Quantum Optics II – Cozumel December 2004 Quantum key distribution with polarized coherent states Quantum Optics Group Instituto de Física “Gleb Wataghin” Universidade Estadual de Campinas Campinas SP Brazil Antonio Vidiella Barranco

Quantum Optics II – Cozumel December 2004 Quantum Optics Group Dr. Antonio Vidiella Barranco Dr. Antonio Vidiella Barranco Dr. José A. Roversi Dr. José A. Roversi Mr. Luis F.M. Borelli Mr. Luis F.M. Borelli Mr. Leandro da S. Aguiar Mr. Leandro da S. Aguiar Mr. Felipe de C. Lourenço Mr. Felipe de C. Lourenço Sponsored by People involved with quantum cryptography

Quantum Optics II – Cozumel December 2004 Outline Introduction Introduction Quantum key distribution protocols Quantum key distribution protocols Stokes variables Stokes variables Polarized coherent states Polarized coherent states An all continuous variable protocol An all continuous variable protocol Conclusions Conclusions

Quantum Optics II – Cozumel December 2004 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)

Quantum Optics II – Cozumel December 2004 such as the first bank transfer using QKDin 2004 such as the first bank transfer using QKD in 2004 A. Poppe et al., Optics Express 12, 3865 (2004) QKD in a “real environment” 1.45 km long fibres Bank ViennaCityHall

Quantum Optics II – Cozumel December 2004 As well as the first quantum network QNet

Quantum Optics II – Cozumel December 2004 What we would like to achieve Use of available sources of light and detectors Use of available sources of light and detectors Higher speed transmission rate Higher speed transmission rate Secure transmission over noisier transmission lines Secure transmission over noisier transmission lines Integration with conventional communication systems Integration with conventional communication systems

Quantum Optics II – Cozumel December 2004 Implementations of QKD 2. Weak laser pulses Most developed; fibres and free-space 1. Single photon sources Difficult to build !

Quantum Optics II – Cozumel December 2004 Implementations of QKD 3. Entangled beams Offers security advantages 4. Continuous variables Emerging field Emerging field

Quantum Optics II – Cozumel December 2004 Quantum protocols Polarization states are at the heart of BB84 It employs highly distinct states It employs highly distinct states or Vertical basis Diagonal basis

Quantum Optics II – Cozumel December 2004 Quantum protocols Security based on the fact that Security based on the fact that If someone (Eve) uses an incompatible basis, a “wrong” state comes out or preparemeasure

Quantum Optics II – Cozumel December 2004 Continuous variables QKD Alternative to single photon schemes Alternative to single photon schemes First schemes were hybrid ones First schemes were hybrid ones M. Hillery, Phys. Rev. A 61, (2000) T.C. Ralph, Phys. Rev. A 61, (R) (2000) Continuous variables but discrete encoding (in bits)

Quantum Optics II – Cozumel December 2004 Continuous variables QKD Quadrature operators Quadrature operators Uncertainty principle Encoding variables

Quantum Optics II – Cozumel December 2004 Continuous variables QKD Squeezed states seemed natural candidates Squeezed states seemed natural candidates X2X2 X1X1 But again, not so easy to generate and transmit!

Quantum Optics II – Cozumel December 2004 Continuous variables QKD Other methods Other methods Bright entangled beams Ch., Silberhorn, N. Korolkova, G. Leuchs, Phys. Rev. Lett. 88, (2002) Effects of losses? Effects of losses? Also uses Synchronization Synchronization

Quantum Optics II – Cozumel December 2004 Continuous variables QKD Readily available sources → laser light No need of squeezed or entangled sources Coherent states key encoding in F. Grosshans and P. Grangier, Phys. Rev. Lett. 88, (2002)

Quantum Optics II – Cozumel December 2004 Continuous variables QKD F. Grosshans et al., Nature 421, 238 (2002). Gaussian modulation Gaussian modulation of signal guarantees secure transmission through lossy lines …but requires …but requires synchronized stations for homodyne detection …can we go further?

Quantum Optics II – Cozumel December 2004 Continuous variables QKD Up to now the key elements have been the amplitude (X 1 ) and phase (X 2 ) quadratures Up to now the key elements have been the amplitude (X 1 ) and phase (X 2 ) quadratures Are there other convenient encoding variables ? Are there other convenient encoding variables ? S. Lorenz, N. Korolkova, G. Leuchs, quant-ph/ L.F.M. Borelli and AVB, quant-ph/ Polarization variables !!!

