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.
XA B
θ = 17.51798...ο
Bell Measurement “1” Rotate it by -90o.
XA B θ = 17.51798...ο
XA B
Single Particle Entanglement
++ Single Electron H=− tcc( 12 + cc 21) ①② 1 ψ =+() 1 2 2 1 or ψ =+() f12 e ef 12 2 Beam Splitter Single Photon ② 1 φ =+()10 01 ① 2 Quantum Teleportation using Single Particle Entanglement
Lee, JK, Qu-ph/0007106; Phys. Rev. A 63, 012305 (2001)
PRL88,070402 (2002) or QP/0204158 "Active" Teleportation of a Quantum Bit S. Giacomini, F. Sciarrino, E. Lombardi and F. De Martini Quantum Superdense Coding
Transmit two bits by sending one qubit
Alice Bob 1 qubit 2 bit XcAB IXYZ,,, HA Entanglement
A B
EPR
Quantum Superdense Coding
⎧+⎫⎧+=+IA :0011 0010(01)0⎫ ⎧ 00 ⎪⎪⎪ ⎪ ⎪ ⎪⎪⎪X :1001++=+ 1101(10)1 ⎪⎪01 00 11A XH : : AB+⇒ AB ⎨⎬⎨ ⇒cAB ⎬⎨ ⇒ A ⎪⎪⎪YA :1001−−=− 1101(10)1 ⎪⎪11 ⎪⎪⎪Z :0011−−=− 0010(01)0 ⎪⎪ ⎩⎭⎩A ⎭ ⎩ 10
Alice’s Bob’s manipulation manipulation Alice sends 4 states the qubit to Bob differentiated 2 bit 1 qubit Zurek, QP/0306072
1 ϕ =+( 34) 2 RR 34 ⎧⎡⎤ ⎪ 11 3 ⎪⎢⎥22 ρϕϕLR ==⎨⎢⎥ 4 ⎪⎢⎥11 ⎩⎪⎣⎦⎢⎥22
3 3 R
4 4 R 1 ϕ =+( 33 44) 2 LR LR
ρϕϕLR =
3 3 3 R L
4 4 L 4 R
C K Hong and T Noh (1998) Y-H Kim and Y Shi (2000)
Delayed Choice Quantum Erasure REVISITED Kim et al., PRL84, 1(2000); C.K.Hong and T.G.Noh, JOSA B15, 1192(1998) 1 ϕ =+33 44 ⎧⎫( LR LR) 1 ⎪⎪⎛⎞⎛⎞123⎟⎟2 124 ⎪⎪⎧⎫ ρϕϕ′ ==−+++−+UU+ ⎪⎪⎪⎪⎪⎪⎜⎜LLL i⎟⎟34 i LLL LR L L ⎨⎬⎨⎬⎜⎜⎟⎟RRhc.. 22⎪⎪⎪⎪⎜⎜ 222⎟⎟ 22 ⎩⎭⎪⎪⎝⎠⎝⎠ρϕϕLR = ⎩⎭⎪⎪ D3 2 BSA D 3 3 1 3 R L D0 BS 4 4 L 4 D2 R BSB 1 D4 ⎡⎤11ii ⎢⎥− ⎢⎥22 22 ⎢⎥ii11− ⎢⎥222 2 U = ⎢⎥ L ⎢⎥i 1 ⎢⎥00 ⎢⎥22 ⎢⎥ ⎢⎥i 001 ⎣⎦⎢⎥22
ρϕϕ′ = 11UU+ RD1 LLLL 1 =−{}34 +ii{} − 3 − 4 8 RRRR 34 D3 2 ⎪⎧⎡⎤1 i 3 ⎪⎢⎥88 BSA = ⎪⎨⎢⎥ 4 ⎪⎢⎥− i 1 ⎪⎣⎦⎢⎥88 D 3 3 ⎩ 1 3 R L D0 BS 4 4 L 4 D2 R BSB 1 D4
ρϕϕ′ = 22UU+ RD2 LLLL 1 =−−−{}ii34{} 3 4 8 RRRR 34 ⎪⎧⎡⎤1 − i 3 ⎪⎢⎥88 = ⎨⎪⎢⎥ 4 ⎪⎢⎥i 1 ⎩⎪⎣⎦⎢⎥88 D3 2 BSA D 3 3 1 3 R L D0 BS 4 4 L 4 D2 R BSB 1 D4
