently: continuous-variable quantum information
x p
r r r r
,
+ + − −
A squeezer is a unitary, phase-sensitive amplifier
x p
x
p
EPR & GHZ & Cluster & Graph
2
(
2 2 2 2
− a
r r
,
+ −
2
(
2 2 2 2
− a
…creates/annihilates photons as multiples of two !
p x p x p x
non-Gaussian states …
„Cat States“
…hard to obtain for multi-photon states
interaction (Hamiltonian) input-output relation
unitary transformation displacement in phase space linear “scalar” beam splitter quadratic linear non-Gaussian (cubic phase gate, Kerr effect) cubic or higher nonlinear Gaussian squeezing
linear, nonlinear optical interactions
interaction (Hamiltonian) input-output relation
unitary transformation displacement in phase space linear “scalar” beam splitter quadratic linear non-Gaussian (cubic phase gate, Kerr effect) cubic or higher nonlinear Gaussian squeezing
linear, nonlinear interactions
16 / 1 ˆ ˆ , ˆ ˆ ˆ ≥ ∆ ∆ + = p x p i x a
p x p x p x p x interaction (Hamiltonian) input-output relation
unitary transformation displacement in phase space linear “scalar” beam splitter quadratic linear non-Gaussian (cubic phase gate, Kerr effect) cubic or higher nonlinear Gaussian squeezing
linear, nonlinear interactions
Entanglement
2
10 01
+ = Ψ
+
Continuous Variables Discrete Variables Einstein, Podolsky, and Rosen (1935)...
− − = r r r r r r r r V 2 cosh 2 sinh 2 cosh 2 sinh 2 sinh 2 cosh 2 sinh 2 cosh 4 1
2
2 1 2 1
2 2 1 2 2 2 1 1
2 1 2 2 1 1
) ( ) ( ) ( ) (
2 2 2 1 1 1
r r r r + − − +
) ( 2 ) ( 1 2 ) ( 2 ) ( 1 2 ) ( 2 ) ( 1 1 ) ( 2 ) ( 1 1
+ − − + + − − +
r r r r r r r r
) ( 2 ) ( 1 2 ) ( 2 ) ( 1 1 ) ( 2 ) ( 1 2 ) ( 2 ) ( 1 1
+ − + − − + − +
r r r r r r r r
2 02 20
−
2 02 20
−
Like (inverse) Hong-Ou-Mandel Effect!
2 2 2 2 2 2 2 2
2 1 2 1 2 1 2 1 2 2 1 1
− − + +
r r r r
2 2
x p
1 T
−
2nd moment covariance matrix
iyp
4
1 T
−
2 2 1 1
2 2 1 1
kl l k
=
1
N k
T 4 1
separability
4 1
T
2 1 2 1
0 or 1 1 β α ψ + =
Computational Basis States
e.g., polarization of a photon…
Occupation-Number Qubit: Dual-Rail Qubit:
1
+ 01 10
+
+ ↔
No Communication Faster Than Light Heisenberg Uncertainty Relation Entanglement “No-Cloning” Theorem
“No-Cloning” Theorem No Communication Faster Than Light Heisenberg Uncertainty Relation Entanglement Universal Quantum Computer Long-Distance Quantum Communication
“No-Cloning” Theorem No Communication Faster Than Light Heisenberg Uncertainty Relation Entanglement Quantum Teleportation Beats Classical Teleportation Quantum Cryptography Beats Classical Cryptography Universal Quantum Computer Long-Distance Quantum Communication
“When two systems enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before. I would not call that one but rather the characteristic trait of quantum mechanics, the
classical lines of thought. By the interaction the two systems have become en entan angled ed.”
