Quantum Entanglement and Decoherence

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1 Quantum Entanglement an Decoherence Jaewan Kim KIAS I think of my lifetime in physics as ivie into three perios. In the first perio I was in the grip of the iea that Everything is Particles, I call my secon perio Everything is Fiels. Now I am in the grip of a new vision, that Everything is Information. John Archibal Wheeler, Geon, Black Holes, an Quantum Foam

2 Yoot = Ancient Korean Binary Dice Qubit ϕ = a + b Qubit = quantum binary igit

Businessweek 8/3/99: Ieas for the st Century Quantum Physics Information Science Quantum Information Science Quantum Parallelism Quantum computing is exponentially larger an faster than igital

3 Businessweek 8/3/99: Ieas for the st Century Quantum Physics Information Science Quantum Information Science Quantum Parallelism Quantum computing is exponentially larger an faster than igital computing. Quantum Fourier Transform, Quantum Database Search, Quantum Many-Boy Simulation (Nanotechnology) No Clonability of Quantum Information, Irrevesability of Quantum Measurement Quantum Cryptography (Absolutely secure igital communication) Quantum Correlation by Quantum Entanglement Quantum Teleportation, Quantum Superense Coing, Quantum Cryptography, Quantum Imaging

4 ϕ = 3 + R 4 ( ) 3 ρlr = ϕ ϕ = 4 R R 4 4 R Parametric Down Conversion : Splitting a photon into two ω ω ω ω

5 ϕ = ρ = ϕ ϕ LR ( ) L R L R ω 3 L R 4 L 4 R C K Hong an T Noh (998) Y-H Kim an Y Shi () Delaye Choice Quantum Erasure REVISITED Kim et al., PRL84, (); C.K.Hong an T.G.Noh, JOSA B5, 9(998)

6 ϕ = ( ) L R L R U U L L L i 3 L L L ρ i 4 hc.. LR = L ϕ ϕ L = ρlr = R ϕ ϕ R D 3 BSA D R L D D BS 4 4 L 4 R D 4 U L BSB i i i i = i i + ρ = U ϕ ϕ U RD L L L L = + 8 { 3 i 4 }{ 3 i 4 } i = 4 i 8 8 R R R R 3 4 D 3 BSA D R L D D BS 4 4 L 4 R D 4 BSB + ρ = U ϕ ϕ U RD L L L L = 8 { i 3 4 }{ i 3 4 } i = 4 i 8 8 R R R R 3 4

7 D 3 BSA D R L D D BS 4 4 L 4 R D 4 BSB i i = + = i i ρ R ( D +D ) ρ = 3 U U + 3 = 3 3 = 4 3 L ϕ ϕ RD L L L R R 4 D 3 BSA D R L D D BS 4 4 L 4 R D 4 BSB ρ = 4 U U + 4 = 4 4 = 4 L ϕ ϕ RD L L L R R 4 4

8 Environment Decoherence D 3 BSA 3 D BS D D 3 L ϕ = ρ = ϕ ϕ LR 3 R ( ) L R L R 4 L 4 4 R D 4 BSB ρ { } R = trlρlr = + = R R R R 4 4 = + + ( ) = ρ R D +D +D +D Entanglement an Decoherence A B or Environment Joos, Zeh, Zurek When system A is entangle with environment, state of A cannot be escribe by a state vector, but by a ensity matrix. Y AB = a A B + b A B = ψ A φ B ρ A = Tr B ρ AB = Tr B Y AB AB Y = a = ψ A A ψ b ba* ab*

When a quantum system is entangle with its environment through some interaction, there is no pure state escription available for the system even

In this case, the system loses its coherence, in other wors, is ecohere or the ecoherence occurre on the system.

