Quantum Key Distribution with Continuous
Variables at Telecom Wavelength
Jerome Lodewyck (1, 2), Thierry Debuisschert (1),Alexei Ourjoumtsev (2), Rosa Tualle-Brouri (2), Philippe Grangier (2)
(1) Thales Research and Technologies, Palaiseau, France(2) Lab. Charles Fabry de l’Institut d’Optique, Orsay, France
in collaboration with :Nicolas Cerf (ULB, Brussels)Raul Garcia-Patron (ULB, Brussels)
with crucial contributions from :F. Grosshans, J. Wenger, G. van Assche, M. Bloch, A. Dantan...
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 1 / 30
Outline
1 Quantum Cryptography with Continuous Variables
2 Implementation in the optical Telecom range
3 Robustness against an Intercept-Resend attack
4 Real-scale implementation : SECOQC project
5 Towards quantum repeaters ?
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 2 / 30
Outline
1 Quantum Cryptography with Continuous Variables
2 Implementation in the optical Telecom range
3 Robustness against an Intercept-Resend attack
4 Real-scale implementation : SECOQC project
5 Towards quantum repeaters ?
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 3 / 30
Quantum key distribution
Alice & Bob want to share a secret message. . .
Alice and Bob establish a secret key through a quantum channel anda classical authenticated channel
This key enables the unconditionnally secure transmission of amessage through a public channel
The key has to be as long as the message and used only once⇒ high key rate needed.
Data Reconciliationhow to correct errors, revealing as less as possible to Eve ?
IAE IBE
IAB
Main idea (Csiszar and Körner 1978, Maurer 1993) :
Alice and Bob can in principle distill, from their correlated key elements, acommon secret key of size S > sup(IAB - IAE , IAB - IBE) bits per key element.
Crucial remark : it is enough that IAB is larger than the smallest of IAE and IBE(i.e. one has to take the best possible case).
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 4 / 30
QKD with coherent states
Alice encodes the key in continuous amplitude and phase by sendingrandomly modulated coherent states with a Gaussian distribution.
Bob detects this state with an homodyne (interferometric) detection.
Échange quantique
Alice
X
P
→
Bob
X
P
Alice envoie une série d'impulsions lumineuses
Modulation gaussienne (variance ' 10 photons)
Bob reçoit une version bruitée du signal : bruit de photon(transmission du canal T ) et excès de bruit ξ.
Bob mesure Q ou P
⇒ Alice et Bob possèdent des variables gaussiennes corrélées
Jérôme Lodewyck (TRT/IOTA) Réconciliation 23 juin 2006 2 / 13
Photodiode
50/50BS
Homodyne detection
Phase control :Measurement of X or P
+ Low-noise- amplifier
Signal
Local Oscillator(classical)
I1 = |ELO|2 + |ES|2 + |ELO| (ES e - i ϕLO + ES* e i ϕLO)
I2 = |ELO|2 + |ES|2 - |ELO| (ES e - i ϕLO + ES* e i ϕLO)
I1 - I2 = 2 |ELO| (ES e - i ϕLO + ES* e i ϕLO)
= 2 |ELO| (ES + ES*) X meas.
= 2 |ELO| i (ES - ES*) P meas.
X
P
Squeezed state
X and P do not commute Heisenberg relation
V(X) V(P) ≥≥≥≥ N02
F. Grosshans et al., Nature 421 238 (2003)
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 5 / 30
QKD with coherent states
Alice encodes the key in continuous amplitude and phase by sendingrandomly modulated coherent states with a Gaussian distribution.
Bob detects this state with an homodyne (interferometric) detection.
Pro & cons of coherent states QKD
No need to produce or detect single photons.
Uses only fast and standard telecom components.
⇒ High key rate achievable in principle
But. . .
Homodyne detection requires a careful design (optics, electronics...).
Data post-processing requires efficient key extraction algorithms.
F. Grosshans et al., Nature 421 238 (2003)
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 6 / 30
Gaussian channel model
To characterize the protocol performance, we measure noise variancesreferred to the input.The coherent states sent in the quantum channel can be altered by
Losses 1− T that decrease the signalamplitude ⇔ ”vacuum” added noiseχ0 = 1/T − 1 (in shot-noise units)Equivalent to photon loss in BB84 schemes
Shot noise
Excess noise ε above the shot noise levelEquivalent to errors in BB84 schemes.