Quantum Optics II – Cozumel December 2004 Stokes variables Classical optics Classical optics Stokes parameters Electromagnetic (polarized) wave

Quantum Optics II – Cozumel December 2004 Stokes operators Quantum optics Quantum optics Stokes operators Non-commuting, e.g.,

Quantum Optics II – Cozumel December 2004 Stokes operators For a two-mode coherent state For a two-mode coherent state The same values as for a classical wave → quasi-classical state

Quantum Optics II – Cozumel December 2004 Stokes operators But exhibits quantum fluctuations in But exhibits quantum fluctuations inpolarization Classical wave + fluctuations fluctuations

Quantum Optics II – Cozumel December 2004 Polarized coherent states Highly polarized beams in x direction Highly polarized beams in x direction Analogue to quadratures Convenient to normalize

Quantum Optics II – Cozumel December 2004 All continuous protocol Key elements Key elements Alice prepares a highly polarized beam, so that S 2 and S 3 are small modulations randomly drawn from a Gaussian distribution with variance V m AVB and L.F.M. Borelli, submitted for publication

Quantum Optics II – Cozumel December 2004 All continuous protocol Bob randomly measures eitherS 2 or S 3 either S 2 or S 3 Alice sends the modulated signal to Bob via a noisy quantum channel PBS LaserEOM MOM λ/2 λ/4 Quantum channel Classical channel While Eve eavesdrops

Quantum Optics II – Cozumel December 2004 All continuous protocol strings of real numbers, instead After several transmissions… Not a secret shared key yet!

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

Quantum Optics II – Cozumel December 2004 All continuous protocol Security under cloning attack UQCM S2AS2AS2AS2A S2BS2BS2BS2B S2ES2ES2ES2E Noise QM

Quantum Optics II – Cozumel December 2004 All continuous protocol If Bob and Eve measure different Stokes parameters, If Bob and Eve measure different Stokes parameters, For convenience we may normalize the deviations to S 0

Quantum Optics II – Cozumel December 2004 All continuous protocol Crossed uncertainty relation even a little noise in Eve’s copy (∆ E) will cause a disturbance on Bob’s one (∆B) obtaining

Quantum Optics II – Cozumel December 2004 All continuous protocol Gaussian noisy channel Gaussian noisy channel Shannon’s formula for Alice and Bob mutual information Bob’s signalvariance Bob’s signal variance Noise variance

Quantum Optics II – Cozumel December 2004 All continuous protocol Gaussian noisy channel Gaussian noisy channel Shannon’s formula for Alice and Eve mutual information Eve’s signalvariance Eve’s signal variance Noise variance

Quantum Optics II – Cozumel December 2004 All continuous protocol Because of the no-cloning theorem, the minimum noise added by Eve will be and therefore We may again normalize

Quantum Optics II – Cozumel December 2004 All continuous protocol In order to have a secure key distillation it must hold (for direct reconciliation) In order to have a secure key distillation it must hold (for direct reconciliation) Using the expressions above for I AB and I AE we obtain

Quantum Optics II – Cozumel December 2004 All continuous protocol If v < 1 Δ I increases as a function of the modulation variance v m If v < 1 Δ I increases as a function of the modulation variance v m Losses are related to the noise in the quantum channel – for a line with transmission η, Losses are related to the noise in the quantum channel – for a line with transmission η, F. Grosshans and P. Grangier, Phys. Rev. Lett. 88, (2002) Secure transmission in a channel with η < 0.5

Quantum Optics II – Cozumel December 2004 Another protocol Key encoding using four Stokes parameters 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 S. Lorenz, N. Korolkova, G. Leuchs, App. Phys. B 79, 273 (2004) Key exchange with η ~ 0.64

Quantum Optics II – Cozumel December 2004 Conclusions New possibilities for continuous variable QKD New possibilities for continuous variable QKD “Key carriers” → polarized coherent states “Key carriers” → polarized coherent states Easy to generate Easy to generate Encoding in Stokes parameters Encoding in Stokes parameters Easy to modulate Easy to modulate Easy to measure Easy to measure But still a lot of work to be done → new reconciliation procedures, attacks, etc. But still a lot of work to be done → new reconciliation procedures, attacks, etc. Experimental work being carried out Experimental work being carried out