⎡⎤⎡⎤11ii− ⎡1 0 ⎤ ⎢⎥⎢⎥88 8 8⎢ ⎥ ρ′ =+=⎢⎥⎢⎥⎢ 4 ⎥ RD+D()12 ⎢⎥⎢⎥− ii11⎢ 0 1⎥ ⎣⎦⎣⎦⎢⎥⎢⎥88 8 8⎣⎢ 4⎦⎥
1 ⎡⎤1 0 ρϕϕ′ ===3333UU+ ⎢⎥4 RD3 LL ⎢⎥ LLRR4 ⎣⎦⎢⎥00
D3 2 BSA D 3 3 1 3 R L D0 BS 4 4 L 4 D2 R BSB 1 D4
1 ⎡⎤00 ρϕϕ′ ===4444UU+ ⎢⎥ RD4 LL ⎢⎥ LLRR4 0 1 ⎣⎦⎢⎥4 1 Environment ϕ =+( 33 44) 2 LR LR Decoherence ρϕϕLR = D3 2 BSA D 3 3 1 3 R L D0 BS 4 4 L 4 D2 R BSB 1 D4
11⎡⎤10 ρρ==tr 3344 + =⎢⎥ RLLR{}RR RR 22⎣⎦⎢⎥01 ⎡⎤11 ⎡⎤ ⎢⎥00⎡⎤⎡⎤1000 ⎢⎥ ρ′ =++=⎢⎥42⎢⎥⎢⎥4 ⎢⎥ R() D1234 +D +D +D ⎢⎥⎢⎥1 ⎢⎥0011⎢⎥⎢⎥00 0 ⎢⎥ ⎣⎦⎢⎥42⎣⎦⎣⎦4 ⎣⎦⎢⎥
Cryptography andGiving QIP disease, 병주고 약주고 Giving medicine. Out with the old, In with the new. • 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 4672856 – key = pass word
[Symmetric] One-Time Pad Vernam • Alice M = 01100110001 ⊕ K = 01010100101 (random) E = 00110010100
• Bob E = 00110010100 ⊕ K = 01010100101 (random) M = 01100110001 • Unconditionally Secure • Impractical: Generation and Distribution Encoding M⊕K: 0⊕0=0, 0⊕1=1, 1⊕0=1, 1⊕1=0 11011001011011001010101010101010010101010 01010101001010100111001000101001010101010 KEY 10010101001010101010010101101010100101010 01010100101001010101001011010010100010010 10010101001010101001010110101010011001111 11011101011000111000100001010010010101010 Cypher 01010001001110101101000010101001010101010 Text 10010001001110100000011110010010100101010 01010000101101011100011001010010100010010 10010010111001011001110101010010011001111
1101110101100011100011011001011011001010100001010010010101010101010101010010101010 KEY 0101000100111010110101010101001010100111000010101001010101010001000101001010101010 1001000100111010000010010101001010101010011110010010100101010010101101010100101010 0101000010110101110001010100101001010101011001010010100010010001011010010100010010 1001001011100101100110010101001010101001110101010010011001111010110101010011001111
[Symmetric] Why is it called One-Time?
