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Quantum Information Quantum Cryptography Quantum Teleportation Quantum Cryptography What hath God wrought. - first message sent by telegraph, by Samuel F. B. Morse Classical Cryptography Cryptography – The art of encoding a message in such a way that only the person to whom it is addressed can read it. The ENIGMA code The key – one-time-pad Θ = Subtraction mod 2: Θ 110111111000001 111011000111001 001100111111000 Solutions to key transfer problem: Multiple use of the same key Public key encryption. EX: RSA Basic Principles Of Quantum Cryptography The purpose: To provide a reliable method for transmitting a secret key and knowing that no one has intercepted it along the way The process of sharing a secret key is called – Quantum Key Distribution. The methods: quantum measurement on single particles entagled states Quantum cryptography does not protect against eavesdropper attacks, but provides a failsafe way of knowing if there has been intercepting Basic Principles Of Quantum Cryptography BS+Amplifier Strong pulse=1 Weak pulse=0 Classically – It is possible to make an exact duplicate of the signal. Quantum mechanics – No-Cloning Theorem. Proof: Suppose a state in system A and B: We can: A e B A e B Perform an observation Control the hamiltonian: We choose: U A e B e B A U †U A A e U B B B A A e B A A B B 2 Basic Principles Of Quantum Cryptography Example: Measure the polarization state of a single photon PBS - Polarizing Beam Splitter On a general case: cos sin 2 P cos 2 P sin Eve can reproduce the same state only for the special case of 0,90 !! Quantum Key Distribution According To The BB84 Protocol A protocol for sharing the private key in a secure way. Basis Binary 1 Binary 0 2 possible bases: The procedure: 1. 2. 3. 4. 5. 0 90 45 135 The sender determines the basis. The receiver determines his basis. The bases are compared in an unsecured communication. The receiver resend a subset of the correct bits. If the error is less then 25% - the transmission was safe and the rest of the correct bits are the private key. Quantum Key Distribution According To The BB84 Protocol Perror PEve has wrong basis PBob gets wrong result 25% System Errors Errors might occur even without eavesdropper: Photon deletion Birefringence Detector dark counts System Errors 1. Photon Deletion Can occur for a number of reasons: Absorption Scattering Detector inefficiency Does not affect information security When comparing selected bases (stage #3) times comparison must be added System Errors 2. Birefringence Such a medium would change the polarization angle Shannon’s noisy channel coding theorem: Ncorrection N log2 (1 )log2 (1 ) N =number of correction bits = error probablility N grows with It can be reduced on the account of the transmission rate System Errors 3. Detector dark counts The original photon never reaches the detector and the detector randomly registers due to thermal noise in the photocathode Error correction can be established as in Birefringence These errors cause a reduction in the length of the private key. It reduces the efficiency of the system, but do not affect it’s security Error correcting The procedure: 1. 2. 3. 4. 5. 6. Permuting a random number of bits Divide the string into blocks of size b Compare their parity discarding the last bit Repeat the procedure for the blocks containing errors with smaller size blocks Correct the error giving only the chosen basis Repeat the total process to eliminate the even error number blocks After 20 parity checks, the probability of an undiscovered eavesdropper is 1 in a million! Identity verification It is impossible to know whether an eavesdropper impersonates the receiver. Only by using the first authenticate private key, it is possible to transmit a newer one. Single Photon Source More then one photon at time would risk giving up information to the eavesdropper The laser must have attenuated beam at a certain frequency The average number of photons per pulse, n, has Poisson distribution A rate of 5% of n>1 to n=1 is alarmingly high! Genuine single-photon source is the solution but still on development Single Photon Source HBT - Hanbury–Brown and Twiss Practical Demonstrations Of Quantum Cryptography Two broad categories: Free-space quantum cryptography Quantum cryptography in optical fibres Practical Demonstrations Of Quantum Cryptography Free-space quantum cryptography Bennett & Brassard ’92: Light Emitting Diode in 550nm Air gap – 0.32m (today – 23km). The goal: communicating with satellites in low earth orbitals. At long range 600-900nm are used in which the atmospheric losses are small, and low-noise detectors with high quantum efficiency are available. These conditions provide two main sources of errors: Air turbulence Stray light Most of these problems are occurred at low heights. Practical Demonstrations Of Quantum Cryptography Quantum cryptography in optical fibers Advantage: The beam does not diverges Disadvantages: signal decay: 850nm introduces greater decay but is used with silicon low-noise SPAD 1300/1550nm energy is below silicon’s band gap and have higher dark count rate, and suffer from