Excess noise
⇒ total added noise χ = χ0 + ε = 1/T − 1 + ε
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 7 / 30
Security analysis
Reverse Reconciliation : the basis for the key is the data received by Bob,not the one sent by Alice. The secret rate is then ∆I = IAB − IBE with
IAB =1
2log2(1 + SNR) =
1
2log2(1 +
ηTVA
1 + ηT ε) Shannon
IBE =1
2log2
ηTVA + 1 + ηT ε
η/[1− T + T ε + T
VA+1
]+ 1− η
Heisenberg
In these formulas all quantities are known or measured by Alice and Bob :
η : quantum efficiency of Bob’s homodyne detection
T : channel transmission
VA : variance of Alice’s modulation
ε : channel excess noise (above shot-noise)
∆I = IAB − IBE > 0 for any value of the transmission T , if ε < VA2(1+VA) .
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 8 / 30
Security analysis
The Reverse Reconciliation protocol was proven secure against a widerange of attacks :
individual, gaussian : Nature 2003 (Heisenberg + Shannon)
finite-size, non-gaussian : PRL 2004 (entropic Heisenberg inequalities)
collective attacks, general : PRL 2005 and 2006 (using Holevo bound)
For a given variance measured by Alice and Bob, the Gaussian attacks aredemonstrated to be optimal for both individual and collective attacks(Grosshans, Navascues, Acin, Cerf, Garcia-Patron).
For a given variance, Alice and Bob are thus always on the safe side byassuming that Eve’s attack is Gaussian ! Eve’s information is then givenby Shannon’s IBE (individual attacks) or Holevo’s χBE (collective attacks).
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 9 / 30
Reconciliation of Gaussian correlated
variables
The quantum transmission leads to correlated quadraturesmeasurements shared by Alice and Bob.
0
0
0
0
00 0 0 0 0 0 0
0 0
0
1
1 1
1 11 1
1 1 1 1 1 11 1
Dec
odin
g A slice reconciliation algorithm bins theGaussian dataError correction is performed with iterativesoft decoding using LDPC codes.
Standard privacy amplification eliminatesany key information known by Eve IIAB
I AE
∆ IIIAB
I AE
∆ I
G. Van Assche et al., IEEE Trans. on Inf. Theory 50(2) 394-400 (2004)M. Bloch et al., arXiv.org:cs/0509041 : LDPC codes (more efficient)
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 10 / 30
Reconciliation performances
The raw key rate is positive for any transmission (if ε small enough) :∆I = IAB − IBE > 0 for all transmission or distance.
The reconciliation efficiency limits the transmission range.∆I = βIAB − IBE < 0 for small transmission / large distance.
Extraction imparfaite
En pratique, Alice et Bob n'extraient qu'une fraction β de IAB⇒ ∆I < 0 si T petit.
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.2 0.4 0.6 0.8 1
Mut
ual
Info
rmat
ion
(bit/
puls
e)
Channel Transmission
∆I eff∆I max
IAB
IBE
β IAB, β = 0.87
Jérôme Lodewyck (TRT/IOTA) Réconciliation 23 juin 2006 4 / 13Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 11 / 30
Outline
1 Quantum Cryptography with Continuous Variables
2 Implementation in the optical Telecom range
3 Robustness against an Intercept-Resend attack
4 Real-scale implementation : SECOQC project
5 Towards quantum repeaters ?
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 12 / 30
Experiment layout
Realization of a CVQKD setup with fiber optics and telecom components
1.55 µm, 1 MHz pulse rate (limited by the acquisition device)
Modulation stage displacing a coherent state in the complex plane.
Detection stage measuring a quadrature of the E.M. field :pulsed, shot noise limited homodyne detector.
LOCAL OSCILLATOR
ALICE
EVE
BOB
Amplitude
Modulator
Amp. & phase SIGNALModulator
−
Phase
Modulator
InGaAs
Photodiodes
DFB diode
1550 nm
HOMODYNE DETECTOR
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 13 / 30
Alice and Bob set-up
Alice
Bob
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 14 / 30
Stability
0 20 40 60 80 100
Qua
drat
ure
(AU
)
Pulse #
Test pulses
Bob’s measurementAlice’s quadrature
Bob’s average (500 meas.)
