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Quantum Teleportation, Quantum Superdense Coding, Quantum Cryptography, Quantum Imaging Quantum Information Processing

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Quantum Teleportation, Quantum Superdense Coding, Quantum Cryptography, Quantum Imaging Quantum Information Processing Quantum Information Science 양자정보과학 量子情報科學 信息 資訊 고등과학원 계산과학부 김재완 ([email protected]) Some stuffs are from Nielsen and Chuang’s book “Quantum computation and quantum information” MEASURING AND MANIPULATING INDIVIDUAL QUANTUM SYSTEMS THE LAUREATES SERGE HAROCHE French citizen. Born 1944 in Casablanca, Morocco. Ph.D. 1971 from Universite Pierre et Marie Curie, Paris, France. Professor at College de France and Ecole Normale Superieure, Paris, France. www.college-de-france.fr/site/en-serge-haroche/biography.htm DAVID J. WINELAND U.S. citizen. Born 1944 in Milwaukee, WI, USA. Ph.D. 1970 from Harvard University, Cambridge, MA, USA. Group Leader and NIST Fellow at National Institute of Standards and Technology (NIST) and University of Colorado Boulder, CO, USA. www.nist.gov/pml/div688/grp10/index.cfm Ion Trap Manipulate charged atom(ion) by light(laser) Cavity Quantum ElectroDynamics Measure light trapped between two mirrors by atoms M/F; S/J No Male No Female ? M/F; S/J No Male No Junior No Female M M/F; S/J Î Polarization M/F; S/J Î Polarization = = Quantum Beyond Nano “There’s plenty of room at the bottom.” -Feynman Moore’s Law :continue to shrinkÆ nanotechnology Æ meets quantum regime Æ uncertainty principle uncertainty in bits (0’s and 1’s) Quantum Technology Æ turns uncertainty into magic Actively utilizing quantumness ‘Let the quantum system solve the quantum manybody problems’ -Feynman I think of my lifetime in physics as divided into three periods. In the first period … I was in the grip of the idea that Everything is Particles, … I call my second period Everything is Fields. … Now I am in the grip of a new vision, that Everything is Information. – John Archibald Wheeler, Geon, Black Holes, and Quantum Foam – Businessweek 8/30/99: 21 Ideas for the 21st Century Quantum Information Physics Science Quantum Information Quantum Parallelism Science Æ Quantum computing is exponentially larger and faster than digital computing. Quantum Fourier Transform, Quantum Database Search, Quantum Many-Body Simulation (Nanotechnology) No Clonability of Quantum Information, Irrevesability of Quantum Measurement Æ Quantum Cryptography (Absolutely secure digital communication) Quantum Correlation by Quantum Entanglement Æ Quantum Teleportation, Quantum Superdense Coding, Quantum Cryptography, Quantum Imaging Quantum Information Processing • Quantum Computer NT – Quantum Algorithms: Softwares • Simulation of quantum many-body systems • Factoring large integers • Database search IT – Experiments: Hardwares •Ion Traps BT •NMR • Cavity QED • Quantum optics • Superconducting qubits, etc. Quantum Information Processing • Quantum Communication – Quantum Cryptography IT • Absolutely secure digital communication • Generation and Distribution of Quantum Key – optical fiber – wirelessÆ secure satellite communication – Quantum Teleportation • Photons • Atoms, Molecules • Quantum Imaging and Quantum Metrology Bit and Logic Gate 0Æ1 1Æ0 “Universal” NOT NAND 00Æ0 01Æ0 “Reversible” 10Æ0 AND 11Æ1 C-NOT 00Æ0 01Æ1 10Æ1 OR 11Æ1 CCN ( Toffoli ) “Universal” 00Æ0 “Reversible” 01Æ1 10Æ1 XOR 11Æ0 C-Exch ( Fredkin ) Classical Computation • Hilbert (1900): 23 most challenging math problems – Is there a mechanical procedure by which the truth or falsity of any mathematical conjecture could be decided? • Turing – Conjecture ~ Sequence of 0’s and 1’s – Read/Write Head: Logic Gates – Model of Modern Computers Turing Machine Finite State Machine: Head q′ ⎧qfqs′ = (,) ⎪ ⎨sgqs′ = (,) q ⎪ ⎩ddqs= (,) d Bit {0,1} ss→ ' Infinitely long tape: Storage Quantum Information •Bit{0,1} 2 2 Î Qubit ab0+ 1 with ab+=1 •N bits ⇒ 2N states, One at a time Linearly parallel computing AT BEST • N qubits Î Linear superposition of 2N states at the same time Exponentially parallel computing Î Quantum Parallelism Deutsch But when you extract result, you cannot get all of them. Quantum Algorithms 1. [Feynman] Simulation of Quantum Physical Systems with HUGE Hilber space ( 2N-D ) e.g. Strongly Correlated Electron Systems 2. [Peter Shor] Factoring large integers, period finding 1/3 tq ∝ Pol (N) tcl ∼ Exp (N ) 3. [Grover] Searching tq ∝√N 〈tcl〉 ~ N/2 Digital Computer N bits 1⋅ N N bits Digital Computers in parallel N bits m{ mN⋅ mN bits Quantum Computer : Quantum Parallelism N bits 2N N bits Quantum Gates Time-dependent Schrödinger Eq. Unitary Transform ∂ Norm Preserving iHh ψψ= ∂t Reversible ψψψ()te==−iHt / h (0) Ut () (0) ⎛⎞10 ⎛⎞10 ⎛⎞ 01 PIX(θ )== = ⎜⎟iθ ⎜⎟ ⎜⎟ ⎝⎠0 e ⎝⎠01 ⎝⎠ 10 ⎛⎞10 ⎛⎞ 0− 11 ⎛⎞ 11 ZYXZH=P()=π ⎜⎟ = = ⎜⎟ = ⎜⎟ ⎝⎠01−− ⎝⎠ 102 ⎝⎠ 11 111⎛⎞⎛⎞⎛⎞11 1 1 H 001===+⎜⎟⎜⎟⎜⎟()Hadamard 222⎝⎠⎝⎠⎝⎠110− 1 111⎛⎞⎛⎞⎛⎞11 0 1 H 101===−⎜⎟⎜⎟⎜⎟() 222⎝⎠⎝⎠⎝⎠111−− 1 Hadamard Gate HH... H 0 0 ... 0 12⊗⊗⊗N 12 ⊗ ⊗⊗N 11 1 =+⊗+⊗+()0 1() 0 1 ...() 0 1 2211 2 2 2NN 1 =+++()012 0 ...0NN 1 12 0 ...0 ... 11 12 ...1 N 2N N−1 1 2 = k()binary expression N ∑ 2 k=0 Universal Quantum Gates General Rotation of a Single Qubit ⎛⎞⎛⎞θθ−iφ ⎛⎞ ⎜⎟cos⎜⎟−ie sin ⎜⎟ ⎝⎠22 ⎝⎠ V(,)θφ =⎜⎟V ⎜⎟+iφ ⎛⎞θθ ⎛⎞ ⎜⎟−ie sin⎜⎟ cos ⎜⎟ ⎝⎠⎝⎠22 ⎝⎠ Xc: CNOT (controlled - NOT) or XOR ⎛⎞⎛⎞⎛⎞⎛⎞10 10 00 01 ⎜⎟⎜⎟⎜⎟⎜⎟⊗+⊗ ⎝⎠⎝⎠⎝⎠⎝⎠00 01 01 10 =⊗+⊗ 00IX 11 Xabc =⊕ aa b CNOT= 00⊗+ I 11 ⊗ X CZ=⊗+⊗00 I 11 Z 00Æ00 01Æ01 10Æ11 XOR 11Æ10 XIX=⊗+⊗00 11 ABAAAA B B Controlled-NOT ⎡⎤⎡⎤⎡⎤⎡⎤10 10 00 01 =⊗+⊗⎢⎥⎢⎥⎢⎥⎢⎥ ⎣⎦⎣⎦⎣⎦⎣⎦00A 01BA 01 10 B ⎡⎤⎡⎤1000 0000 ⎢⎥⎢⎥0100 0000 =+⎢⎥⎢⎥ ⎢⎥⎢⎥0000 0001 ⎢⎥⎢⎥ 0000 0010 ≡ ⎣⎦⎣⎦AB AB X ⎡⎤1000 ⎢⎥0100 = ⎢⎥ ⎢⎥0001 ⎢⎥ ⎣⎦0010AB Qubit Copying Circuit? x x ψ =+ab01 x x ψ =+ab01 ab00+ 11 0 ψ =+ab01 0 y xy⊕ x Deutsch Algorithm Global property of f Quantum Fourier Transform Quantum Fourier Transform 2 2 ~n =(log2N) vs N log2N ***can be used only as a subroutine, used for factoring Grover Search Grover Search Quantum Network DiVincenzo, Qu-Ph/0002077 Scalable Qubits Unitary evolution : Deterministic Initial State : Reversible Universal Gates |0〉 X C M …X1H2 X13|Ψ〉 |Ψ〉 |0〉 H M |0〉 t M Quantum measurement Cohere, : Probabilistic Not Decohere : Irreversible change Quantum Key Distribution√ Single-Qubit Gates ~ 3 [BB84,B92] QKD[E91] √ Single- Quantum Repeater ~ 3 & Two-Qubit Gates Quantum Teleportation√ Quantum Error Correction Single- Quantum Computer & Two-Qubit Gates 10-Qubit QC√ Physical systems actively considered for quantum computer implementation •Liquid-state NMR • Electrons on liquid He • NMR spin lattices • Superconducting Qubits; Josephson junctions • Linear ion-trap spectroscopy – “charge” qubits • Neutral-atom optical –“flux” qubits lattices • Spin spectroscopies, • Cavity QED + atoms impurities in semiconductors • Linear optics with • Coupled quantum dots single photons – Qubits: spin,charge, • Nitrogen vacancies in excitons diamond – Exchange coupled, cavity coupled 15=× 3 5 Chuang Nature 414, 883-887 (20/27 Dec 2001) OR QP/0112176 Concept device: spin-resonance transistor R. Vrijen et al, Phys. Rev. A 62, 012306 (2000) Quantum-dot array proposal Ion Traps • Couple lowest centre-of-mass modes to internal electronic states of N ions. Quantum Error Correcting Code Three Bit Code Encode Recover Decode channel φ φ 0 Um1m2 0 noise 0 M 0 M Entanglement EPR & Nonlocality Alice Bob 1 ()10−≠⊗ 01 ψ φ 2 AB AB A B Local Hidden Variable Æ Bell’s Inequality Aspect’s Experiment Æ Quantum Mechanics is nonlocal! 1 GHZ State: ()0 0 0+ 1 1 1 2 ABC ABC Charles Bennett’s Hippie analogy I don’t know what I think. You don’t know what you think. But we know what we think. Molding a Quantum State |0〉 X C M …X1H2 X13|Ψ〉 |Ψ〉 |0〉 H M |0〉 t M Molding Sculpturing a Quantum State - Cluster State Quantum Computing - |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 |+〉 1. Initialize each qubit in |+〉 state. 2. Contolled-Phase between the neighboring qubits. 3. Single qubit manipulations and single qubit measurements only [Sculpturing]. No two qubit operations! Entanglement EPR & Nonlocality Alice Bob 1 ()10−≠⊗ 01 ψ φ 2 AB AB A B Local Hidden Variable Æ Bell’s Inequality Aspect’s Experiment Æ Quantum Mechanics is nonlocal! 1 GHZ State: ()0 0 0+ 1 1 1 2 ABC ABC Bell’s inequality, CHSH inequality (Classical) Violation of inequality (Quantum) Tripartite Qubit Entanglement Z-basis: 0 with +−1, 1 with 1 Not expectation values X-basis: a≈+≈−− 0 + 1 with 1, b 0 1 with 1 0a+b,≈≈− 1ab Yi-basis: c≈+≈−− 0 + 1 with 1, d 0 i 1 with 1 0≈≈−− c + d , 1i () c b 1 GHZ =+{}000 111 : Z-basis 2 A ≈++++−−−()()()()()()ababab ababab ≈+++aaa abb bab bba : xxx = +1 ≈+(abcdcd )( + )( +−− ) ( abcdcd )( − )( − ) BC≈++acd adc bcc+ bdd : xyy =−1 xxx = +1 xyy = −1 ()xy22( xy 22)( xy 22 ) = 1 yxy =−1 ABC? yyx = −1 Bipartite Qubit Entanglement Not expectation values Ψ=abcd00 + 01 + 10 + 11 with abc222 + + =1 { =0 00′′Ψ= 0 =ab 0 ′′ 0 + 0 ′′ 1 BBBB 00′′′ 0ac 0 0 0 1 Ψ= =AAAA + 2 2 22pp(1−− p )(1 p ) pa=Ψ=00′′′ 0 ′ 0 0 ′′ 0 =AB A B ABAB 1424314243 1− ppAB ==ppAB Maximize p ! ∂∂pp ==⇒−+012ppp22 =−+ 12 ppp = 0 ∂∂pp AAB AAB counterfactual argument AB −±15 51 − pp==1,⇒==pp = g : golden mean AB 22AB pg=≈5 9 % Entanglement Alice Quantum correlation Bob x y Classical physics: x and y are decided when picked up.0AB 1 or 1 A 0 B Quantum physics: x and y are decided when measured. {0,1} basis Æ 0 or 1 1 Ψ=()01 − 10 {+,-} basis Æ + or - 2 AB A B 1 =+−−−+() 2 AB AB 1 = ()01′ ′′′− 10 2 AB A B 11 +=()01, + −= ()01− 22 Entanglement and Two-Qubit Gate a |0〉 + b |1〉 a |0〉|0〉 + b |1〉|1〉 |0〉 a |0〉 + b |1〉 a |0〉|0〉|0〉 + b |1〉|1〉|1〉 |0〉 |0〉 Classical Teleportation? Alice m Bob 1 bit X ⊕aÆm b ⊕m⊕1 ÆX 1 bit “Correlated” Pair a b x ⊕ a ⊕ b ⊕ 1 1 bit 1 bit 123 X = 0 or 1 m = x ⊕⊕1 1 1 bit {0,1} = x Quantum Teleportation • Transportation – Continuous movement through space P P • Quantum Teleportation B1, B2 X A BX Bennett Quantum Teleportation Transmit an unknown qubit without sending the qubit Ω=αβ01 + 2 Bits AliceClassical Information Bob Bell State Measurement Unitary Transform ΦΨ±±, I,,ZX ,− Y Entangled Pair 132 unknown Ω=αβ01 + EPR-Source θ = 17.51798...ο 0 0 3 3 1 1 2 2 X AB θ = 17.51798...ο 0 0 3 1 3 1 2 2 X AB θ = 17.51798...ο 0 0 3 1 3 1 2 2 XA B θ = 17.51798...ο Bell Measurement “0” Leave it as it is.
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