Quantum Information Science Jaewan Kim jaewankias re kr

Quantum Information Science Jaewan Kim (jaewan@kias. re. kr) School of Computational Sciences Korea Institute for Advanced Study Seoul, Korea

2012 Nobel Prize in Physics Haroche Wineland

In the Serge Haroche laboratory in Paris, in vacuum and at a temperature of almost absolute zero, the microwave photons bounce back and forth inside a small cavity between two mirrors. The mirrors are so reflective that a single photon stays for more than a tenth of a second before it is lost. During its long life time, many quantum manipulations can be performed with the trapped photon without destroying it.

David Wineland’s laboratory in Boulder, Colorado, electrically charged atoms or ions are kept inside a trap by surrounding electric fields. One of the secrets behind Wineland’s breakthrough is mastery of the art of using laser beams and creating laser pulses. A laser is used to put the ion in its lowest energy state and thus enabling the study of quantum phenomena with the trapped ion.

Twenty Questions • It is an animal. • With four legs? • Herbivorous? . • Tiger? Yes(1); No(0). . .

It from Bit A. Wheeler Yin(--)and Yang(━) … Iching (oriental classic)

x x x Yoot(Korean dice) x x x 1 0 1 1

Korean National Flag 1 0 Qubi t Qubit=quantum bit(binary digit)

Businessweek 8/30/ 21 Ideas for the 21 st Century

Quantum Physics Information Science Quantum Parallelism 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 Quantum Cryptography (Absolutely secure digital communication) Quantum Correlation by Quantum Entanglement 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 IT • Database search – Experiments: Hardwares • Ion Traps • NMR • Cavity QED, etc. BT

Quantum Information Processing • Quantum Communication – Quantum Cryptography • Absolutely secure digital communication • Generation and Distribution of Quantum Key – optical fiber – wireless secure satellite communication – Quantum Teleportation • Photons • Atoms, Molecules • Quantum Imaging and Quantum Metrology IT

Bit and Logic Gate 0 1 1 0 NOT NAND 00 0 01 0 10 0 11 1 OR 00 0 01 1 10 1 11 1 XOR “Universal” 00 0 01 1 10 1 11 0 “Reversible” C-NOT CCN ( Toffoli ) C-Exch ( Fredkin ) “Universal” “Reversible”

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 Bit Infinitely long tape: Storage

Quantum Information • Bit Qubit • N bits 2 N states, One at a time Linearly parallel computing AT BEST • N qubits Linear superposition of 2 N states at the same time Exponentially parallel computing 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 ( 2 N-D ) e. g. Strongly Correlated Electron Systems 2. [Peter Shor] Factoring large integers, period finding tq ∝ Pol (N) tcl ∼ Exp (N 1/3) 3. [Grover] Searching tq ∝ N tcl ~ N/2

Digital Computer N bits Digital Computers in parallel N bits Quantum Computer : Quantum Parallelism N bits

? No Male F No Female F

? No Male No Young No Female F O M

Female / Male vs. Y F O M F Young / Old No Young No Male No Female

Y F O M No Young F O No Male M F No Female Y O F O M

Simplest form of quantum uncertainty principle

3 -D Bloch sphere Pure states on the surface of a Bloch sphere

Quantum Gates Time-dependent Schrödinger Eq. Hadamard Unitary Transform Norm Preserving Reversible

Hadamard Gate

Universal Quantum Gates General Rotation of a Single Qubit V Xc: CNOT (controlled - NOT) or XOR

Quantum Network Di. Vincenzo, Qu-Ph/0002077 Scalable Qubits Unitary evolution : Deterministic : Reversible Universal Gates Initial State |0 |Y |0 |0 X H t Cohere, Not Decohere …X 1 H 2 CX M 13|Y M M Quantum measurement : Probabilistic : Irreversible change

√ Quantum Key Distribution [BB 84, B 92] Single-Qubit Gates QKD[E 91] √ Single- Quantum Repeater & Two-Qubit Gates √ Quantum Teleportation Quantum Error Correction Single- Quantum Computer √ & Two-Qubit Gates ~ 3

Physical systems actively considered for quantum computer implementation Liquid-state NMR spin lattices Ion-trap Neutral-atom optical lattices • Cavity QED + atoms • Linear optics with single photons • Nitrogen vacancies in diamond • • • Electrons on liquid He • Josephson junctions – “charge” qubits – “flux” qubits • Spin spectroscopies, impurities in semiconductors • Coupled quantum dots – Qubits: spin, charge, excitons – Exchange coupled, cavity coupled