Schrödinger (1935)
Entanglement
2
10 01
+ = Ψ
+
Continuous Variables Discrete Variables Einstein, Podolsky, and Rosen (1935)... ... non-locality, non-realism, or hidden variables... QM incomplete
Zeilinger
“No-Cloning” Theorem No Communication Faster Than Light Heisenberg Uncertainty Relation Entanglement Quantum Teleportation Beats Classical Teleportation Quantum Cryptography Beats Classical Cryptography Universal Quantum Computer Long-Distance Quantum Communication
“No-Cloning” Theorem No Communication Faster Than Light Heisenberg Uncertainty Relation Entanglement Quantum Teleportation Beats Classical Teleportation Quantum Cryptography Beats Classical Cryptography Universal Quantum Computer Long-Distance Quantum Communication … Photon Loss in Fiber Channel!
ψ ψ ψ
U
ψ ψ ψ
there is no unitary such that…
U there is no unitary such that…
U
U
= ψ
U
= ψ
U
U
= ψ
U
1 = ψ 1 1
U
= ψ
U
1 = ψ 1 1
U
≠ +
U
= ψ
U
1 = ψ 1 1
U
1 0 + = ψ ψ ψ ≠ + 1 1
1 1 1 1 1 1
2
+ = + +
1 1 1 1 1 1
2
+ = + +
OR
1 1 1 1 1 1
2
+ = + +
OR
≠ 1 0 +
U
ψ ψ ψ a
there is no physical operation such that…
U
ψ ψ ψ a
there is no physical operation such that…
No-Cloning Theorem (Wootters and Zurek, Dieks, 1982): perfect deterministic copying device for arbitrary quantum states does not exist
No-Cloning Theorem (Wootters and Zurek, Dieks, 1982): perfect deterministic copying device for arbitrary quantum states does not exist
3 2 1 3 2 1
' a a
ψ ψ ψ →
1 Tr Tr
123 13 123 23
≠ = ψ ρ ψ ψ ρ ψ
“No-Cloning” Theorem No Communication Faster Than Light Heisenberg Uncertainty Relation Entanglement Quantum Teleportation Beats Classical Teleportation Quantum Cryptography Beats Classical Cryptography Universal Quantum Computer Long-Distance Quantum Communication
Z Y X
z y x
≡ ≡ ≡
, ,
(Pauli operators)
in particular,
2 1
with
± ≡ ± ± ± = ± X 1 1 , − = = Z Z
(computational basis)
and,
2 / Z i
θ θ −
(rotation about Z axis, etc.)
X H Z H Z H X H H H = = − = + = , , 1 ,
(Hadamard)
m n m n C
nm Z
1 − =
controlled sign
4 / 2 /
, , ,
π π
Z C Z H
Z
universal set
m n m n C
nm Z
1 − =
controlled sign
4 / 2 /
, , ,
π π
Z C Z H
Z
Clifford set universal set
2 modes
… any single-qubit gate easy with linear elements !
Adami, Cerf (1998)
d modes
Adami, Cerf (1998)
d modes
… bad scaling of optical elements… N qubit space needs control of modes
N
11 11 , 10 10 , 01 01 , 00 00
ˆ ˆ ˆ ˆ
→ → → →
U U U U
b b a a i
e U
ˆ ˆ ˆ ˆ
ˆ
π
=
Cross Kerr …
Z
11 11 , 10 10 , 01 01 , 00 00
ˆ ˆ ˆ ˆ
→ → → →
U U U U
b b a a i
e U
ˆ ˆ ˆ ˆ
ˆ
π
=
Cross Kerr … …hard to obtain for single-photon states
Z
) 1 ˆ ˆ ( ˆ ˆ ) 2 / (
−
a a a a i
π
) 1 ˆ ˆ ( ˆ ˆ ) 2 / (
−
a a a a i
