Quantum key distribution with continuous variables at telecom wavelength

Quantum Key Distribution with Continuous
Variables at Telecom Wavelength

J´rˆme Lodewyck (1, 2), Thierry Debuisschert (1),
eo
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)
Ra`l Garcia-Patr`n (ULB, Brussels)
u o
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 and
a classical authenticated channel
This key enables the unconditionnally secure transmission of a
message through a public channel
The key has to be as long as the message and used only once
⇒ high key rate needed.
Data Reconciliation
how to correct errors, revealing as less as possible to Eve ?

IAE IBE

IAB

QKD with coherent states
Homodyne detection

I1 = |ELO|2 + |ES|2 + |ELO| (ES e - i ϕLO + ES* e i ϕLO) P

- i ϕLO i ϕLO)
|ELO|2 |2
Alice encodes= the keyS in |ELO| (ES e
I2 + |E - continuous + ES e *
amplitude and phase by sending
X
randomly modulated|ELO| (ES e - i ϕLO + ES* e with a Gaussian distribution.
I1 - I2 = 2 coherent states i ϕLO)
Squeezed state
Bob detects this stateLO| (ES + an*)homodyne (interferometric) detection.
= 2 |E with ES X meas.

= 2 |ELO| i (ES - ES*) P meas.

nge quantique X and P do not commute
Heisenberg relation
Alice Bob 50/50 + Low-noise
P V(X) V(P) ≥ N02
P
BS - amplifier
→ Signal
X X Photodiode
Local Oscillator Phase control :
(classical) Measurement of X or P

ce envoie une série d'impulsions lumineuses
odulation gaussienne (varianceGrosshans et al., Nature 421 238
F. 10 photons) (2003)
b reçoit une version bruitée du signal : bruit de photon
ansmission du canal T ) et excès de bruit ξ .
bThales &Q ou P d’Optique (CNRS)
mesure Institut QKD with Continuous Variables 5 / 30

QKD with coherent states

Alice encodes the key in continuous amplitude and phase by sending
randomly 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 variances
referred to the input.
The coherent states sent in the quantum channel can be altered by
Losses 1 − T that decrease the signal Shot noise

amplitude ⇔ ”vacuum” added noise
χ0 = 1/T − 1 (in shot-noise units)
Equivalent to photon loss in BB84 schemes

Excess noise

Excess noise above the shot noise level
Equivalent to errors in BB84 schemes.

⇒ total added noise χ = χ0 + = 1/T − 1 +


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
1 1 ηTVA
IAB = log2 (1 + SNR) = log2 (1 + ) Shannon
2 2 1 + ηT
1 ηTVA + 1 + ηT
IBE = log2 Heisenberg
2 T
η/ 1 − T + T + VA +1 + 1 − η

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
: channel excess noise (above shot-noise)
VA
∆I = IAB − IBE > 0 for any value of the transmission T , if < 2(1+VA ) .


Security analysis

The Reverse Reconciliation protocol was proven secure against a wide
range 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 are
demonstrated 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 by
assuming that Eve’s attack is Gaussian ! Eve’s information is then given
by Shannon’s IBE (individual attacks) or Holevo’s χBE (collective attacks).


Reconciliation of Gaussian correlated
variables

The quantum transmission leads to correlated quadratures
measurements shared by Alice and Bob.
0 1
A slice reconciliation algorithm bins the

Decoding
1 0 1 0
Gaussian data
1 0 1 0 1 0 1 0
Error correction is performed with iterative
1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 soft decoding using LDPC codes.

Standard privacy ampliﬁcation eliminates
∆I
∆I
any key information known by Eve IAB
IAB
I AE I AE

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)

Reconciliation performances

The raw key rate imparfaite any transmission (if small enough) :
Extraction is positive for
∆I = IAB − IBE > 0 for all transmission or distance.
The pratique, Alice et eﬃciency limits the transmissionIAB
En reconciliation Bob n'extraient qu'une fraction β de range.
∆I =I βIABsi− IBE < 0
⇒ ∆ < 0 T petit. for small transmission / large distance.

1.6
Mutual Information (bit/pulse)

IAB
1.4 IBE
1.2 β IAB, β = 0.87

1 ∆I max
∆I eff
0.8

0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8 1
Channel Transmission

Thales & Jérôme Lodewyck (TRT/IOTA) QKD withRéconciliationVariables
Institut d’Optique (CNRS) Continuous 23 juin 2006 4 / 13 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.

