Quantum key distribution with continuous variables at telecom wavelength

1,164 views

Published on

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
1,164
On SlideShare
0
From Embeds
0
Number of Embeds
5
Actions
Shares
0
Downloads
33
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Quantum key distribution with continuous variables at telecom wavelength

  1. 1. 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
  2. 2. 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
  3. 3. 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
  4. 4. 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 IABThales & Institut d’Optique (CNRS) QKD with Continuous Variables 4 / 30
  5. 5. 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 dimpulsions lumineusesodulation gaussienne (varianceGrosshans et al., Nature 421 238 F. 10 photons) (2003) b reçoit une version bruitée du signal : bruit de photonansmission du canal T ) et excès de bruit ξ . bThales &Q ou P d’Optique (CNRS) mesure Institut QKD with Continuous Variables 5 / 30
  6. 6. 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)Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 6 / 30
  7. 7. 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 +Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 7 / 30
  8. 8. 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 ) .Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 8 / 30
  9. 9. 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).Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 9 / 30
  10. 10. 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)Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 10 / 30
  11. 11. 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 nextraient quune 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 TransmissionThales & Jérôme Lodewyck (TRT/IOTA) QKD withRéconciliationVariables Institut d’Optique (CNRS) Continuous 23 juin 2006 4 / 13 11 / 30
  12. 12. 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
  13. 13. 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 − PhotodiodesThales & Institut d’Optique (CNRS) QKD with Continuous Variables 13 / 30
  14. 14. Alice and Bob set-up Alice BobThales & Institut d’Optique (CNRS) QKD with Continuous Variables 14 / 30
  15. 15. 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.Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 15 / 30
  16. 16. 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 inaccuraciesThales & Institut d’Optique (CNRS) QKD with Continuous Variables 16 / 30
  17. 17. 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)Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 17 / 30
  18. 18. 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)Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 18 / 30
  19. 19. 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
  20. 20. 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) BobThales & Institut d’Optique (CNRS) QKD with Continuous Variables 20 / 30
  21. 21. Noise analysis χA→B = χA→E + χE →B Ève Ève Bob + = Alice Bob AliceThales & Institut d’Optique (CNRS) QKD with Continuous Variables 21 / 30
  22. 22. 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 !Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 21 / 30
  23. 23. 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.Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 22 / 30
  24. 24. 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)Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 23 / 30
  25. 25. 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 / 1Thales & Institut d’Optique (CNRS) QKD with Continuous Variables 24 / 30
  26. 26. 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
  27. 27. 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
  28. 28. QBB Demonstrator in the SIEMENS Glass Fiber Network Vienna, Sept. 2008• 5 QKD Technologies• 5 QBB-Nodes / 7 QBB-Links
  29. 29. QBB Demonstrator inthe SIEMENS GlassFiber NetworkVienna, Sept. 2008
  30. 30. 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. ´ ˆThales meInstitut d’Optique (CNRS) Jero & Lodewyck (TRT/IOTA) QKD with Continuous Variables QKD with coherent states May 21, 2006 27 2 / 1 / 30
  31. 31. 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 channelThales & 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
  32. 32. 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
  33. 33. 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
  34. 34. 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 XLOs 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 !
  35. 35. Quantum repeaters for continuous variables For quantum cryptography only « virtual » entanglement is required But quantum repeaters do need « real » entanglement. EPR sourceXA , 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
  36. 36. 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) !
  37. 37. How to produce QCV entangled beams ? QIPC1. 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
  38. 38. Homodyne detection, Wigner Function and Quantum Tomography QIPCLocal 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
  39. 39. « 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
  40. 40. Experimental Set-up QIPCSpecial feature: Femtosecond Ti-Sapph laser : pulsed time- Pulse duration 180 fs , rep. rate 800 kHzdomain 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 APDGlobal quantum efficiency : = 80% Spatial & spectral filtering
  41. 41. 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
  42. 42. Gaussian entangled EPR beams Jérôme Wenger, Alexei Ourjoumtsev et al., EPJD 2004 Measured covariance matrix : [ (0)s from symmetry arguments ] Gaussianentangled stateMode 1 Duan-Simon criterion : Mode 2 Entropy of formation : [G.Giedke et al, PRL 91 ,107901 (2003) ]
  43. 43. Coherent Photon Subtraction Gaussian entangled state Coherent Photon Subtraction Expected state structure : Mode 1 Mode 2X1 , P1 « Delocalized » squeezed state X2 , P2 « Delocalized » Homodyne detections Schrödinger Kitten state Is this true ?
  44. 44. Delocalized Schrödinger Kitten Mode 1 Quantum tomography on homodyne data : Mode 2X1 , P1 X2 , P2 Ws = Wsqueezed Wc = Wkitten
  45. 45. SetupPhase control : Most phases fixed with waveplates 1 and 2 compensate each other only one phase to controlMeasure the two-mode correlation variance (800 kHz rep. rate reasonably fast)
  46. 46. 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 dBWigner 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)
  47. 47. Increasing the Entanglement up to 3 dB A. Ourjoumtsev et al, Phys. Rev. Lett. 98, 030502 (2007) CrossoverYES! 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
  48. 48. What next ?
  49. 49. 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)
  50. 50. Violation of Bell’s Inequalities « Aspect Experiment », Orsay, 1981- 19824p2 1S0 * Polarisation-entangled pairs of photons emitted by an atomic cascade excited by two lasers. 1 = 551nm * Remote polarisations measurement on the two4p4p 1P1 photons are very strongly correlated and cannot be described by any « local realistic » model. 2 = 422nm Violation of Bell’s Inequalities4s2 1S0 1982 expt : first test of « locality loophole » Calcium 40 atomic beam
  51. 51. A new violation of Bell ’s inequalities ? R. Garcia-Patron et al, Phys. Rev. Lett. 93, 130409 (2004) QIPCXA , 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 andmeasure 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 theoryNo violation here ! (the Wigner function provides a local hidden variable model !)
  52. 52. A new violation of Bell ’s inequalities ? R. Garcia-Patron et al, Phys. Rev. Lett. 93, 130409 (2004) QIPC APD APDXA , 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 ?
  53. 53. 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 ?
  54. 54. Conclusion QIPCMany 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...) …
  55. 55. « Degaussification » of a squeezed state J. Wenger & al., PRL 92, 153601 (2004) QIPCA squeezed state can be « degaussified » by photon subtraction (one single photon in the APD beam) Wigner function Wigner function APDX P X P R<<1Squeezed vacuum : Non-gaussian state : |0 + |2 + |4 + … |1 + 2 (1-R) |3 + …
  56. 56. 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)
  57. 57. Measured Probability Distributions after photon subtraction Measured probability distributions of the quadratures components as function of the LO phase Anti-squeezed quadrature Squeezed quadratureDip in the squeezed quadrature : hint for a negative Wigner function !
  58. 58. Wigner function of the « raw » measured state (no correction) Analytic Model Numerical Radon TransformRadon transform clearly negative ! (no hypothesis, no correction) A. Ourjoumtsev et al, Science 312: 83, 7 april 2006

×