π
00 00 00 00 → → →
10 10
2 01 10 2 01 10
→ → →
+ +
) 1 ˆ ˆ ( ˆ ˆ ) 2 / (
−
a a a a i
π
) 1 ˆ ˆ ( ˆ ˆ ) 2 / (
−
a a a a i
π
01 01
2 01 10 2 01 10
→ → →
− −
) 1 ˆ ˆ ( ˆ ˆ ) 2 / (
−
a a a a i
π
11 11
2 20 02 2 02 20
− → → →
− −
x t i p s i
e t Z e s X
ˆ 2 ˆ 2
) ( , ) ( ≡ ≡
−
(WH operators)
t p p t Z p p s X
p s i
e
+ = =
−
) ( , ) (
2
(position/computational basis)
s x x s X x x t Z
x t i
e
+ = = ) ( , ) (
2
x t i p s i
e t Z e s X
ˆ 2 ˆ 2
) ( , ) ( ≡ ≡
−
(WH operators)
t p p t Z p p s X
p s i
e
+ = =
−
) ( , ) (
2
(position/computational basis)
s x x s X x x t Z
x t i
e
+ = = ) ( , ) (
2
) ˆ (x f i
x t i p s i
e t Z e s X
ˆ 2 ˆ 2
) ( , ) ( ≡ ≡
−
(WH operators)
t p p t Z p p s X
p s i
e
+ = =
−
) ( , ) (
2
(position/computational basis)
s x x s X x x t Z
x t i
e
+ = = ) ( , ) (
2
) ˆ (x f i
x F p F p F x F x p x F ˆ ˆ , ˆ ˆ , etc. , = − = = =
x x i Z
ˆ ˆ 2 ⊗
(controlled Z gate) (QND gate)
(universal set)
∈ R λ κ
λ κ
, , ; , , ), ( ,
3 2
ˆ ˆ
t e C e t Z F
x i Z x i
“cubic phase gate”
x x i Z
ˆ ˆ 2 ⊗
(controlled Z gate) (QND gate)
U
input
ancilla
anc in anc in in
U
input
ancilla
Gaussian
U
input
ancilla
If input state has N modes, the most general Gaussian CPTP map has at most 2N ancilla modes
T +
T
unitary nonunitary
BS
input
n thermal,
η
4 1
BS
input
n thermal,
η
If input state has 1 mode, the most general Gaussian CPTP map has at most 2 ancilla modes
TMS
2 01 10
±
" "+ " "−
10 01
− + + β α β α
2 1 2 1 01 10
linear optics…
( )
B A B A
↔ ↔ +
2 1
S S
e α α
ϕ sin
cos
i
+ ↔
Innsbruck Experiment
− =
1 1 1 1
1 1 1 1 2 1 b a d c − =
2 2 2 2
1 1 1 1 2 1 b a d c
( )
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( )
[ ] { }0
2 2 2 2 2 1 2 1 2 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 + + + + + + + + + + + + + + + + + + + + + + + +
− ± − → ± = ↔ ↔ ± − → − = ↔ − ↔ − → + = ↔ + ↔ d c d c b a b a d c c d b a b a d d c c b a b a
A S A S A S A S A S A S
Analysis:
distinct click pattern (two separate clicks) undistinguishable click pattern 50% success rate S A B Result U(i) a1 a2 b1 b2 c1 c2 d1 d2 PBS BS PBS
Naimark extension:
µ µ µ
ν µ ν µ
for POVM elements:
µ µ µ
u u E = ˆ
k H =
signal
dim
n ... 1 = µ
i n k i i v
+ =
1 µ µ i k i i v
=
1 µ µ
= +
=
n i i i a
U w
1
ˆ
µ µ
= + + = +
= = = →
n j j j n j i j i j i
a a a U U w
U
1 1 , * lin
ˆ ˆ ˆ
µ µ µ µ
δ
linear optics
α = A
α ± α 2 α 2 − " "+ " "−
„Unambiguous State Discrimination“
2
2α
−
2
α
−
inconcl
α = A
α ± α 2 α 2 − " "+ " "−
H
Hadamard Fourier transform:
position momentum momentum
2
1 F p x y dy p
ipy
e
= = =
π
( ) ( )
y x y x y x − + →
2 1 2 1
C-NOT beam splitter:
S B Result D(x,p)
AB
n n
n n
− , 1
2
λ λ
e.g.