Chuang Nature 414, 883 -887 (20/27 Dec 2001) OR QP/0112176

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 U m 1 m 2 noise M M

Cryptography and QIP Giving disease (Q Comp), Giving medicine (QKD). • Public Key Cryptosystem (Asymmetric) – Computationally Secure – Based on unproven mathematical conjectures – Cursed by Quantum Computation • One-Time Pad (Symmetric) – Unconditionally Secure – Impractical – Saved by Quantum Cryptography

“Quantum Key Distribution is a major paradigm shift in the development of cryptography. Conventional and quantum cryptography are a powerful combination in making a secure communications a reality. ” -Burt Kaliski, Chief Scientist, RSA Laboratories

[Symmetric] One-Time pad Alice Bob Tell me the password. Pass word + key = 4672856 Eve 4672856 – key = pass word

[Symmetric] One-Time Pad Vernam • Alice M = 0110001 K = 01010100101 (random) E = 00110010100 • Bob E = 00110010100 K = 01010100101 (random) M = 0110001 • Unconditionally Secure • Impractical: Generation and Distribution

Encoding M K: 0 0=0, 0 1=1, 1 0=1, 1 1=0 KEY 1101100101010101001010100111001010010101010101101001010100101001011010001001010101001010101001111 110101100010000101001001010 Cypher 0101001110100001010101010 10010011101000000111100100101010 Text 0101000010111000110010100010010010110011101001001111 KEY 1101100101010101001010 11010110001000010100100101001010100111001010010101001110100001010101010 1001010101010110100101010 1001001110100000011110010010101001010010110100010010 010100001011100011001010001001010101001010101001111 10010010110011101001001111

[Symmetric] Why is it called One-Time? • E 1 = M 1 K • E 2 = M 2 K • E 1 E 2 = ( M 1 K ) ( M 2 K ) = M 1 M 2 ( K K ) = M 1 M 2 ( 0 ) = M 1 M 2

[Asymmetric] Public Key Cryptosystem Ali ce 2 lice 1 A Diffie, Hellman 3 Alice RSA : Rivest, Shamir, Adleman Alice 1, Alice 2, … Message [m] b Bo Eve Cryptogram [c] Encryption Key Bob [k] • Computationally Secure • Could be broken, especially by Quantum Computers Message [m] Decryption Key [d=f(k)]

[Asymmetric] RSA Cryptosystem • Message M • n = p q , e d = 1 mod (p-1)(q-1) – n, e : Public Key (Encryption) – n, d : Secret Key (Decryption) • Alice : E = Me mod n • Bob : M = Ed mod n • M Message

RSA Example • Bob’s Keys – p = 11, q = 13, n = p q = 143 – d = 103, e = 7 – d e mod (p-1)(q-1) = 103 x 7 mod 120 = 1 • Alice’s Message: – M = 9 – E = Me mod n = 97 mod 143 = 48 • Bob – M=Ed mod n = 48103 mod 143 = 9

How Hard is Factoring? • Almost exponentially complex with the number of digits, L • RSA 129 (1977) – Factored 17 years later using 1, 600 computers • 2, 000 Digit Number Impossible to factor even Vazirani – With as many digital computers as the number of particles in the Universe (1080) – In as long time as the age of the universe (1018 sec)

No Cloning Theorem An Unknown Quantum State Cannot Be Clon <Proof> Zurek, Wootters Diks

XOR 00 00 01 01 10 11 11 10 Controlled-NOT X

Qubit Copying Circuit?

IF an unknow quantum state CAN be cloned … • Quantum States can be measured as accurately as possible ? ? ? | | , | … measure, … • Communication Faster Than Light? | = |1 A |0 B - |0 A |1 B for “ 0” = |+ A |- B - |- A |+ B for “ 1”

Communication Faster Than Light when unknown quantum state can be copied. • Alice wants to send Bob “ 1. ” Alice mesures her qubit in {|+ , |- }. • Alice’s state will become |+ or |-. • Bob’s state will become |- or |+. Let’s assume it is |+. • Bob makes many copies of this. He measures them in {|+ , |- }, and gets 100% |+. He measures them in |0 , |1 }, and gets 50% |0 and 50% |1. Thus Bob can conclude that Alice measured her state in {|+ , |- }.