0
1
2
3
4
5
6
0 20 40 60 80 100 120 140
−0.4
−0.2
0
0.2
0.4
Rel
ativ
e ph
ase
Bal
anci
ng
Time (s)
An arbitrary modulation can be applied.
Test pulses are used for synchronization and measuring relative phase.
Automated, real-time, continuous acquisition software.
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 15 / 30
Noise analysis
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Noi
se r
efer
red
to th
e in
put
(S
NL
units
)
Channel transmission (T)
χχ0
ε
Noises
χ0 = 1/T − 1χ = χ0 + ε
5 to 10 % of excess noisecoming from
Laser phase noise
Electronic noise
Modulation inaccuracies
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 16 / 30
Reverse Reconciliation performances
Shannon raw key rate : 270 kb/s @ 15 km, 145 kb/s @ 25 km
The reconciliation efficiency limits the transmission range.∆I = βIAB − IBE : currently β = 87% → 35 kb/s @ 25 km.
The reconciliation processing speed limits the key rate → typically1 kb/s @ 25 km. (200 000 data points decoded in a few seconds)
Secret bit rate (bit/s) for β = 0.87 (current), 0.925 (doable), 1 (ideal).
20 40 60 80 100
100
1000
10000
100000
essaiRec.nb 1
Distance (km)Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 17 / 30
Reverse Reconciliation performances
Holevo raw key rate : 230 kb/s @ 15 km, 120 kb/s @ 25 km
The reconciliation efficiency limits the transmission range.∆I = βIAB − χBE : currently β = 87% → 13 kb/s @ 25 km.
The reconciliation processing speed limits the key rate → typically1 kb/s @ 25 km. (200 000 data points decoded in a few seconds)
Secret bit rate (bit/s) for β = 0.87 (current), 0.94 (doable), 1 (ideal).
20 40 60 80 100
100
1000
10000
100000
essaiRec.nb 1
Distance (km)Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 18 / 30
Outline
1 Quantum Cryptography with Continuous Variables
2 Implementation in the optical Telecom range
3 Robustness against an Intercept-Resend attack
4 Real-scale implementation : SECOQC project
5 Towards quantum repeaters ?
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 19 / 30
Intercept-Resend attack
Usual beam splitting attacks (implemented by attenuating the signal) onlyintroduce ”vacuum” added noise χ0 = 1/T − 1
Bob
Alice
P
X S
Eve
We implemented an intercept-resend attack whichintroduces 2 shot noise units of excess noise : χ = χ0 + 2.
Experiment in 3 steps :
Alice sends, Eve (using Bob) mesures X
Alice sends, Eve (using Bob) mesures P
Eve (using Alice) resends (x , p)
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 20 / 30
Noise analysis
χA→B = χA→E + χE→B
+ =
Bob AliceAlice
Ève
BobÈve
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 21 / 30
Noise analysis
ε = χ− χ0 = 2 + εTwhere χ is measured and χ0 = 1/T − 1
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Noi
se r
efer
red
to th
e in
put
(S
NL
units
)
Channel transmission (T)
χχ0
ε
Entanglement-breaking attack → no secret key generated !
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 21 / 30
Partial Intercept-Resend attack
Limitations of the IR attack
The amount of excess noise is fixed to 2 SNL.
Breaks the entanglement limit : Alice and Bob get no no secret key.
Partial IR attack
Eves make an IR attack on a random data subset of variables size µ. Onthe remaining data, she performs a standard BS attack.
Bob
X
P S
Eve
µ1−µAlice IR
BS
Properties of the partial IR
Eve can introduce an arbitrary amount of excess noise.
It is a simple non-Gaussian attack.
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 22 / 30
Noise analysis
0
0.5
1
1.5
2
2.5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Exce
ss n
ois
e r
efe
rre
d t
o t
he
in
pu
t (
ξ in
SN
L u
nits)
Intercepted−reemitted pulse fraction (µ)
Theoretical excess noiseExperimental excess noise
Excess noise:ε = µ(εIR+εT )+(1−µ)εTε = 2µ + εT
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 23 / 30
Information analysis
Information rates can be computed from experimental data (T = 0.25)
Information analysis
Information rates can be computed from experimental data (T = 0.25)
1.1
1.2
1.3
1.4
1.5
0 0.2 0.4 0.6 0.8 1
0 0.4 0.8 1.2 1.6 2
Info
rmat
ion
rate
(bi
ts/p
ulse
)
Fraction of IR pulses
Excess noise
Information rates
IAB experimentalIBE Gaussian modelIBE Beam SplitterIBE experimentalIBE non-GaussianSecurity margins have to be considered to takeinto account statistical fluctuations
J. Lodewyck, R. Garcıa-Patron et al., in preparation
Jerome Lodewyck (TRT/IOTA) QKD with coherent states May 21, 2006 20 / 1
Eve’s information rates
IBE ,G : optimal Gaussian attackIBE ,NG : a Non-Gaussian attack : sub-optimal !IBE ,IR : implemented attack : not so bad !IBE ,BS : Beam-Splitter only : much weaker !