• E1 = M1 ⊕ K • E2 = M2 ⊕ K • E1 ⊕ E2 = ( M1 ⊕ K ) ⊕ ( M2 ⊕ K ) = M1 ⊕ M2 ⊕ ( K ⊕ K ) = M1 ⊕ M2 ⊕ ( 0 ) = M1 ⊕ M2 [Asymmetric] Public Key Cryptosystem
Diffie, Hellman RSA : Rivest, Shamir, Adleman
Alice1, Alice2, … Bob Message [m] Eve Message [m] Cryptogram Encryption Decryption [c] Encryption Key Decryption Key [k] [d=f(k)] • Computationally Secure • Could be broken, especially by Quantum Computers
[Asymmetric] RSA Cryptosystem • Message Î 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 Î 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 • RSA129 (1977) – Factored 17 years later using 1,600 computers
• 2,000 Digit Number Vazirani Impossible to factor even – 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 Cloned.
Zurek, Wootters
Qubit Copying Circuit?
x x ψ =+ab01 x x ψ =+ab01 ab00+ 11 0 ψ =+ab01 0 y xy⊕ x If an unknow quantum state Can be cloned … • Quantum States can be measured as accurately as possible ??? |ψ〉 ⇒ |ψ〉 , |ψ〉 , |ψ〉 , |ψ〉 … measure, measure, …
• Commnunication Faster Than Light? |ψ〉 = |0〉A |1〉B + |1〉A |0〉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 “道可道 非常道 名可名 非常名” -老子- |ψ〉= a|0〉 + b|1〉 = c|0'〉 + d|1'〉
Measure in {|0〉, |1〉} Î |0〉 or |1〉
Measure in {|0'〉, |1'〉} Î |0'〉 or |1'〉 I ∝ cos2 θ
b D Polarizing Beam Splitter
Quantum Cryptography Quantum Key Distribution BB84, B92, E91 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. DiVincenzo, Qu-Ph/0002077 BB84: 4 Polarizations D1 0o,90o 0o,45o 0o,45o
Laser PC1 PC2 PC3 PBS D0 • Alice A1 = 0 1 1 0 0 1 0 0 0 0 1 1 A2 = ⊗⊕⊗⊕⊗⊕⊗⊕⊗⊕⊗⊕ P = 174↔ 171↔ 1 ↔ 47
• Bob B = ⊕⊕⊕⊗⊕⊕⊗⊗⊕⊕⊗⊕ D = 0 1 0 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1
BB84 full implementation “Authenticated Public Discussion”
Mesurement in Same No Discard random basis basis? Null result Yes or different basis
Check random No Transmission of sample Stop raw data: Error < Q% Random coding Yes on single photons Reconciliation Estimate information of bit strings leakage Parity check n bits k bits Initial Authentication NEEDED Secret Privacy key Amplification s bits Keep the parities World’s first QKD (1989): Bennett(IBM) et al. 32 cm in free space with 4 polarizations
Phase Coding
2 ⎛⎞φAB−φ sin ⎜⎟ ⎝⎠2 Mach-Zehnder Interferometer D2
D1
Single φA φB Photon
Cf. Delayed Choice (Wheeler) Phase Coding
φ φ Single A B SL or LS Photon L S D1
LL SS
D2
QKD via QE Ekert 91
Alice 1 Bob ()↔−↔bb 2 A BAB
- Measurement of same polarization: Anticorrelation - Measurement of different polarization : Bell’s Inequality to check the eavesdropping Quantum Key Distribution over 67 km with a plug&play system Gisin, Zbinden, QP/0203118
Practical free-space quantum key distribution over 10 km in daylight and at night 30km 45km Richard J. Hughes, Jane E. Nordholt, Derek Derkacs and Charles G. Peterson QP/0206092 http://www.eqcspot.org/ Autumn 2001: 23.4km Qinetiq-MPQ joint free space key exchange trial between Zugspitze and Karwendel
Autumn 2000: Key exchange over 1.9km between Qinetiq site and the local pub!
