Quantum cryptography: BB84 and uncertainty relations Ekert and entanglement no cloning theorem BB84 Ekert Implementations Eve: optimal individual attack Error correction, privacy amplification, advantage distillation Quantum Teleportation

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Quantum Communication Nicolas Gisin

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Quantum cryptography: BB84 and uncertainty relations Ekert and entanglement no cloning theorem BB84 Ekert Implementations Eve: optimal individual attack Error correction, privacy amplification, advantage distillation Quantum Teleportation principle, connection to optimal state estimation and cloning experiments quantum relays and quantum repeaters Optimal and generalized quantum measurements optimal quantum cloning POVMs (tetrahedron, unambiguous state discrimination) weak measurements Quantum CommunicationNicolas Gisin Q crypto: RMP 74, 145-195, 2002 Q cloning: RMP 77, 1225-1256, 2005

Quantum cryptography: a beautiful idea • Basic Quantum Mechanics: • A quantum measurement perturbs the system QM limitations • However, QM gave us the laser, micro-electronics, superconductivity, etc. • New Idea:Let's exploit QM for secure communications

If Eve tries to eavesdrop a "quantum communication channel", she has to perform some measurements on individual quanta (single photon pulses). • But, quantum mechanics tells us: every measurement perturbs the quantum system. • Hence the "reading" of the "quantum signal" by a third party reduces the correlation between Alice's and Bob's data. • Alice and Bob can thus detect any undesired third party by comparing (on a public channel) part of their "quantum signal".

The "quantum communication channel" is not used to transmit a message (information), only a "key" is transmitted (no information). • If it turns out that the key is corrupted, they simply disregard this key (no information is lost). • If the key passes successfully the control, Alice and Bob can use it safely. • Confidentiality of the key is checked before the message is send. • The safety of Quantum Cryptography is based on the root of Quantum Physics.

Modern CryptologySecrecy is based on: • Complexity theory • The key is public • The public key contains the decoding key, but it is very difficult to find (one way functions) • The security is not proven (no one knows whether one way functions exist) • Example: • 127 x 229 = 29083 Information theory The key is secrete The key contains the decoding key: Only the two partners have a copy ! The security is proven (Shannon theorem) Example: Message: 011001001 Key: 110100110 Coded message: 101101111

Alice Bob Eve Security from Heisenberg uncertainty relations P(X, Y, Z) Theorem 1: (I. Csiszàr and J. Körner 1978, U. Maurer 1993) If I(A:B) > min{I(A:E),I(B:E)}, then Alice & Bob can distil a secret key using 1-way communication over an error free authenticated public channel. where I(A:B) = Shannon mutual information = H(A)-H(A|B) = # bits one can save when writing A knowing B

Finite-coherent attacks • Theorem 2(Hall, PRL 74,3307,1995) • Heisenberg uncertainty relation in Shannon-information terms: • I(A:B) + I(A:E) < 2.log(d.c) • where c=maximum overlap of eigenvectors and d is the dimension of the Hilbert space. For BB84 with n qubits, d=2n and c=2^(-n/2). Hence, Theorem 2 reads: I(A:B) + I(A:E) < n It follows from Csiszàr and Körner theorem that the security is guaranteed whenever I(A:B) < 1/2 (per qubit) This corresponds exactly to the bound of the Mayers et al. proofs, i.e. QBER<11% Note: same reasoning valid for 6-state protocols, and for higher dimensions (M. Bourennane et al.).

Quantum Communication is the art of transferring a Q state from one place to another. Example: Q cryptography Q teleportation Quantum Information is the art of turning a Q paradox into a potentially useful task. Example: Q communication: from no-cloning to Q crypto Q computing: from superpositions to Q parallelism Note that entanglement and Q nonlocality are always present, at least implicitely. Though their exact power is not yet fully understood Quantum Communication

Theorem: let If AB is pure, then Ekert protocol (E91) a=x,z b=x,z source a b 0,1 0,1

Proof #1: Proof #2: (by contradiction) Source of entangled particules Alice Bob } * M clones Arbitrary fast signaling ! No cloning theorem and thecompatibility with relativity No cloning theorem: It is impossible to copy an unknown quantum state, /

universal no.signaling achievable by the Hillery-Buzek UQCM Optimal Universal non-signaling Quantum Cloning symmetric and universal N.Gisin, Phys. Lett.A 242, 1-3, 1998

source source Alice Indistinguishable from a single photon source. The qubit is coded in the a-basis And holds the bit value given by Alice results. BB84 E91 a=x,z b=x,z a b 0,1 0,1

Experimental Realization • Single photon source • Polarization or phase control during the single photon propagation • Single photon detection • laser pulses strongly attenuated ( 0.1 photon/pulse) • photon pair source (parametric downconversion) • true single-photon source • parallel transport of the polarization state (Berry topological phase) no vibrations • fluctuations of the birefringence thermal and mechanical stability • depolarization polarization mode dispersion smaller than the source coherence • Stability of the interferometers coding for the phase • avalanche photodiode (Germanium or InGaAs) in Geiger mode dark counts • based on supraconductors requires cryostats

Telecommunication wavelengths • Attenuation ( transparency) • Chromatic dispersion • Components available l [mm] a [dB/km] T10km 0.8 2 1% 1.3 0.35 44% 1.55 0.2 63% Two windows

" 0 or 1 or 2 or..." rather tha n 1 • Simple, handy, uses reliable technology Þ today’s best solution Single Photon Generation (1) • Attenuated Laser Pulse Poissonian Distribution Attenuating Medium 100% 80% Mean = 1 Mean = 0.1 60% Probability 40% 20% 0% 0 1 2 3 4 5 Number of photons per pulse

Avalanche photodiodes • Single-photon detection with avalanches in Geiger mode macroscopic avalanche triggered by single-photon Silicon: 1000 nm Germanium: 1450 nm InGaAs/InP: 1600 nm

Noise sources • Charge tunneling across the junction not significant • Band to band thermal excitation reduce temperature • Afterpulses release of charges trapped during a previous avalanche increase temperature Optimization !!!