The system entangle with environment is not a pure state any more an is calle a mixe state, which cannot be escribe by a ket vector or a state

9 When a quantum system is entangle with its environment through some interaction, there is no pure state escription available for the system even though pure state escription of the system plus environment as a whole might be possible. In this case, the system loses its coherence, in other wors, is ecohere or the ecoherence occurre on the system. When the whole is entangle, the part is ecohere. The system entangle with environment is not a pure state any more an is calle a mixe state, which cannot be escribe by a ket vector or a state function, but by a ensity matrix. While the conventional iea for the emergence of classical physics from quantum physics is base on the smallness of planck constant an has nothing to o with the existence of external system or environment, the ecoherence iea requires the interaction between the system an its environment. Zurek, QP/367

10 Ol Quantum Theory Iθ = nh Einstein s forgotten question: How woul you quantize non-integrable systems? New quantization x xˆ p pˆ xp ˆˆ px ˆˆ xp ˆˆ px ˆˆ = ihˆ

11 Mysterious Connection Between Linearity of QM & Special Relativity Weinberg: Can QM be nonlinear? Experiments: Not so positive result. Polchinski, Gisin: If QM is nonlinear, communication faster than light is possible. No Cloning Theorem An Unknown Quantum State Cannot Be Clone. <Proof> ( α ) ( ) Zurek, Wootters Diks U = α α U β = β β α β Let γ = ( α + β ). Then U ( γ ) = ( α α + β β ) γ γ

12 If an unknow quantum state Can be clone Communication Faster Than Light when unknown quantum states can be copie.

15 Quantum Measurement Is the moon there when noboy looks? -Einstein Decoherence Is the moon classical only when someboy looks?

16 Alice x Entanglement Quantum correlation Bob y Classical physics: x an y are ecie when picke up. or A B A B Quantum physics: x an y are ecie when measure. {,} basis or {+,-} basis + or - Ψ = ( A B A B) = ( + + A B A B) = ( A B A B) + = +, = ( ) ( ) Entanglement EPR & Nonlocality Alice Bob ( ) ψ A B A B A B φ Local Hien Variable Bell s Inequality Aspect s Experiment Quantum Mechanics is nonlocal! ( + ) GHZ State: A B C A B C

17 Charles Bennett s Hippie analogy I on t know what I think. You on t know what you think. But we know what we think.

20 Purification Church of Larger Hilbert space Measure of Entanglement bipartite: C, S tripartite: GHZ, W multipartite: shoul not be a scalar Bipartite Qubit Entanglement Hary s Nonlocality Not about the expectation values Bell s inequality, CHSH bipartite qubit entanglement inequality for the expectation values GHZ tripartite qubit entanglement not about the expectation values Hary s nonlocality: The SIMPLEST bipartite qubit entanglement not about the expectation values

21 Bell s inequality, CHSH Bell s inequality, CHSH inequality (Classical) Violation of inequality (Quantum) from Nielsen an Chuang s book GHZ: Tripartite Qubit Entanglement Z-basis: with, with + X-basis: a + with, b with + a + b, a b Y-basis: c + i with, i with + ( ) c +, i c b Not expectation values A GHZ = { + } : Z-basis ( a + b)( a + b)( a + b) + ( a b)( a b)( a b) aaa + abb + bab + bba : xxx = B C xxx xyy yxy yyx ( a + b)( c + )( c + ) ( a b)( c )( c ) ac + ac + bcc + b : xyy =+ = = + =+ = + ( ) ( ) ( ) = A B C xy xy xy?

22 Hary s nonlocality Bipartite Qubit Entanglement counterfactual argument Not expectation values Q=, T= R=, S= Q=, T= Q=, T= Q=, T= R=, S= R=, S= Q=, S= Q=, S= Q=, S= Q=, S= Q=, T=! Q=, S=! R=, S=! R=, T=?