ALICE
Amp. & phase SIGNAL
Modulator
1550 nm Amplitude
DFB diode Modulator

LOCAL OSCILLATOR

EVE

BOB HOMODYNE DETECTOR

Phase
Modulator
InGaAs
− Photodiodes


Alice and Bob set-up

Alice

Bob


Stability

Bob’s measurement
Alice’s quadrature 6
0.4
Bob’s average (500 meas.)
5
Quadrature (AU)

0.2

Relative phase
4

Balancing
3 0

2 −0.2
Test pulses 1
−0.4
0 20 40 60 80 100 0
0 20 40 60 80 100 120 140
Pulse #
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

6
χ Noises
χ0
Noise referred to the input

5 χ0 = 1/T − 1
ε
4 χ = χ0 +
(SNL units)

3
5 to 10 % of excess noise
2
coming from
1
Laser phase noise
0 Electronic noise
0 0.2 0.4 0.6 0.8 1
Channel transmission (T) 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 → typically
1 kb/s @ 25 km. (200 000 data points decoded in a few seconds)
essaiRec.nb 1

Secret bit rate (bit/s) for β = 0.87 (current), 0.925 (doable), 1 (ideal).
100000

10000

1000

100

20 40 60 80 100
Distance (km)

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 → typically
1 kb/s @ 25 km. (200 000 data points decoded in a few seconds)
essaiRec.nb 1

Secret bit rate (bit/s) for β = 0.87 (current), 0.94 (doable), 1 (ideal).

100000

10000

1000

100

20 40 60 80 100
Distance (km)

Outline







Intercept-Resend attack

Usual beam splitting attacks (implemented by attenuating the signal) only
introduce ”vacuum” added noise χ0 = 1/T − 1

Alice

We implemented an intercept-resend attack which
Eve introduces 2 shot noise units of excess noise : χ = χ0 + 2.
P
Experiment in 3 steps :
Alice sends, Eve (using Bob) mesures X
X S
Alice sends, Eve (using Bob) mesures P
Eve (using Alice) resends (x, p)
Bob


Noise analysis

χA→B = χA→E + χE →B
Ève

Ève

Bob
+ =

Alice Bob Alice


Noise analysis

= χ − χ0 = 2 + T
where χ is measured and χ0 = 1/T − 1

6
χ
χ0
Noise referred to the input

5
ε
4
(SNL units)

3

2

1

0
0 0.2 0.4 0.6 0.8 1
Channel transmission (T)
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. On
the remaining data, she performs a standard BS attack.

Eve
BS P S Bob

1−µ µ
X
Alice IR

Properties of the partial IR
Eve can introduce an arbitrary amount of excess noise.
It is a simple non-Gaussian attack.

Noise analysis
Excess noise referred to the input

2.5
Theoretical excess noise
Experimental excess noise
2
(ξ in SNL units)

1.5
Excess noise:
1 = µ( IR + T )+(1−µ) T
= 2µ + T
0.5

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Intercepted−reemitted pulse fraction (µ)


Information analysis
Information analysis

Information rates can be computed from experimental data (T = = 0.25)
Information can be computed from experimental data (T 0.25)
Excess noise
0 0.4 0.8 1.2 1.6 2
1.5 Eve’s information rates
Information rates attack
IBE ,G : optimal Gaussian
IIBE ,NG : a Non-Gaussian attack : sub-optimal !
Information rate (bits/pulse)

1.4 AB experimental
IBE ,IR : implemented attack : not so bad !
IBE Gaussian model
IIBE ,BS : Beam-Splitter only : much weaker !
BE Beam Splitter
1.3
IBE experimental
IBE non-Gaussian
1.2 Security margins have to be considered to take
Bob’s information rate
into account statistical ﬂuctuations
IAB : measured on the experiment
1.1
(security margins have been included to take
0 0.2 0.4 0.6 0.8 1 into account statistical ﬂuctuations).
Fraction of IR pulses

J. J. Lodewyck, R. Garc` Rev.nLett. 98, preparation
ıa-Patr` et al., in 030503 (2007)
Lodewyck et al., Phys. o
J´rˆ me Lodewyck (TRT/IOTA)
e o QKD with coherent states May 21, 2006 20 / 1

Outline







SECOQC Network Paradigm:
„Quantum Back Bone“
• Binary QBB-Links

• QBB-Nodes with multiple
QBB-Links to neighbouring
QBB-Nodes

• Hop-by-hop distribution of
secrets

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
BuildingContinuous Variables quantum key distribution setup fiber
Build a a coherent states quantum key distribution setup with
optics
with fiber optics and off-the-shelf telecom components
Noise & robustness& robustness
Characterize noise characterization.
Exploring (and detect !) real attacks
Simulate new physics (attacks simulation)
Building19” rack prototype for the the SECOQC european project.
Build a a 19 inches prototype for SECOQC european project.