Rarity
144 km free-space, Zeilinger (2007)
AQIS 2008 hosted by KIAS KIAS-ETRI, December 2005. 25 km Quantum Cryptography
Quantum Key Distribution Entanglement Scheme Ekert 91
Alice 1 Bob ()↔−↔bb 2 A BAB
- Same Polarization: Anticorrelation - Different Polarization: Bell’s Inequality to check the eavesdropping - Entanglement Purification Quantum Repeater 1. Quantum Repeater ~100 km Æ Indefinite distance Briegel, Durr, Cirac, and Zoller, quant-ph/9803056 Transmission: Photon Storage, Processing: Atomic Physics, etc.
& Entanglement Purification 2. Multiuser Quantum Network 1:1 Æ multiusers Phoenix, Barnett, Townsend, and Blow, JMO 42, 1155 (1995) Biham, Huttner, and Mor, Phys. Rev. A 54, 2651 (1996)
Quantum Repeater
Vrijen, R.; Yablonovitch, E. A spin-coherent semiconductor photodetector for quantum communication. Physica E, vol.10, (no.4), Elsevier, June 2001. p.569-75.
Yablonovitch Vrijen, R.; Yablonovitch, E. A spin-coherent semiconductor photodetector for quantum communication. Physica E, vol.10, (no.4), Elsevier, June 2001. p.569-75.
Yablonovitch
Vrijen, R.; Yablonovitch, E. A spin-coherent semiconductor photodetector for quantum communication. Physica E, vol.10, (no.4), Elsevier, June 2001. p.569-75. Yablonovitch Optical Imaging by Two-Photon Entanglement Shih et al., PRA52, R3429 (1995).
no yes
yes no Quantum Holographic Imaging (Saleh et al. at Boston University)
www.bu.edu/qil
PRL 87, 123602 (Sep 2001); Optics Express 9, 498 (Nov 2001) Complementary image of absorbing channel
x 2 px22() x1 E k x h px() 2 22 ≠ px22() y1
absorbing channel
With state-transforming channels,
Bucket detector for transformed states
x2 x1 Ek px22() x h2 px22()
y1
state-transforming channel 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. Dowling1,* PRL85, 2733 (2000) Two-Photon Diffraction and Quantum Lithography Milena D’Angelo, Maria V. Chekhova,* and Yanhua Shih PRL87, 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 coincidence- counting 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]
(from Michael Brooks) Quantum Information Processing Roadmap (1987-1998) Research Goals Basic Research • Quantum interference Applications • Ion traps • Cavity QED Now • Quantum dots • Photon counters • NMR schemes • Precision range finding • Si:P • Optical time-domain • Shor’s algorithm reflectometry • Quantum cryptography • Grover’s algorithm • Metrology • Single photon detector • Secure optical • Error correction communications (from Michael Brooks) Quantum Information Processing Roadmap (1998-2003) Research Goals Near Term Goals Applications • NMR to do 8 qubit QC • 3-5 photon entanglement Near Future • 3-5 qubit ion trap • Quantum dot gate • Quantum cryptography • Silicon-based gate demonstrators • Few qubit applications • General quantum • New algorithms communications • Fault tolerant QC • Random number generators • Novel QC • 2-photon metrology • Quantum repeaters • DNA sequencing • Quantum amplifiers • Single molecule sensors • Detectors • Simulation of quantum systems • Sources • Quantum gyroscopes
(from Michael Brooks) Quantum Information Processing Roadmap (2003-2015) Research Goals Applications Mid Term Goals • 10 - 20 qubit computation Far Future • Demonstrator systems • Quantum repeaters • Novel algorithms • Quantum amplifiers • Macroscopic superposition • Quantum computers • Bose condensate - factoring connections - non-structured information • Quasi-classical few particle retrieval gates and bits • Molecular simulation QUIPU [kí:pu]
1. Quantum Information Processing Unit
2. A device consisting of a cord with knotted strings of various colors attached, used by the ancient Peruvians for recording events, keeping accounts, etc.
* Peruvian Information Processing Unit for Communication and Computation
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 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