Mysterious Connection Between QM & Relativity • Weinberg: Can QM be nonlinear? • Experiments: Not so positive result. • Polchinski, Gisin: If QM is nonlinear, communication faster than light is possible.

Irreversability of Quantum Measurement “If you describe it as that, that is not it anymore” -老子 - | = a|0 + b|1 = c|0' + d|1' Measure in {|0 , |1 } |0 or |1 Measure in {|0' , |1' } |0' or |1'

Quantum Cryptography Quantum Key Distribution BB 84, B 92, E 91 No Cloning & Irreversability of Quantum Measurement EVE (eavesdropper) Alice Bob “Single Photon, Ent. , Coh. ” 1. Interconvertibility between stationary and flying qubits. 2. Faithful transmission of flying qubits. Di. Vincenzo, Qu-Ph/0002077

BB 84: 4 Polarizations Laser • Alice A 1 A 2 P • Bob B D 0 o, 90 o 0 o, 45 o PC 1 PC 2 PC 3 A 1 A 2 B D 1 PBS D 0 = = = 0 1 1 0 0 0 0 1 1 = = ↔ ↔ ↔ 0 1 0 0 0 0 1 1 1 1

BB 84 full implementation “Authenticated Public Discussion” Mesurement in random basis Same basis? No Yes Transmission of raw data: Random coding on single photons Initial Authentication NEEDED Null result or different basis Check random sample No Discard Stop Error < Q% Yes Reconciliation of bit strings Parity check n bits Secret key s bits Estimate information leakage k bits Privacy Amplification Keep the parities

World’s first QKD (1989): Bennett(IBM) et al. 32 cm in free space with 4 polarizations

KIAS-ETRI, December 2005. 25 km Quantum Cryptography

Deutsch Algorithm Global property of f

Quantum Fourier Transform ~n 2=(log 2 N)2 vs N log 2 N ***can be used only as a subroutine, used for factoring

Molding a Quantum State |0 |Y |0 |0 X H …X 1 H 2 t Molding CX M 13|Y M M

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 Local Hidden Variable Bell’s Inequality Aspect’s Experiment Quantum Mechanics is nonlocal!

Bell’s inequality, CHSH inequality (Classical) Violation of inequality (Quantum)

Entanglement Alice Quantum correlation x 0 1 Classical physics: x and y are decided when picked up. Quantum physics: x and y are decided when measured. {0, 1} basis 0 or 1 {+, -} basis + or - Bob y

Entanglement and Two-Qubit Gate a |0 + b |1 a |0 |0 + b |1 |1 |0 a |0 + b |1 |0 |0 a |0 |0 |0 + b |1 |1 |1

Classical Teleportation? Alice Bob m 1 bit X a m b m 1 X 1 bit “Correlated” Pair a b 1 bit X = 0 or 1 1 bit {0, 1}

Quantum Teleportation • Transportation – Continuous movement through space P P • Quantum Teleportation B 1 X B A 2 B 1, B 2 B X

Bennett et al. Quantum Teleportation Transmit an unknown qubit without sending the qubit Alice 2 Bits Bob Classical Information Bell State Measurement Unitary Transform Entangled Pair 1 unknown 2 3 EPR-Source

o q = 17. 51798. . . 0 0 3 3 1 1 2 2 X A B

o q = 17. 51798. . . 0 3 X 0 1 3 1 2 2 A B

o q = 17. 51798. . . 0 3 X 0 1 3 1 2 2 A B

o q = 17. 51798. . . Bell Measurement “ 0” Leave it as it is. 0 0 3 3 1 1 2 2 X A B

o q = 17. 51798. . . Bell Measurement “ 1” 0 0 1 1 1 2 3 2 2 3 3 0 Rotate it by -90 o. X A B

o q = 17. 51798. . . 2 3 3 0 1 2 0 1 X A B

Single Particle Entanglement Single Electron ① ② Single Photon Beam Splitter ② ①

Quantum Teleportation using Single Particle Entanglement Lee, JK, Qu-ph/0007106; Phys. Rev. A 63, 012305 (2001)

PRL 88, 070402 (2002) or QP/0204158 "Active" Teleportation of a Quantum Bit S. Giacomini, F. Sciarrino, E. Lombardi and F. De Martini

Quantum Teleportation of Single Particle Entanglement Lee, Qu-ph/0104097; Phys. Rev. A 64, 014302 (2001)

Quantum Superdense Coding Transmit two bits by sending one qubit Alice Bob 1 qubit 2 bit Entanglement A B EPR

Quantum Superdense Coding Bob’s manipulation Alice sends the qubit to Bob 1 qubit 4 states differentiated 2 bit

Tripartite Qubit Entanglement Not expectation values A B C ?