Phase Coding • Single-photon interference Basis 1: fA = 0; p Basis 2: fA = p/2; 3 p/2 Basis: fB = 0; p/2 Incompatible: Alice and Bob ?? Compatible: Alice fA Di Bob Di fA Bases (fA-fB = np) (fA-fB = p/2)

Time Window short - long + long - short Coincidences long -long short -short 0 -3 -2 -1 0 1 2 3 Time (ns) • Problems: • stabilization of the path difference active feedback control • stability of the interfering polarization states Difficulties with Phase Coding • Stability of a 20 km long interferometer?

Faraday rotator • standard mirror ( incidence) • Faraday rotator FM Independent of Faraday mirrors

RMP 74, 145-195, 2002, Quant-ph/0101098 QC over 67 km, QBER 5% D. Stucki et al., New Journal of Physics 4, 41.1-41.8, 2002. Quant-ph/0203118 + aerial cable (in Ste Croix, Jura) !

Company established in 2001 • Spin-off from the University of Geneva • Products • Quantum Cryptography (optical fiber system) • Quantum Random Number Generator • Single-photon detector module (1.3 mm and 1.55 mm) • Contact information email: info@idquantique.com web: http://www.idquantique.com

lp ls,i laser nonlinear birefringent crystal filtre Photon pairs source • Parametric fluorescence • Energy and momentum conservation • Phase matching determines the wavelengths and propagation directions of the down-converted photons

2-photon Q cryptography:Franson interferometer Two unbalanced interferometers no first order interferences photon pairs possibility to measure coincidences One can not distinguish between "long-long" and "short-short" Hence, according to QM, one should add the probability amplitudes interferences (of second order)

F L P Laser KNbO 3 Photon pairs source (Geneva 1997) • Energy-time entanglement • diode laser • simple, compact, handy 40 x 45 x 15 cm3 • Ipump = 8 mW • with waveguide in LiNbO3 with quasi phase matching, Ipump 8 W 655nm output 1 output 2 crystal lens filter laser

j1 y2 j2 y1 analyzer analyzer Quantum non locality • the statistics of the correlations can‘t be described by local variables Quantum non locality single counts single counts _ b b y1 j1 y2 j2 a-b

f i y = a + b s e l Alice Bob 1 1 0 0 j f D 0 n h D 1 switch switch varia ble coupler variable coupler The qubit sphere and the time-bin qubit • qubit : • different properties : spin, polarization, time-bins • any qubit state can be created and measured in any basis

FM C 2 d 1 3 FM The interferometers • single mode fibers • Michelson configuration • circulator C : second output port • Faraday mirrors FM: compensation of birefringence • temperature tuning enables phase change

f s A l A variable coupler non-linear crystal l B B s • depending on coupling ratio and phase f, maximally and non-maximally entangled states can be created • extension to entanglement in higher dimensions is possible • robustness (bit-flip and phase errors) depends on separation of time-bins entangled time-bin qubit

KNbO 3 test of Bell inequalities over 10 km Bellevue 4.5 km FM d 1 Z FM + quantum channel APD 1 Genève 8.1 km - APD 1 R++ F P L R-+ 10.9 km & laser classical channels R+- R-- - APD 2 9.3 km + APD 2 FS quantum channel Z d 2 FS 7.3 km Bernex

1.0 0.5 0.0 correlation coefficient -0.5 V = (85.3 0.9)% ± raw V = (95.5 1) % ± net. 0 1000 4000 7000 10000 13000 time [sec] results • 15 Hz coincidences • Sraw = 2.41 Snet = 2.7 • violation of Bell inequalities by 16 (25) standard-deviations • close to quantum-mechanical predictions • same result in the lab

Violation of Bell inequalities by more than 15s Bell test over 50 km • With phase control we can choose four different settings a = 0°or 90°andb = -45° or 45° • Violation of Bell inequalities:

PRL 93, 010503, 2004 Bell Violation I(lhv) = 2 < I(2) = 2.829 < I(3) = 2.872 I = 2.784 +/- 0.023

Coincidences Da-Db (red) and Da-Db’ (blue) as function of time while varying the phase a D. Stucki et al., quant-ph/0502169 Two-photon Fabry-Perot interferometer Aim : direct detection of high dimensional entanglement NLC : non linear crystal

Plasmon assisted entanglement transfer polarization direction polarization direction fiber fiber BCB BCB 15 15 20nm 20nm BCB BCB 15 15 Si-waffer Si-waffer phase SS+LL 1 cm s t n LS SL e v e TAC difference of detection time a short lived phenomenon like a plasmon can be coherently excited at two times that differ by much more than its lifetime. At a macroscopic level this would lead to a “Schrödinger cat” in superposition of living at two epochs that differ by much more than a cat’s lifetime.

Experimental QKD with entanglement cw source Alice Bob NL crystal J. Franson, PRL 62, 2205, 1989 W. Tittel et al., PRL81, 3563-3566, 1998

G. Ribordy et al., Phys. Rev. A 63, 012309, 2001 S. Fasel et al.,European Physical Journal D, 30, 143-148, 2004 P.D. Townsend et al., Electr. Lett. 30, 809, 1994R. Hughes et al., J. Modern Opt. 47, 533-547 , 2000 A. Shields et al., Optics Express 13, 660, 2005 QKD Alice Bob N. Gisin & N. Brunner, quant-ph//0312011