´ ˆ
Thales meInstitut d’Optique (CNRS)
Jero & Lodewyck (TRT/IOTA) QKD with Continuous Variables
QKD with coherent states May 21, 2006 27 2 / 1
/ 30

Time multiplexing
Both signal and phase reference have to travel through the same fibre :
2

Alice

I
Signal
EOM − Amplitude EOM − Phase A0
Reference 99/1
DFB
Polarization
EOM 40 m controller
99/1

Bob I Channel
S 40 m 25 km
I
LO EOM − Phase 90%

Polarization 10%
controller

−
FIG. 1: Experimental setup. Alice generates modulated signal pulses, and Bob measures a random quadrature with a pulsed,
This solves phase and polarization fluctuations during propagation.
shot noise limited homodyne detector. The first EOM (left side) slices 100 ns pulses, and the EOM denoted as “A0 ” sets the
variance of Alice’s modulation. At the detection stage the signal S and local oscillator LO are overlapped using a delay line.

This involves:
III. IMPLEMENTATION

Introducing delay lines (40 m long) A. Experimental setup.

A scheme of the at shown on →It fiber It displaces a setup, of pulsedat 1550 nm and within10%
Demultiplexingset-up isBob’sFig. 1. components. coupler 90% (signal) exclusively (LO)
assembled with fiber optics and fast telecom
is a coherent-state QKD
train
working
coherent states
/ the
complex plane, with arbitrary amplitude and phase, randomly chosen from a two-dimensional Gaussian distribution
Controllingoscillator (LO), .with ∼ 10 width is 100 ns.the channelarbitrarywith a strong phase reference
– or local
the polarization atpulse. Bob can sent to an along measurement phase with controller)
with variances V ∼ 12 N The pulse
A 0
photons per9
The signal is
select
Bob output (active
a
phase modulator placed on the LO path. The selected quadrature is measured with an all-fiber shot noise limited,
Tested on an installed fiber (750 m) and on a fiber coil (25 km) : OK !
time-resolved homodyne detector. A key transmission is composed of independent blocks of 50000 pulses, sent at a
rate of 500 kHz, among which 10000 test pulses with agreed amplitude and phase are used to synchronize Alice and
Bob and to determine the relative phase between the signal and LO (see [2] for more details), and 5000 for channel
Thales & Institut d’Optique “effective” pulse rate usedwith the secret bit ratesVariables is thus 350 kHz.
evaluation. The (CNRS) QKD in for Continuous quoted below 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 !


Outline







Where is the entanglement ?
EPR versus coherent protocol
(XA + XB ) and (PA - PB )
P
EPR are squeezed
source X
Bob
LO Alice measures
XA or PA on half The state received by Bob is prepared in a
an EPR beam squeezed state, conditional to Alice ’s result

50-50 BS P
EPR
source
Bob X
LOs Alice measures
( /2) XA and PA on half The state received by Bob is prepared in a
an EPR beam coherent state, conditional to Alice ’s result

EPR protocol equivalent to our coherent state protocol !
Cf BB84 vs entangled pair (Ekert) protocol - Crucial for the security proofs !

Quantum repeaters for continuous variables

For quantum cryptography only « virtual » entanglement is required
But quantum repeaters do need « real » entanglement.

EPR source
XA , P A ? EPR source ? XB , P B

EPR source

Is it possible to carry some operation on several EPR beams
to 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

Continuous-variables EPR beams
QIPC

XA , PA EPR XB , PB
source

(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) !

How to produce QCV entangled beams ?
QIPC
1. Combine two orthogonally squeezed beams on a 50-50 beamsplitter

P XC , PC
X XC squeezed XA , PA XC = ( XA + XB )/ 2
PC = ( PA + PB )/ 2
entangled ! XD = ( XA - XB )/ 2
P PD = ( PA - PB )/ 2
X XD , PD XB , PB
PD squeezed OK !

2. Use a Non-degenerate Optical Parametric Amplifier (NOPA)
P XA , PA
Vacuum state
X
Pump
NOPA
(khi-2 entangled !
beam
P crystal)
Vacuum state
X XB , PB

Homodyne detection,
Wigner Function and Quantum Tomography
QIPC

Local Oscillator
(classical) Signal (quantum) Homodyne detection :
Measures X = X*cos( )+P*sin( )
50/50
BS
Marginals of W(X,P)
Quadrature distributions Π(X )

V1-V2 X Quadrature distributions Π(X )
W(X,P) : tomography

Specific quantum
W(X,P) states :
W(X,P)
negative Wigner Π(X)
function!
Π(X)
Many interesting
properties for P X
quantum
information
P X processing

« Schrödinger Kitten »
QIPC

• 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

Wigner function of a Fidelity between the kitten
Wigner function of a
Photon-subtracted and the most similar
small Schrödinger cat
squeezed state photon-subtracted state

Experimental Set-up
QIPC

Special feature: Femtosecond Ti-Sapph laser :
pulsed time- Pulse duration 180 fs , rep. rate 800 kHz
domain analysis

