Quantum key distribution with polarized coherent states

1. Quantum key distribution with polarized coherent states Antonio Vidiella Barranco vidiella@ifi.unicamp.br Quantum Optics Group Instituto de Física “Gleb Wataghin” Universidade Estadual de Campinas 13083-970 Campinas SP Brazil

2. Quantum Optics Group People involved with quantum cryptography • Dr. Antonio Vidiella Barranco • Dr. José A. Roversi • Mr. Luis F.M. Borelli • Mr. Leandro da S. Aguiar • Mr. Felipe de C. Lourenço Sponsored by

3. Outline • Introduction • Quantum key distribution protocols • Stokes variables • Polarized coherent states • An all continuous variable protocol • Conclusions

4. Introduction Quantum key distribution (QKD) After twenty years of BB84 we have witnessed a few advances, J.A. Smolin, IBM J. Res. & Dev. 48, 47 (2004)

5. such as the first bank transfer using QKDin 2004 Bank QKD in a “real environment” 1.45 km long fibres Vienna City Hall A. Poppe et al., Optics Express 12, 3865 (2004)

6. As well as the first quantum network QNet http://www.bbn.com/networking/quantumcryptography.html

7. What we would like to achieve • Use of available sources of light and detectors • Higher speed transmission rate • Secure transmission over noisier transmission lines • Integration with conventional communication systems

8. Single photon sources • Difficult to build ! Most developed; fibres and free-space Implementations of QKD 2. Weak laser pulses

9. 4. Continuous variables Emerging field Implementations of QKD 3.Entangled beams Offerssecurity advantages

10. or or Quantum protocols Polarization states are at the heart of BB84 • It employs highly distinct states Vertical basis Diagonal basis

11. or Quantum protocols • Security based on the fact that If someone (Eve) uses an incompatible basis, a “wrong” state comes out prepare measure

12. Continuous variables QKD • Alternative to single photon schemes • First schemes were hybrid ones Continuous variables but discrete encoding(in bits) M. Hillery, Phys. Rev. A61, 022309 (2000) T.C. Ralph, Phys. Rev. A 61, 010303(R) (2000)

13. Encoding variables Continuous variables QKD • Quadrature operators Uncertainty principle

14. X2 X1 Continuous variables QKD • Squeezed states seemed natural candidates But again, not so easy to generate and transmit!

15. Also uses Continuous variables QKD • Other methods Bright entangled beams Effects of losses? Synchronization Ch., Silberhorn, N. Korolkova, G. Leuchs, Phys. Rev. Lett. 88, 167902 (2002)

16. Continuous variables QKD Coherent states Readily available sources → laser light No need of squeezed or entangled sources key encoding in F. Grosshans and P. Grangier , Phys. Rev. Lett. 88, 057902 (2002)

17. Continuous variables QKD • Gaussian modulation • of signal guarantees • secure transmission • through lossy lines • …but requires • synchronized stations • for homodyne detection • …can we go further? F. Grosshans et al., Nature421, 238 (2002).

18. Continuous variables QKD • Up to now the key elements have been the amplitude (X1) and phase (X2) quadratures • Are there other convenient encoding variables ? Polarization variables !!! S. Lorenz, N. Korolkova, G. Leuchs, quant-ph/0403064 L.F.M. Borelli and AVB, quant-ph/0403076

19. Stokes variables • Classical optics Stokes parameters Electromagnetic (polarized) wave

20. Stokes operators • Quantum optics Stokes operators Non-commuting, e.g.,

21. Stokes operators • For a two-mode coherent state The same values as for a classical wave → quasi-classical state

22. Classical wave + fluctuations Stokes operators • But exhibits quantum fluctuations in polarization

23. Polarized coherent states • Highly polarized beams in x direction Convenient to normalize Analogue to quadratures

24. All continuous protocol • Key elements Alice prepares a highly polarized beam, so that S2and S3 are small modulations randomly drawn from a Gaussian distribution with variance Vm AVB and L.F.M. Borelli, submitted for publication

25. λ/2 λ/4 Laser EOM MOM λ/2 PBS While Eve eavesdrops All continuous protocol Classical channel Bob randomly measures eitherS2or S3 Alice sends the modulated signal to Bob via a noisy quantum channel Quantum channel

26. strings ofreal numbers, instead All continuous protocol After several transmissions… Not a secret shared key yet!

27. Classical channel All continuous protocol Reconciliation and privacy amplification → classical operations on numbers ? secret binary key N.J. Cerf, S. Iblisdir and G. Van Assche, Eur. Phys. J. D 18 211 (2002); G. Van Assche, J. Cardinal and N.J. Cerf, IEEE Transac on Inf. Theory 50, 394 (2004)

28. UQCM S2A S2B S2E Noise All continuous protocol Security under cloning attack QM

29. All continuous protocol • If Bob and Eve measure different Stokes parameters, For convenience we may normalize the deviations to S0

30. All continuous protocol obtaining Crossed uncertainty relation even a little noise in Eve’s copy (∆E) will cause a disturbance on Bob’s one (∆B)

31. Bob’s signalvariance Noise variance All continuous protocol • Gaussian noisy channel Shannon’s formula for Alice and Bob mutual information

32. Eve’s signalvariance Noise variance All continuous protocol • Gaussian noisy channel Shannon’s formula for Alice and Eve mutual information

33. All continuous protocol We may again normalize Because of the no-cloning theorem, the minimum noise added by Eve will be and therefore

34. All continuous protocol • In order to have a secure key distillation it must hold (for direct reconciliation) Using the expressions above for IAB and IAEwe obtain

35. All continuous protocol • If v<1 ΔI increases as a function of the modulation variancevm • Losses are related to the noise in the quantum channel – for a line with transmission η, Secure transmission in a channel with η < 0.5 F. Grosshans and P. Grangier , Phys. Rev. Lett. 88, 057902 (2002)

36. Another protocol • Key encoding using four Stokes parameters Overlap of polarization states due to quantum noise makes it impossible to Eve to distinguish among them. Bob simply discards events below a certain threshold →post-selection Key exchange with η ~ 0.64 S. Lorenz, N. Korolkova, G. Leuchs, App. Phys. B 79, 273 (2004)

37. Conclusions • New possibilities for continuous variable QKD • “Key carriers” → polarized coherent states • Easy to generate • Encoding in Stokes parameters • Easy to modulate • Easy to measure • But still a lot of work to be done → new reconciliation procedures, attacks, etc. • Experimental work being carried out