23 Ψ = a + b + c + with a + b + c = { = Ψ = = a + b Ψ = = a + c p= Ψ = a A B = A B Maximize p! p p A B B A A B A A B A A A B A A B B = p = p p = = p + p p = p + p p = p B papb( pa)( pb) p p ± 5 5 pa = pb =, pa = pb = = g: golen mean 5 p= g 9 % A B Ψ = a + b + c + with = = a + b + c + with a = = a + b + c + with a = = α + β + γ + δ with α? with a + b + c = for normalization. Since a =, Ψ = a + c =. Since a =, Ψ = a + b =. α = Ψ { } = Ψ = a+ b+ c = a a a = a

24 Let us call the probability of R=+ an T=+ p. Let s= an t =. Since a + b + c =, a + c =, an a + b =, s( s) t( t) p = α =. st To fin the s an t maximizing p, p t( t) = ( s+ s t) = s ( st) an p s( s) = ( t+ t s) =. t ( st) Thus p is maximize when + 5 s = t = = g (golen mean) 5 p= g 9% Contrary to the classical case with reality assume, quantum mechanical case coul give nonzero probability of R=T=+ case. This is the first showup of golen mean in quantum mechanics. Limite amount of Classical Information Alice Some Correlation Bob

25 bit Classical Teleportation? Alice X a m X = or bit bit Correlate Pair a m {,} b bit Bob b m X bit 3 x a b m = x = x Quantum Teleportation Transportation Continuous movement through space P P Quantum Teleportation X A B, B BX

26 Quantum Teleportation Transmit an unknown qubit without sening the qubit Bits Alice Classical Information Bob Bell State Measurement Φ ±, Ψ ± Unitary Transform I, Z, X, Y Entangle Pair 3 unknown Ω = α + β Bennett Ω = α + β EPR-Source

27 ( a b Q Q) + A B A B + xx + yy ( ax + by ) a a b b = xxx + xyy + yxx + yyy Ξ xx + yy xx yy = Φ Ψ + ( ax + by) ( ax by) xy + yx xy yx Φ ( ay + bx) + ( ay bx) Ψ Q A B A B A B ( a + b ) ( ax + by) Q Q + xx + yy x x xx xy x = ( a b) ( x y) / = ( a b) / y y yx yy y xx x xy x = ( a b) / + ( a b) / yy y yx y ax ay = ( xx yy ) / + ( xy yx) / by bx ax = ( xx yy ) HH / + xy by ( ) ay yx HH / bx ax + by ay + bx xx + yy xx yy xy + yx xy yx = + ax by ay bx xx + yy xx yy = + + ( ax by ) ( ax by ) xy + yx xy yx ( ay bx) ( ay bx) H = H = + H = ( ) ( )

28 Bell Measurement H Q X AQ xx + yy xx yy ( ax + by ) + ( ax by ) xy + yx xy yx + ( ay + bx) + ( ay bx ) xx + yx xx yx ( ax + by ) + ( ax by ) = HQ xy + yy xy yy + ( ay + bx) + ( ay bx ) X I + X AQ Q Q A Q Q A = an X = ( ) HQ = + Q Q Q an H = x + y x y ( ) ( ) x ax + by + x ax by = HQ x + y x y + y( ay + bx) + y( ay bx ) = xx ax + by + yx ax by + xy ay + bx + yy ay bx ( ) ( ) ( ) ( ) X A A A A A A ( ) Q Q Q Q θ = ο 3 3 X A B

29 θ = ο 3 3 X A B θ = ο 3 3 X A B

30 θ = ο Bell Measurement Leave it as it is. X A B θ = ο Bell Measurement Rotate it by -9 o. X A B

31 θ = ο X A B Quantum Teleportation using Single Particle Entanglement Lee, JK, Qu-ph/76; Phys. Rev. A 63, 35 ()

32 PRL88,74 () or QP/458 "Active" Teleportation of a Quantum Bit S. Giacomini, F. Sciarrino, E. Lombari an F. De Martini Quantum Teleportation of Single Particle Entanglement Lee, Qu-ph/497; Phys. Rev. A 64, 43 ()

33 Generalize QTP: quit x, x,... x D D ax + ax ad xd xx + xx xd xd D H M M m m ax ax X m Z m - Generalize Bell Measurements - Moulo-D Summation - Generalize Haamar Transformation ad xd Scalable Qubits Initial State Quantum Network DiVincenzo, Qu-Ph/77 Unitary evolution : Deterministic : Reversible Universal Gates Ψ H t X X H C X 3 Ψ M M M Cohere, Not Decohere Quantum measurement : Probabilistic : Irreversible change

34 Moling a Quantum State Ψ H t X X H C X 3 Ψ M M M Moling Sculpturing a Quantum State - Cluster State Quantum Computing Initialize each qubit in + state.. Contolle-Phase between the neighboring qubits. 3. Single qubit manipulations an single qubit measurements only [Sculpturing]. No two qubit operations!

