Continuous-variable QKD over long distances

Anthony Leverrier, Télécom ParisTech

Feynman Festival, June 2009

joint work with Philippe Grangier,Institut d’Optique

Outline of the talk

• Continuous-variable QKD- quick overview –

• Towards long distance CVQKD- with a discrete modulation –

Continuous-variable QKD

- Quick overview -

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

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)

Gaussian channel modelThe coherent states sent in the quantum channel can be

altered by:

Losses 1-Tdecrease the signal amplitude« vacuum » added noise 1/T-1

Losses 1-Tdecrease the signal amplitude« vacuum » added noise 1/T-1

Excess noise εAbove the shot noise limitEquivalent to errors in BB84

Excess noise εAbove the shot noise limitEquivalent to errors in BB84

Total noise 1/T-1+ ε Total noise 1/T-1+ ε

Security proofs (1/2) Prepare & measure protocol

Equivalent entanglement-based protocol

- Used in practice- Alice sends coherent states

with a Gaussian modulation

- Used in practice- Alice sends coherent states

with a Gaussian modulation

- Used for security proofs- Alice measures one half of an

EPR pair and projects the other half on a coherent state

- 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)

K = β IAB - IBEK = β IAB - IBE

Security proofs (2/2)

Directly observed

Upper bound for IBE ?

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)

Extremality of Gaussian states

Extremality of Gaussian states R. García-Patrón and N. Cerf, PRL 97, 190503 (2006)

Pros & Cons- No need to produce nor detect single photons- Uses only fast and standard telecom components- High key rate achievable in principle

but …

- 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-correctionWhy ? Because of error-correction

Discrete Variables

Continuous Variables

Support of information

Single photons(or attenuated

coherent states)Coherent states

Detection technique

Photon countingHomodyne detection

Effect of losses Deletion Noise

Signature of an eavesdropper

Errors (QBER) More noise !!

PerformancesLong distance (100-200 km)

High key rateAt short

distance (30-50 km)

Main limitationTechnology (detectors)

Postprocessing (error

correction)

Towards long distance CVQKD

- With a discrete modulation -

Impact of reconciliation efficiencyK = β IAB – IBE

Impact almost negligeable while β ≈ 80%Long distance one needs to work at low SNR

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

Gaussian variables are difficult to reconcile at low SNR that’s why CVQKD with Gaussian modulation is limited to short distances

Gaussian or discrete modulation ?

• K = β IAB - IBE

• One wants to maximize β IAB

• A Gaussian modulation maximizes IAB

… but not β IAB

• 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). At low SNR, IAB(discrete) ≈ IAB(Gaussian).

Binary variables are easy to reconcile

A discrete modulation takes care of the reconciliation problem !A discrete modulation takes care of the reconciliation problem !

Gaussian modulation discrete modulation

P

Q

P

Q

N0

-A

-A

1 = /4

A

A

Alice’s modulationAlice’s modulation

After the channelAfter the channel

The new Prepare & Measure protocolThe new Prepare & Measure protocol

Bob measures a random quadrature

Raw key Bob sends the absolute value to Alice

Works well, even for VERY noisy data

What about the security of the new protocol ?What about the security of the new protocol ?

Alice performs a projective measurement on the first half of . This projects the second half on one of the four coherent states.

Alice performs a projective measurement on the first half of . This projects the second half on one of the four coherent states.

Entanglement based version of the protocol:

Coherent statesOrthogonal states

There exist s. t.

For small variance

Hence, IBE(discrete) ≈ IBE(Gaussian)

Performances

same as discrete-variable QKD !!same as discrete-variable QKD !!

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)

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

SimilaritiesSimilarities

DifferencesDifferences