Bob’s information rate
IAB : measured on the experiment(security margins have been included to takeinto account statistical fluctuations).
J. Lodewyck et al., Phys. Rev. Lett. 98, 030503 (2007)
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 24 / 30
Outline
1 Quantum Cryptography with Continuous Variables
2 Implementation in the optical Telecom range
3 Robustness against an Intercept-Resend attack
4 Real-scale implementation : SECOQC project
5 Towards quantum repeaters ?
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 25 / 30
• Binary QBB-Links
• QBB-Nodes with multiple
QBB-Links to neighbouring
QBB-Nodes
• Hop-by-hop distribution of
secrets
SECOQC Network Paradigm:
„Quantum Back Bone“
QBB Demonstrator in the
SIEMENS Glass Fiber Network
Vienna, Sept. 2008
• 5 QKD Technologies
• 5 QBB-Nodes / 7 QBB-Links
QBB Demonstrator in
the SIEMENS Glass
Fiber Network
Vienna, Sept. 2008
Objectives for CV-QKD
Build a Continuous Variables quantum key distribution setupwith fiber optics and off-the-shelf telecom componentsCharacterize noise & robustnessSimulate (and detect !) real attacksBuild a 19” rack prototype for the SECOQC european project.
Objectives
Building a coherent states quantum key distribution setup with fiberopticsNoise & robustness characterization.Exploring new physics (attacks simulation)Building a 19 inches prototype for the SECOQC european project.
Jerome Lodewyck (TRT/IOTA) QKD with coherent states May 21, 2006 2 / 1Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 27 / 30
Time multiplexing
Both signal and phase reference have to travel through the same fibre :2
DFB
I
I
I
Polarizationcontroller
Polarizationcontroller
EOM99/1
Signal
Reference
BobChannel
−
Alice
10%
90%
EOM − Amplitude EOM − Phase A0
EOM − Phase
25 km
99/1
LO
S
40 m
40 m
FIG. 1: Experimental setup. Alice generates modulated signal pulses, and Bob measures a random quadrature with a pulsed,shot noise limited homodyne detector. The first EOM (left side) slices 100 ns pulses, and the EOM denoted as “A0” sets thevariance of Alice’s modulation. At the detection stage the signal S and local oscillator LO are overlapped using a delay line.
III. IMPLEMENTATION
A. Experimental setup.
A scheme of the set-up is shown on Fig. 1. It is a coherent-state QKD setup, working at 1550 nm and exclusivelyassembled with fiber optics and fast telecom components. It displaces a train of pulsed coherent states within thecomplex plane, with arbitrary amplitude and phase, randomly chosen from a two-dimensional Gaussian distributionwith variances VA ∼ 12 N0. The pulse width is 100 ns. The signal is sent to Bob along with a strong phase reference– or local oscillator (LO), with ∼ 109 photons per pulse. Bob can select an arbitrary measurement phase with aphase modulator placed on the LO path. The selected quadrature is measured with an all-fiber shot noise limited,time-resolved homodyne detector. A key transmission is composed of independent blocks of 50000 pulses, sent at arate of 500 kHz, among which 10000 test pulses with agreed amplitude and phase are used to synchronize Alice andBob and to determine the relative phase between the signal and LO (see [2] for more details), and 5000 for channelevaluation. The “effective” pulse rate used in for the secret bit rates quoted below is thus 350 kHz.