35 Cont-Z an + - Commuting with each other - Symmetric w.r.t. control an target B. C. Saners an G. J. Milburn, Phys. Rev. A 45, 99 (99). M. Paternostra et al., Phys. Rev. A 67, 38 (3). Wang W.-F. et al., Chin. Phys. Lett. 5, 839 (8) Nguyen B. A. an J. Kim, Phys. Rev. A 8, 436 (9). Even superposition of computational basis states + + Z + + = Z + + Interaction) = Z + Z + = + = + + Bell State ZZ = Z 3 Z Z 3 Z = GHZ state 3 3 Opt Comm 337, 79-8 (5) Jaewan Kim, Juhui Lee, Se-Wan Ji, Hyunchul Nha, Petr M. Anisimov, Jonathan P. Dowling (Cross Kerr Nonlinear Optical e x f k An Interesting Observation of Exponential Function ( x) x x x = L +!! (! ) terms + + x x x x L +!! ( +! ) ( +! ) ( ) + + x x x ( )! ( + )! ( + )! ( 3 ) k= ( ) f ( x) f ( ) L f ( x) = f x + + x + + = L f k m= ( x) ( k + m ) L k+ m x = for k =,, L,! + x 3 Opt Comm 337, 79-8 (5) Jaewan Kim, Juhui Lee, Se-Wan Ji, Hyunchul Nha, Petr M. Anisimov, Jonathan P. Dowling (Cross Kerr Nonlinear Optical Interaction)!

36 Oe π x ( ) Conjugate Relations ( ) s ks ( ) = ω ( ) πi ω x e x e with ω = e. s fk x es x s= ks es( x) = ω fk ( x) k= i ω = e = ω πi

37 α with An Interesting Observation of a Coherent State = e α n= n α n n! α α α α α α α α = e + e + e + L + e!!! (! ) π α ( ) Oe ( ) α α + α + α α α α α e e e L e (! ( + )! ( + )! ( )! ) α α + α + α 3 α α α α + e + e + + e + + L + e ( 3 ) ( )! ( + )! ( + )! ( 3 )! + L = L + = k k = α k+ m α k = e k + m m=! ( α ) k+ m m= ( + ) ( k + m) k k α = e as α. k m! As α, k k = an k l = δ. kl practically α Pseuo-Number Basis an Pseuo-Phase Basis l Define l% = ωα. k = l% = l= l= ω kl lk ω k l% X l% = l ω α α =% X seuo-number an an Even Superposition of Computational basis th is the P state ( ) Pseuo Phase States. (Coherent States; Conjugate Basis States) α = % is the th Pseuo-phase state ( Conu j gate basis) an an Even Superposition of Pseuo Number States. ( Computational Basis States)

38 Qubits an Quits Computational Basis { {, } Conjugate Basis +, } + = + = ( ) ( ) Computational Basis : pseuo-number basis { { πi s= k = ( ) },, L, Conjugate Basis : pseuo-phase basis ( ) } % = α, % = ωα, L, ω = e k = l% = ω kl lk ω l% k Qubit Operators an Quit Operators X = Z = CNOT = X = I + X = C-Z = Z = I + Z = H = = + + X Z O O O = O πi L ω = e ω O ω O O M k = k ˆ k n = ω = ω O k = ω ω = Z = I + Z + Z + L kl nn ˆˆ = k k ω l l = ω k= l= L L 3 ω ω ω L 4 6 ω ω ω = 3 6 k k ω ω = % M O k = M M O ( ) ω C-Z H Cf. D. L. Zhou et al., Phys. Rev. A 68, 633 (3)

Generalize Controlle-Z Operator H nn ˆ ˆ = χ Cross Kerr Interaction C. F. Roos et al., Phys. Rev. A 77, 43(R) (8). Michael Siomau, Ali A.