Bipartite Qubit Entanglement Not expectation values counterfactual argument

Zurek, QP/0306072

3 4

Parametric Down Conversion : Splitting a photon into two

3 4 C K Hong and T Noh (1998) Y-H Kim and Y Shi (2000)

Delayed Choice Quantum Erasure REVISITED Kim et al. , PRL 84, 1(2000); C. K. Hong and T. G. Noh, JOSA B 15, 1192(1998)

D 3 2 BSA 3 D 1 D 2 BS 4 BSB D 4 1 D 0

D 3 2 BSA 3 D 1 D 2 BS 4 BSB D 4 1 D 0

D 3 2 BSA 3 D 1 D 2 BS 4 BSB D 4 1 D 0

D 3 2 BSA 3 D 1 D 2 BS 4 BSB D 4 1 D 0

Environment Decoherence D 3 2 BSA 3 D 1 D 2 BS 4 BSB D 4 1 D 0

Entanglement and Decoherence Zurek B or Environment A When system A is entangled with environment, state of A cannot be described by a state vector, but by a density matrix. |Y AB = a |0 A |0 B + b |1 A |1 B = | A | B A = Tr. B AB = Tr. B |Y AB AB Y| = = | A A | |a|2 0 0 |b|2 ba* ab*

Quantum Repeater 1. Quantum Repeater ~100 km Indefinite distance Briegel, Durr, Cirac, and Zoller, quant-ph/9803056 Transmission: Photon Storage, Processing: Atomic Physics, etc. 2. Multiuser Quantum Network 1: 1 multiusers & Entanglement Purification Phoenix, Barnett, Townsend, and Blow, JMO 42, 1155 (1995) Biham, Huttner, and Mor, Phys. Rev. A 54, 2651 (1996)

Optical Imaging by Two-Photon Entanglement Shih et al. , PRA 52, R 3429 (1995).

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Quantum Lithography

Quantum Interferometric Optical Lithography: Exploiting Entanglement to Beat the Diffraction Limit Agedi N. Boto, 1 Pieter Kok, 2 Daniel S. Abrams, 1 Samuel L. Braunstein, 2 Colin P. Williams, 1 and Jonathan P. Dowling 1, * PRL 85, 2733 (2000) Two-Photon Diffraction and Quantum Lithography Milena D’Angelo, Maria V. Chekhova, * and Yanhua Shih PRL 87, 13602 (2001)

Quantum Metrology We are applying our understanding of the quantum nature of entangled-photon pairs for the characterization of optical-detector quantum efficiency, source radiation density, and photonic-material characteristics, all without the necessity of employing a reference. The use of this unique light source provides particular advantages over the classical counterparts that are traditionally used. Another example is provided by a technique we call quantum ellipsometry. The high accuracy required in traditional ellipsometric measurements necessitates the absolute calibration of both the source and the detector. These requirements can be circumvented by using spontaneous parametric down-conversion in conjunction with a novel polarization interferometer and coincidencecounting detection scheme. Entangled-photon ellipsometry A. F. Abouraddy, K. C. Toussaint, Jr. , A. V. Sergienko, B. E. A. Saleh, and M. C. Teich J. Opt. Soc. Am. B. 19, 656 -662 (2002). [PDF] Ellipsometric measurements by use of photon pairs generated by spontaneous parametric downconversion Ayman F. Abouraddy, Kimani C. Toussaint, Jr. , Alexander V. Sergienko, Bahaa E. A. Saleh, and Malvin C. Teich Opt. Lett. 26, 1717 -1719 (2001). [PDF]

Quantum Physics Information Science Quantum Parallelism 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 Quantum Cryptography (Absolutely secure digital communication) Quantum Correlation by Quantum Entanglement Quantum Teleportation, Quantum Superdense Coding, Quantum Cryptography, Quantum Imaging