B. Time multiplexing.
In order to propagate over long distances with negligible phase noise, the LO and signal are time-multiplexed using40 m delay lines, as shown on Fig. 1. Since the signal photon number is very small compared to the LO photonnumber, and since the LO should keep enough intensity, the time multiplexer uses two 99/1 splitters which keep themain part of the beam in the LO. At the reception stage, we use a passive 10/90 splitter which now keeps 90 % of thesignal, beating against 10 % of the initial LO. For proper operation of the system, it is crucial that the backgroundlevel of the DFB laser be extremely weak at the place (in time) where the weak signal (∼ 10 photons) overlaps thelight from the L.O. channel (∼ 109 photons). For that purpose, the laser diode operates in a pulsed mode, with pulseslong enough to avoid the transient regime. Then, an EOM slices the required 100 ns pulses, and by using appropriatedelay lines and synchronization we obtain the required rejection of the background. The signal level sent by Alice,and the LO level received by Bob, are continuously monitored for channel characterization and verification. This isdone in real time by using additional A/D channels of the data acquisition boards (National Instrument).
The phase stabilization system based on test pulses [2] ensures good operation of the system in laboratory conditions,in presence of the two 40 m delay lines realized with non-polarization-preserving fibers, followed by polarizationcontrollers (see Fig. 1). At the packaging step (Milestone 3) it is envisioned that all fibered optics at Alice and Bobsites will be realized with polarization-preserving fibers.
This solves phase and polarization fluctuations during propagation.This involves:
Introducing delay lines (40 m long)
Demultiplexing at Bob’s → fiber coupler 90% (signal) / 10% (LO)
Controlling the polarization at the channel output (active controller)
Tested on an installed fiber (750 m) and on a fiber coil (25 km) : OK !
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 28 / 30
Present status of the CV-QKD set-up
Done
built a complete QKD setup with coherent states
implemented partial intercept-resend attacks
time-multiplexed transmission over 750 m (installed) or 25 km (coil)
Outlook
automatize and optimize transmission over 25 km
improve algorithms : better efficiency, faster
be ready for the show in Vienna in 2008 !
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 29 / 30
Outline
1 Quantum Cryptography with Continuous Variables
2 Implementation in the optical Telecom range
3 Robustness against an Intercept-Resend attack
4 Real-scale implementation : SECOQC project
5 Towards quantum repeaters ?
Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 30 / 30
Where is the entanglement ?EPR versus coherent protocol
LO
Bob
EPRsource
Alice measuresXA or PA on half
an EPR beam
EPR protocol equivalent to our coherent state protocol !Cf BB84 vs entangled pair (Ekert) protocol - Crucial for the security proofs !
The state received by Bob is prepared in asqueezed state, conditional to Alice ’s result
X
P
LOs( /2)
Bob
EPRsource
Alice measuresXA and PA on half
an EPR beamThe state received by Bob is prepared in a
coherent state, conditional to Alice ’s result
X
P50-50 BS
(XA + XB ) and (PA - PB )are squeezed
Quantum repeaters for continuous variables
Is it possible to carry some operation on several EPR beamsto increase the final entanglement ?
Theorem (Fiurasek, Eisert, Cirac…):This cannot be done if all states and all operations are gaussian !
Non-gaussian states and/or operations are required
EPR source
XA , PA XB , PBEPR source
EPR source
? ?
For quantum cryptography only « virtual » entanglement is requiredBut quantum repeaters do need « real » entanglement.
Continuous-variables EPR beams
Q I P C
(XA + XB ) and (PA - PB ) are squeezed (commuting operators !)then (PA + PB ) and (XA - XB ) are anti squeezed
If Alice measures XA , she will know XB
If Alice measures PA , she will know PB
and for a large enough squeezing we have :
V(XB|XA) V(PB|PA) < N02 !!!
« apparent » violation of Heisenberg relations V(XB) V(PB) N02
If the squeezing goes to infinity : original EPR state (1935) !
EPRsource
XA , PA XB , PB
How to produce QCV entangled beams ?
Q I P C
XC , PC
XC squeezed
XD , PD
PD squeezed
X
P
X
P
XC = ( XA + XB )/ 2PC = ( PA + PB )/ 2XD = ( XA - XB )/ 2PD = ( PA - PB )/ 2
OK !
1. Combine two orthogonally squeezed beams on a 50-50 beamsplitter
XA , PA
XB , PB
entangled !