afterpulsing. Birefringence of the fiber: In long distances thermal and mechanical induction changes the fiber birefringence. A solution – optical phase encoding – Mach-Zender. Practical Demonstrations Of Quantum Cryptography Mach-Zender The achievements: 850nm – 100kbps over 4.2 km 1550nm – 122km. Gentlemen do not read each other's mail Henry Stimson, U.S. Secretary of State Quantum Teleportation "God does not play dice with the universe" -- Albert Einstein Entangled States Entangled state – the wave function can not be factorized into a product of the wave functions of the of the individual particles. EPR – Correlated photon pairs: The polarization of either photon measured independently of the other is random The polarization of the pair of photons is perfectly correlated: 1 01 , 02 11 ,12 1 01,12 11, 02 2 2 The measurement of one photon can determine in 100% the state of the other photon Entangled States Shroedinger’s Cat Paradox 1 live,1 dead , 2 2 Generation Of Entangled Photon Pairs 1967 - Kocher and Commins The initial and final state are both J 0 states → the photons emitted are with J 0 Both have even parity The photon pair have the polarization correlation properties required for EPRB 1 01 , 02 11 ,12 2 Generation Of Entangled Photon Pairs Down-conversion Using non-linear optics Phase matching condition: 1. 0 1 2 2. k0 k1 k2 Entangled state occur only on intersection If 1 2 20 the process is called- degenerate The dispersion relation Vs. the birefringence produces the one of the matching: Type 1 – Same polarization Type 2 – Different polarization Generation Of Entangled Photon Pairs Type 2 example: UV from a pump laser is focused into a -barium borate crystal The photon is converted into 2 red degenerate photons: 1 1 , 2 2 ei 1 , 2 By gauge of one can achieve either state. This method is widely used today Single-Photon Interference Experiment 1987 Hong-Ou-Mandel Interferometer Coherence length - The distance over which interference will occur. Coherence length of an optical source is affected by the size of the source, spatial coherence, the phase purity of the source, temporal coherence, and the spectral bandwidth of the light. Single-Photon Interference Experiment 1991- Zou-Wang-Mandel The mere possibility of which-path information is sufficient to destroy the single photon interference Bell’s Ineqaulity The conditions: 1 2 P11 (1 ,2 ) P10 (1 ,2 ) 0.5 P11 ( , ) 0.5 P01 (1 ,2 ) P00 (1 ,2 ) 0.5 P10 ( , ) 0 P11 (1 ,2 ) P01 (1 ,2 ) 0.5 P01 ( , ) 0 P10 (1 ,2 ) P00 (1 ,2 ) 0.5 P00 ( , ) 0.5 1 cos 2 2 1 P10 (1 , 0) sin 2 2 1 P01 (1 , 0) sin 2 2 1 P00 (1 , 0) cos 2 2 P11 (1 , 0) Bell’s Ineqaulity Quantum Machanics LHV 1 1 2 2 2 2 P11 (1 , 2 ) cos 2 (1 2 ) P ( , ) (sin sin cos cos 2 ) 11 1 2 1 2 1 2 2 1 2 1 2 2 2 2 P10 (1 , 2 ) sin (1 2 ) P ( , ) (sin cos cos sin 2 ) 10 1 2 1 2 1 2 2 1 1 P01 (1 , 2 ) sin 2 (1 2 ) P01 (1 , 2 ) (cos 2 1 sin 2 2 sin 2 1 cos 2 2 ) 2 2 1 1 2 P ( , ) (cos 2 1 cos 2 2 sin 2 1 sin 2 2 ) P00 (1 , 2 ) cos (1 2 ) 00 1 2 2 2 E (1 ,2 ) P11 (1 ,2 ) P00 (1 ,2 ) P10 (1 ,2 ) P01 (1 , 2 ) S E (1, 2 ) E (1, 2 ) E (1, 2 ) E (1, 2 ) Bell’s Inequality: Inequality violation: 1 0 2 22.5 2 S 2 1 45 2 67.5 S 2 2 Experimental Confirmation Of Bell's Theory 1981-82 Alain Aspect 3 experiments: Different angels Limits of S Non-local correlation The violation of Bell’s inequality has been confirmed Principles Of Teleportation The basic idea: To transfer the quantum state of one photon to another that is physically separated from it. To exchange quantum information without direct transformation of Qubits. General principles: 1. 2. 3. 4. The input photon must either be destroy or lose its initial state in an irretrievable way (according to the no-cloning theorem. The more info gleaned about - the less fidelity gained. No matter is teleported between the input and output, only quantum information. It is impossible transmit information faster then the speed of light. Principles Of Teleportation 1993 – Bennett – Teleporting of photon polarization 1 C0 0 1 C1 1 1 C0 C1 1 2 2 23 1 0 2 2 1312 0 3 Principles Of Teleportation The process: The full wave function of the 3 particles: 1 C0 0 1 C1 1 1 0 2 1 3 1 2 0 3 123 2 1 C0 0 1 0 2 1 3 C0 0 1 1 2 0 3 C1 1 1 0 2 With Bell’s notation: 123 2 1 3 C1 1 1 1 2 0 C0 1 C1 0 3 3 12 1 12 C0 1 3 C1 0 3 2 C0 0 C1 1 3 3 12 C0 0 3 C1 1 3 12 3 2 remarks: The protocol can only work after transmitting the result classically. Photons 1 and 2 are entangled and there is no information about the coefficients, hence there was no duplication. Experimental Demonstration of Teleportation 1997-8 Bouwmeester Bibliography Mark Fox, “Quantum Optics”, Oxford University Press, 2006 http://www.ai.sri.com/~goldwate/quantum.html#PrivAmp http://www.csa.com/discoveryguides/crypt/overview.php Bennett, C. H., Bessette, F., Brassard, G., Salvail, L., and Smolin, J., "Experimental Quantum Cryptography", Journal of Cryptology, vol. 5, no.1, 1992, pp. 3-28. A. Muller, T. Herzog, B. Huttner,a) W. Tittel, H. Zbinden, and N. Gisin ‘‘Plug and play’’ systems for quantum cryptography, 1996 www.wikipedia.org A. Poppe, A. Fedrizzi, H. Hübel, R. Ursin, A. Zeilinger “Entangled State Quantum Key Distribution and Teleportation”