S
α
A a1 b1 BS x p c1 d1 Caltech Experiment
Alice Bob
entanglement generation joint measurement conditional transformation classical channel
entanglement generation Bell measurement conditional transformation classical channel
( )
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( )
[ ] { }0
2 2 2 2 2 1 2 1 2 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 + + + + + + + + + + + + + + + + + + + + + + + +
− ± − → ± = ↔ ↔ ± − → − = ↔ − ↔ − → + = ↔ + ↔ d c d c b a b a d c c d b a b a d d c c b a b a
A S A S A S A S A S A S
distinct click pattern (two separate clicks) undistinguishable click pattern 50% success rate
( )
B A B A
↔ ↔ +
2 1
S S
e α α
ϕ sin
cos
i
+ ↔
S A B Result U(i) a1 a2 b1 b2 c1 c2 d1 d2 PBS BS PBS
Bob Alice
( )
B A B A
↔ ↔ +
2 1
S S
e α α
ϕ sin
cos
i
+ ↔
Innsbruck Experiment, 1997 (Zeilinger)
− =
1 1 1 1
1 1 1 1 2 1 b a d c − =
2 2 2 2
1 1 1 1 2 1 b a d c
( )
( ) ( )
( )
( ) ( )
( )
( ) ( ) ( ) ( ) ( )
[ ] { }0
2 2 2 2 2 1 2 1 2 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 1 2 2 1 2 1 2 1 + + + + + + + + + + + + + + + + + + + + + + + +
− ± − → ± = ↔ ↔ ± − → − = ↔ − ↔ − → + = ↔ + ↔ d c d c b a b a d c c d b a b a d d c c b a b a
A S A S A S A S A S A S
Analysis:
distinct click pattern (two separate clicks) undistinguishable click pattern 50% success rate S A B Result U(i) a1 a2 b1 b2 c1 c2 d1 d2 PBS BS PBS
S B Result D(x,p)
AB
n n
n n
− , 1
2
λ λ
e.g.
S
α
A a1 b1 BS x p c1 d1
Alice Bob
H
Hadamard Fourier transform:
position momentum momentum
2
1 F p x y dy p
ipy
e
= = =
π
( ) ( )
y x y x y x − + →
2 1 2 1
CNOT beam splitter:
S B Result D(x,p)
AB
n n
n n
− , 1
2
λ λ
e.g.
S
α
A a1 b1 BS x p c1 d1 Caltech Experiment, 1998 (Kimble)
S B Result D(x,p)
AB
n n
n n
− , 1
2
λ λ
e.g.
S
α
A BS x p
) ( 1 2 1 ) ( 2 2 1
r r − −
1 1
in v in u
2 2 2 2 v u
Alice Bob Eve
1 + −
“0” “1”
2 1
±
1 + −
“0” “1”
2 1
±
Alice sends value 0 1 1 0 1 1 1 0 0 Bob measures result
1 + −
“0” “1”
2 1
±
Alice sends value 0 1 1 0 1 1 1 0 0 Bob measures result
Alice sends value 0 1 1 0 1 1 1 0 0 Bob measures result Eve measures result
Alice Bob
~ 1000 km
“No-Cloning” Theorem No Communication Faster Than Light Heisenberg Uncertainty Relation Entanglement Quantum Teleportation Beats Classical Teleportation Quantum Cryptography Beats Classical Cryptography Universal Quantum Computer Long-Distance Quantum Communication … Photon Loss in Fiber Channel!