NOPA(khi-2
crystal)
2. Use a Non-degenerate Optical Parametric Amplifier (NOPA)
X
P
Vacuum stateX
P
Vacuum stateXA , PA
XB , PB
entangled !Pumpbeam
Signal (quantum)
Local Oscillator
(classical)
V1-V2 X
Homodyne detection :Measures X = X*cos( )+P*sin( )
Marginals of W(X,P)
Quadrature distributions Π(X )
Quadrature distributions Π(X )
W(X,P) : tomography
50/50
BS
Specific quantum
states :
negative Wigner
function!
Many interesting
properties for
quantum
information
processing
Homodyne detection,Wigner Function and Quantum Tomography
Q I P C
W(X,P) W(X,P)
P
P
X
X
Π(X)
Π(X)
Q I P C
• Odd : | = c ( | | ) = an |2n+1
• Look at small | | ~ 1• Very similar to a photon-subtracted squeezed vacuum state• Very similar to a squeezed single-photon state
« Schrödinger Kitten »
Wigner function of asmall Schrödinger cat
Wigner function of aPhoton-subtracted
squeezed state
Fidelity between the kitten and the most similar
photon-subtracted state
Experimental Set-up
Femtosecond Ti-Sapph laser :Pulse duration 180 fs , rep. rate 800 kHz
Frequence doubling in KNbO3Single pass efficiency : SHG = 30%
Parametric amplifier in KNbO3Typical (single pass) squeezing : 3 dB
Pulsed Homodyne DetectionGlobal quantum efficiency : = 80%
IR Filter
Spatial & spectralfiltering
R=10%
APD
Q I P C
Special feature:pulsed time-
domain analysis
Q I P C
W0 = - 0.13 ± 0.01s = 0.56, | | 0.9
Wideal kitten = - 0.32
Wigner function of the Kitten(corrected for homodyne efficiency)
A. Ourjoumtsev et al, Science 312: 83, 7 april 2006
= 0.70 (pure kitten) + 0.29 (pure squeezed) + 0.01 (residuals)
Gaussian
entangled state
Mode 1
Mode 2
Entropy of formation :[G.Giedke et al, PRL 91 ,107901 (2003) ]
Gaussian entangled EPR beamsJérôme Wenger, Alexei Ourjoumtsev et al., EPJD 2004
Measured covariance matrix :[ (0)s from symmetry arguments ]
Duan-Simon criterion :
Gaussian
entangled state
Coherent
Photon
Subtraction
Homodyne detections
Expected state structure :
« Delocalized »
squeezed state
« Delocalized »
Schrödinger
Kitten state
Is this true ?
Mode 1
Mode 2
X1 , P1
X2 , P2
Coherent Photon Subtraction
Delocalized Schrödinger Kitten
Mode 1
Mode 2
Ws = Wsqueezed Wc = Wkitten
X1 , P1
X2 , P2
Quantum tomography on homodyne data :
Setup
Phase control : Most phases fixed with waveplates
1 and 2 compensate each other only one phase to control
Measure the two-mode correlation variance (800 kHz rep. rate reasonably fast)
Experimental ResultsA. Ourjoumtsev et al, Phys. Rev. Lett. 98, 030502 (2007)
Wigner function W Density matrix (Fock basis, 20 photons : 400 400)
Entanglement measure : Negativity
(absolute value of the sum of negative eigenvalues of the partially transposed density matrix)
Several projections of two measured two-mode Wigner
functions, corrected for homodyne losses, with R = 10% :
1.3 dB
3.2 dB
Initial
squeezing :
Increasing the Entanglement up to 3 dBA. Ourjoumtsev et al, Phys. Rev. Lett. 98, 030502 (2007)
YES! increased entanglement
NO :initial states too fragile
to resist an imperfect
photon subtraction
Crossover
at R = 3% for
3 dB gain
For high gain (> 3dB)
small experimental
improvements may
have a strong effect
on the position of the
crossover.
What next ?
• Need to share highly entangled states (cryptography..)
• Solution : Entanglement Distillation :
• Problem : losses
• But impossible to distill Gaussian
entanglement with Gaussian means use non-gaussian operations !
(such as photon subtraction)
Large
number of
weakly
entangled
states
Small
number of
strongly
entangled
states
Long-Distance Quantum Communications
Violation of Bell’s Inequalities« Aspect Experiment », Orsay, 1981- 1982
* Polarisation-entangled pairs of photons emitted byan atomic cascade excited by two lasers.* Remote polarisations measurement on the twophotons are very strongly correlated and cannot bedescribed by any « local realistic » model.
