Continuous-variable QKD over long distances

1. Continuous-variable QKD over long distances Anthony Leverrier, Télécom ParisTech joint work with Philippe Grangier, Institut d’Optique Feynman Festival, June 2009

2. Outline of the talk • Continuous-variable QKD - quick overview – • Towards long distance CVQKD - with a discrete modulation –

3. Continuous-variable QKD - Quick overview -

4. Quantum Key Distribution • QKD  Alice & Bob can share a secret key. • This key can be used for classical cryptography (one-time pad, AES) • If IAB > IAE (or IAB > IBE) then A & B can distill a secret key • QM imposes tradeoffs between IAB, IAE & IBE

5. QKD with coherent states • Alice encodes information onto the quadratures of the EM field • Coherent states with a Gaussian modulation • Bob detects this state with an homodyne (interferometric) detection  Continuous-Variable QKD. F. Grosshans et al., Nature 421 238 (2003)

6. Gaussian channel model The coherent states sent in the quantum channel can be altered by: • Losses 1-T • decrease the signal amplitude • « vacuum » added noise 1/T-1 • Excess noise ε • Above the shot noise limit • Equivalent to errors in BB84 • Total noise 1/T-1+ ε

7. Security proofs (1/2) • Prepare & measure protocol • Used in practice • Alice sends coherent states with a Gaussian modulation • Equivalent entanglement-based • protocol • Used for security proofs • Alice measures one half of an EPR pair and projects the other half on a coherent state F. Grosshans, et al, Quantum Inf. Comput. 3, 535 (2003)

8. Security proofs (2/2) K = β IAB - IBE Directly observed Upper bound for IBE ? Extremality of Gaussian states R. García-Patrón and N. Cerf, PRL 97, 190503 (2006) • State of Alice and Bob: ρAB • IBE = f(ρAB) • f is unknown but is such that: • For any state ρ,f(ρ) ≤ f(ρG) where ρG is the Gaussian state with the same covariance matrix Γ as ρ • IBE≤ f’(ΓAB), which only depends on T and ε (=accessible experimental parameters)

9. Pros & Cons • No need to produce nor detect single photons • Uses only fast and standard telecom components • High key rate achievable in principle • but … V. Scarani et al., arxiv 0802.4155 (Review of Modern Physics) Why ? Because of error-correction

11. Towards long distance CVQKD - With a discrete modulation -

12. Impact of reconciliation efficiency K = β IAB – IBE Impact almost negligeable while β ≈ 80% Long distance  one needs to work at low SNR Gaussian variables are difficult to reconcile at low SNR  that’s why CVQKD with Gaussian modulation is limited to short distances

13. Gaussian or discrete modulation ? • K = β IAB - IBE • One wants to maximize β IAB • A Gaussian modulation maximizes IAB … but not β IAB  At low SNR, IAB(discrete) ≈ IAB(Gaussian).

14. Binary variables are easy to reconcile Gaussian modulation discrete modulation A discrete modulation takes care of the reconciliation problem !

15. The new Prepare & Measure protocol P Alice’s modulation A N0 P 1 = /4 -A A Q Q -A After the channel • Bob measures a random quadrature • Raw key • Bob sends the absolute value to Alice • Works well, even for VERY noisy data

16. What about the security of the new protocol ? Entanglement based version of the protocol: Coherent states Orthogonal states Alice performs a projective measurement on the first half of . This projects the second half on one of the four coherent states. • There exist s. t. • For small variance • Hence, IBE(discrete) ≈ IBE(Gaussian)

17. Performances • For small variance: • IAB(discrete) ≈ IAB(Gaussian) • IBE(discrete) ≈ IBE(Gaussian) • But β(discrete) ≈ 80% • K = βIAB-IBE > 0, even at long distance ! A.L and P. Grangier, PRL 102, 180504 (2009) same as discrete-variable QKD !!

18. Perspectives: CV vs DV protocols • Homodyne detection vs photon counting • DV: lots of erasures, but small QBER (< 10%) • CV: no erasure  high error rate (manageable with discrete modulation, not with a Gaussian modulation) • Same performances (long distance ! ) • Same support of information: coherent states with less than one photon per pulse Differences Similarities