Frequence doubling in KNbO3
Single pass efficiency : SHG = 30%

Parametric amplifier in KNbO3 IR Filter
Typical (single pass) squeezing : 3 dB

R=10%
Pulsed Homodyne Detection APD
Global quantum efficiency : = 80%

Spatial & spectral
filtering

Wigner function of the Kitten
(corrected for homodyne efficiency)
QIPC

W0 = - 0.13 ± 0.01
s = 0.56, | | 0.9
Wideal kitten = - 0.32

= 0.70 (pure kitten)
+ 0.29 (pure squeezed)
+ 0.01 (residuals)

A. Ourjoumtsev et al, Science 312: 83, 7 april 2006

Gaussian entangled EPR beams
Jérôme Wenger, Alexei Ourjoumtsev et al., EPJD 2004

Measured covariance matrix :
[ (0)s from symmetry arguments ]

Gaussian
entangled state

Mode 1 Duan-Simon criterion :
Mode 2

Entropy of formation :
[G.Giedke et al, PRL 91 ,107901 (2003) ]

Coherent Photon Subtraction

Gaussian
entangled state

Coherent
Photon
Subtraction
Expected state structure :
Mode 1

Mode 2

X1 , P1
« Delocalized »
squeezed state

X2 , P2 « Delocalized »
Homodyne detections Schrödinger
Kitten state

Is this true ?

Delocalized Schrödinger Kitten

Mode 1
Quantum tomography on homodyne data :
Mode 2
X1 , P1

X2 , P2

Ws = Wsqueezed Wc = Wkitten

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 Results
A. Ourjoumtsev et al, Phys. Rev. Lett. 98, 030502 (2007)
Several projections of two measured two-mode Wigner
functions, corrected for homodyne losses, with R = 10% :
Initial
squeezing :
1.3 dB

3.2 dB

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)

Increasing the Entanglement up to 3 dB
A. Ourjoumtsev et al, Phys. Rev. Lett. 98, 030502 (2007)

Crossover
YES! increased entanglement
at R = 3% for
3 dB gain

For high gain (> 3dB)
small experimental
improvements may
have a strong effect
on the position of the
NO : crossover.
initial states too fragile
to resist an imperfect
photon subtraction

Long-Distance Quantum Communications

• Need to share highly entangled states (cryptography..)
• Problem : losses
• Solution : Entanglement Distillation :

Large
number of
weakly Small
entangled number of
states strongly
entangled
states

• But impossible to distill Gaussian
entanglement with Gaussian means
use non-gaussian operations !
(such as photon subtraction)

Violation of Bell’s Inequalities
« Aspect Experiment », Orsay, 1981- 1982

4p2 1S0 * Polarisation-entangled pairs of photons emitted by
an atomic cascade excited by two lasers.
1 = 551nm * Remote polarisations measurement on the two
4p4p 1P1 photons are very strongly correlated and cannot be
described by any « local realistic » model.
2 = 422nm Violation of Bell’s Inequalities
4s2 1S0 1982 expt : first test of « locality loophole »
Calcium 40 atomic beam

A new violation of Bell ’s inequalities ?
R. Garcia-Patron et al, Phys. Rev. Lett. 93, 130409 (2004)
QIPC

XA , PA XB , PB
EPR
source

(XA + XB ) and (PA - PB ) are squeezed : original EPR state (1935) !

* 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 P
sx = Sign(X) = ± 1, sp = Sign(P) = ± 1
and they compute the S parameter for Bell CHSH inequalities
S = < 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 !)

A new violation of Bell ’s inequalities ?
R. Garcia-Patron et al, Phys. Rev. Lett. 93, 130409 (2004)
QIPC

APD APD
XA , PA XB , PB
EPR
source

(XA + XB ) and (PA - PB ) are squeezed : original EPR state (1935) !

* Now « degaussify » by using two APDs (« event ready » detectors)

* Apply the same procedure (… but now the Wigner function of the generated
state 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 ?

BI test : increasing the value of S ?

• 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
S = 2.68
| g(x) = 0.848 | n = 1 + 0.152 | n = 5

f(x) g(x)

=> higher violation, but (very ?) difficult to prepare :
best compromise still to be found ?

Conclusion
QIPC

Many potential uses for Quantum Continuous Variables…
* Quantum cryptography
* Coherent states protocols using reverse reconciliation,
secure against any (gaussian or non-gaussian) collective attack

* Conditional preparation of « squeezed » non-gaussian pulses / 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 (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 state
J. Wenger & al., PRL 92, 153601 (2004)
QIPC

A squeezed state can be « degaussified » by photon subtraction
(one single photon in the APD beam)
Wigner function Wigner function

APD
X P X P
R<<1

Squeezed vacuum : Non-gaussian state :
|0 + |2 + |4 + … |1 + 2 (1-R) |3 + …

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 Distributions
after photon subtraction
Measured probability
distributions of the
quadratures components
as 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

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