Violation of Bell’s Inequalities1982 expt : first test of « locality loophole »
4p2 1S0
4p4p 1P1
4s2 1S0
1 = 551nm
2 = 422nm
Calcium 40 atomic beam
A new violation of Bell ’s inequalities ?R. Garcia-Patron et al, Phys. Rev. Lett. 93, 130409 (2004)
(XA + XB ) and (PA - PB ) are squeezed : original EPR state (1935) !
EPRsource
XA , PA XB , PB
* Alice and Bob perform homodyne detections on each side and measure either XA or PA (for Alice), and XB or PB (for Bob).
* Then they « digitize » the data by taking the sign ( ± ) of the value of X or Psx = Sign(X) = ± 1, sp = Sign(P) = ± 1
and they compute the S parameter for Bell CHSH inequalitiesS = < sxA sxB > + < sxA spB > + < spA sxB > < spA spB >
* According to Bell ’s theorem, | S | 2 for any local hidden variables theory
No violation here ! (the Wigner function provides a local hidden variable model !)
Q I P C
A new violation of Bell ’s inequalities ?R. Garcia-Patron et al, Phys. Rev. Lett. 93, 130409 (2004)
(XA + XB ) and (PA - PB ) are squeezed : original EPR state (1935) !
EPRsource
XA , PA XB , PB
* Now « degaussify » by using two APDs (« event ready » detectors)
* Apply the same procedure (… but now the Wigner function of the generatedstate can take negative values : not a valid LHVT !)
* Violation ! S = 2.02 > 2 [ 6 dB squeezing, (APD) = 30%, (hom) = 95% ]
« Loophole -free » test, all events are taken into account, feasible ?
APDAPD Q I P C
• It is possible to find other QCV states with S up to 2 2 = 2.828
maximal violation of BI, see M. Hafezi et al, PRA 67, 012105 (2003)
• Ex : entangled state | f(x1) f(x2) + i | g(x1) g(x2)
| f(x) = 0.585 | n = 0 - 0.415 | n = 4
| g(x) = 0.848 | n = 1 + 0.152 | n = 5
=> higher violation, but (very ?) difficult to prepare :
best compromise still to be found ?
BI test : increasing the value of S ?
S = 2.68
f(x) g(x)
Conclusion
Q I P C
Many potential Many potential uses for Quantumuses for Quantum Continuous Continuous VariablesVariables……* QuantumQuantum cryptography cryptography* Coherent states protocols using reverse reconciliation,
secure against any (gaussian or non-gaussian) collective attack
* Conditional preparation of Conditional preparation of «« squeezedsqueezed »» non-gaussian non-gaussian pulses /pulses / cats cats* Big family of phase-dependant negative Wigner function* First step towards : entanglement distillation procedures ?
new tests of Bell’s inequalities ?* See also new experimental results by the groups of
A. Lvovsky, M. Bellini, E. Polzik, A. Furusawa, M. Sasaki...
* Many other proposed schemes Many other proposed schemes (Sam Braunstein, Tim Ralph)…* « Growing up the cat » (A.P. Lund et al, Phys. Rev. A 70, 020101 (R) (2004))* Universal quantum computing (QCV version of KLM...) …
« Degaussification » of a squeezed stateJ. Wenger & al., PRL 92, 153601 (2004)
Q I P C
A squeezed state can be « degaussified » by photon subtraction(one single photon in the APD beam)
Squeezed vacuum : |0 + |2 + |4 + …
Non-gaussian state :
|1 + 2 (1-R) |3 + …
APD
R<<1
Wigner function
X P
Wigner function
X P
Perspectives… growing up the kitten
• Larger squeezing will create a larger non-gaussian state, but not a cat• Requires a “breeding” process (interference, detection and post-selection)
see A.P. Lund et al, Phys. Rev. A 70, 020101 (R) (2004)
Measured Probability Distributionsafter photon subtraction
Measured probabilitydistributions of the
quadratures componentsas function of the LO phase
Anti-squeezed quadrature
Squeezed quadrature
Dip in the squeezed quadrature : hint for a negative Wigner function !
Wigner function of the « raw »measured state (no correction)
Analytic Model
Numerical Radon Transform
Radon transform clearly negative ! (no hypothesis, no correction)A. Ourjoumtsev et al, Science 312: 83, 7 april 2006