Europe-USA 41 Kanäle mit 10Gb/s
Ideal Wavelength 1.5µm
repeater stations for
wave packets 1mW @ 1.5µm corresponds to 7.106 photons per 0.1ns pulse plus in- and out-coupling losses: transmission through 300km fiber ~100 photons
Alice Bob Alice Bob http://www.idquantique.com http://www.magiqtech.com maximal distance: 50-100km; already after 1 km fiber: photon loss of about 5% exponentially decreasing transmission probability: 10GHz @ 1km 1Hz @ 500km 0.01Hz @ 600km 10-10Hz @ 1000km, 1 photon every 300 years
Intermediate Quantum Amplifiers, Quantum Repeater
α β α´ β´
α β α´ β´
beam-splitter model for photon loss:
U
input
ancilla
anc in anc in in
BS
η
anc
n k n k n
k k n k n k
− −
∞ = −
η η 1
Error model is amplitude damping:
= →
k k k
A A η α η α α α α α
coherent state:
k n k
k
ˆ
Error model is amplitude damping:
= →
k k k
A A η α η α α α α α
coherent state:
k
Error model is amplitude damping:
= →
k k k
A A η α η α α α α α
coherent state:
No-(Re-)Amplification due to No-Cloning Entanglement Quantum Teleportation Long-Distance Quantum Key Distribution
α β α´ β´
split amplified coherent state at a (50:50) beam splitter:
Amp Amp Amp
α η
α η
α η
larger distances (1000 km) ?
Alice Bob
1 1 + ≈ classical channel
ψ
Superconductors Ion Traps Photons
Cavity-QED Free-Space-QED
Entanglement Purification Entanglement Distribution Entanglement Swapping Quantum Memories 1 1 + ≈
distribute known, entangled states distribute different copies in each segment QED/entanglement purification quantum memories two-way classical communication
distr
distr
swap
swap 2 distr
distr
distr
swap
1 / swap / distr
−
L L L L
Bell Meas.
1 / / PDC 2
−
L L L L
1 / / PDC 2
−
L L L L
swap distr
distr
distr
swap
) 3 / 2 ( log ) / ( log swap distr
swap 2 2
P L L
distr
distr
swap
Essence of subexponential scaling: some form of quantum error detection and quantum memories
distr
distr
swap
distr
distr
swap
distr
distr
swap
Quantum Error Detection/Correction
η
2 2 distr
η
L.M. Duan, M.D. Lukin, J.I. Cirac, P. Zoller, Nature 414, 413 (2001)
η
2 2 distr
η
2
no-loss space
η
2 2 distr
η
2
loss space: only QED, not QEC!
E.g. error model is amplitude damping:
e.g., dual-rail qubit:
1
… simplest optical quantum error detection code!
1
in
β α ψ + =
in
ψ
ρ
1 . c . H 1
2
η β η β α ρ − + × + =
att
exp L L − = η L
01 10
in
β α ψ + =
in
ψ
ρ
00 00 1
in in
η ψ ψ η ρ − + =
in
in att
exp ψ ρ ψ η = = − = F L L L
01 10
in
β α ψ + =
in
ψ
in in PS
ψ ψ η ρ =
PS
succ att
Tr exp
ρ η = = − = P L L L
QED
PS
ρ
01 10
in
β α ψ + =
in
ψ
QED
01 10
in
β α ψ + =
in
ψ
QED
….need to detect the qubit non-destructively
+
φ
BM
+
φ
+
φ L L L
Bell measurement detects syndrome and „recovers“ in one step: no loss = 2-photon detection, photon lost =1-photon detection
classical channel
in
ψ
+
φ
BM
+
φ
+
φ L L L
classical channel
in
ψ
Complications: on-demand generation of local Bell states Bell measurement with unit success probability never beats direct transmission
+
φ
BM
+
φ
+
φ L L L
classical channel
in
ψ
att att BM succ
exp exp
/
L L L L P P
L L
− ≤ − = L
( for any )
Initial doubts: no-cloning and continuous errors
Solution: encode globally into larger, entangled state Solution: „discretize“ errors through measurements
THIRD SERIES, VOLUME 52, NUMBER 4 OCTOBER 1995
RAPID COMMUNICATIONS
The Rapid Communications section is intended for the accelerated publication of important new results. Since manuscripts submitted to this section are given priority treatment both in the editorial office and in production, authors should explain in their submittal letter why the work justifies this special handling. A Rapid Communication should be no langer than 4 printed pages and must be accompanied by an
Scheme for reducing decoherence in quantum computer memory
Peter W. Shor*
