quantum key distribution with continuous variables at telecom wavelength

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 ?


Outline







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).


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)


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)


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


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) .


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).


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)


arXiv.org:cs/0509041

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







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


Alice and Bob set-up

Alice

Bob


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.


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


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







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)


Noise analysis

χA→B = χA→E + χE→B

+ =

Bob AliceAlice

Ève

BobÈve


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 !


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.


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


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)


Outline







• 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 !


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 !


Outline







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

quantum key distribution with continuous variables at telecom wavelength

Technology

e continuous

es es

continuous amplitude

continuous variables1

continuous variables4

continuous variables6

continuous variables7

continuous variables8