AT&T Bell l.Aboratories, Room 2D-149, 600 Mountain Avenue, Murray Hill, New Jersey 07974 (Received 17 May 1995) Recently, it was realized that use of the properties of quantum mechanics might speed up certain computa tions dramatically. Interest has since been growing in the area of quantum computation. One of the main difficulties of quantum computation is that decoherence destroys the information in a superposition of states contained in a quantum computer, thus making long computations impossible. lt is shown how to reduce the effects of decoherence for information stored in quantum memory, assuming that the decoherence process acts independently on each of the bits stored in memory. This involves the use of a quantum analog of error correcting codes. PACS number(s): 03.65.Bz, 89.70.+c
Shor code for qubits: Shor-type code for qumodes:
p p p p p d , , ) ( ψ
∝
x x x e dx p
ixp
, ,
2
with
2 111 000
± = ±
with
dual-rail qubit (error detection code)
c0 0L + c1 1L
Bosonic code (error correction code)
= c0 01 + c1 10 c0 0L + c1 1L = c0 04 + 40 2 + c1 22
NOON code (error correction code)
c0 0L + c1 1L = c0 02 + 20 2 ⊗ 02 + 20 2 + c1 1111
…
Bosonic code:
21 03 ) 1 ( 2 " 12 30 ) 1 ( 2 " 22 2 04 40 1
1 1 2
β α η η η β α η η η β α η β α
− = ⊗ − = ⊗ + = + ⊗ A A A A A A
N N N BS N N N BS ⊗ − = ⊗ + =
2 1 , 2
… each codeword has photons and code corrects up to losses 2
: 2 = N
1111 1 , 2 220 2002 202 20 2 = + + + =
Before, without quantum error correction code, we had:
N
error no
Including quantum error correction (Bosonic code), we now have:
3 4 error no
error no
The more photons we have, the more likely it is that at least one gets lost:
N
error
The more photons we have, the more likely it is that at least one does not get lost:
N
error
m m m m n n m m m n
⊗ ⊗ ⊗
) , (
Quantum Parity Code (QPC):
T.C. Ralph, A.J.F. Hayes, and A. Gilchrist., PRL 95, 100501 (2005)
) , ( ) , ( ) , ( ) , ( ) , ( ) , (
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QPC(n,n) corrects (n – 1) photon losses
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Ultraf rafas ast L Long-Di Distance Qu e Quantum Communication
with Static L Linea ear Op Optics
Fabian Ewert,* Marcel Bergmann,† and Peter van Loock‡
Institute of Physics, Johannes Gutenberg-Universität Mainz, Staudingerweg 7, 55128 Mainz, Germany (Received 29 April 2015; revised manuscript received 9 September 2016; published 14 November 2016) We propose a projection measurement onto encoded Bell states with a static network of linear optical elements. By increasing the size of the quantum error correction code, both Bell measurement efficiency and photon-loss tolerance can be made arbitrarily high at the same time. As a main application, we show that all-optical quantum communication
photon detectors. As another application, generalizing state teleportation to gate teleportation for quantum computation, we find that in
minimal amount of feedforward has to be added.
DOI: 10.1103/PhysRevLett.117.210501
PRL 11 117, 210501 (2016) P H Y S I C A L R E V I E W L E T T E R S
week ending 18 NOVEMBER 2016
) , ( in m n
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All-optical = No more memories, no more nonlinear light-matter gates Ultrafast = No more two-way classical communication, approaching classical rates
No-Cloning prevents building quantum repeaters like classical ones Fundamental elements of quantum information allow for scaling up quantum communication to large distances Such elements are especially entanglement and quantum teleportation that also allow for quantum computation Quantum optical systems and methods are most useful
QuOReP, Q.com
Standard quantum repeaters are scalable in principle, work with fairly simple quantum states, but are rather slow requiring good memories New generation of quantum repeaters using quantum error correction significantly improve the rates Ultrafast loss-code-based schemes are implementable with linear optics Important elements are integration and hybridization
QuOReP, Q.com