!) nˆˆ n U = e = e = ω = Z nn ˆˆ Z α α = ω k l k= l= kl ω k l k= l= = = l% l = k k% l= k= Maximal Entanglement of Pseuo-Number State an

Dowling (Cross Kerr Nonlinear Optical Interaction) Deterministic Generation of a Quit Cluster State Opt Comm 337, 79-8 (5) Jaewan Kim, Juhui

39 Generalize Controlle-Z Operator H nn ˆ ˆ = χ Cross Kerr Interaction C. F. Roos et al., Phys. Rev. A 77, 43(R) (8). Michael Siomau, Ali A. Kamli, Sergey A. Moiseev, an Barry C. Saners, Phys. Rev. A 85, 533(R) () χ L πi nn π ˆ, ˆ = < α iχ Lnˆˆ n ~ ( ~?!) nˆˆ n U = e = e = ω = Z nn ˆˆ Z α α = ω k l k= l= kl ω k l k= l= = = l% l = k k% l= k= Maximal Entanglement of Pseuo-Number State an Pseuo-Phase St t Opt Comm 337, 79-8 (5) Jaewan Kim, Juhui Lee, Se-Wan Ji, Hyunchul Nha, Petr M. Anisimov, Jonathan P. Dowling (Cross Kerr Nonlinear Optical Interaction) Deterministic Generation of a Quit Cluster State Opt Comm 337, 79-8 (5) Jaewan Kim, Juhui Lee, Se-Wan Ji, Hyunchul Nha, Petr M. Anisimov, Jonathan P. Dowling (Cross Kerr Nonlinear Optical Interaction) Tayloring Scissors: measurements in pseuonumber basis (Z) Stitches: measurements in pseuophase basis (X)

40 Bell State Tayloring Scissors: measurements in pseuo-number basis (Z) Stitches: measurements in pseuo-phase basis (X) GHZ State Tayloring Scissors: measurements in pseuo-number basis (Z) Stitches: measurements in pseuo-phase basis (X)

41 Entanglement of pure states Wootters, PRL8, 45(998) Entanglement of mixe states Wootters, PRL8, 45(998)

42 Entanglement Measure Discor Olivier & Zurek () Henerson & Veral ()

System Classical an Quantum Corresponence Quantum to Classical Habib,

43 Decoherence Thermalization? Srenicki, PRL7, 666(993) Decoherence in a Chaotic Driven Double-Well System Classical an Quantum Corresponence Quantum to Classical Habib, Shizume, an Zurek Decoherence, Chaos, an the Corresponence Principle Phys. Rev. Lett. 8, 436 (998) Zurek, QP/367

44 Preskill s Lecture Note on QI&Comp

45 Localization Joos et al.(3) Localization rate Joos et al.(3)

46 Chaotic Dynamics of a Rotator Classical vs. Quantum Simplest, faithful, nonlinear Stanar map (Chirikov) I = I Ksinφ φ n+ n n = φ + I n+ n n+ K = from wikipeia

47 Simplest, faithful, nonlinear Rotor Rotation aroun y-axis (parameter a) Twist aroun z-axis (parameter b) z x y Classical ynamics

48 Unitary ynamics Simplest, faithful, nonlinear Rotor Rotation aroun y-axis (parameter a) Twist aroun z-axis (parameter b) Collapse to spin coherent state (ecoherence) z x y

49 Unitary + ecoherence Unitary + ecoherence

Entanglement of System an Environment with preferre basis in the interaction Decoherence of the System Emergence of classical physics QUIPU [kí:pu]. Quantum Information Processing Unit.

50 Entanglement of System an Environment with preferre basis in the interaction Decoherence of the System Emergence of classical physics QUIPU [kí:pu]. Quantum Information Processing Unit. A evice consisting of a cor with knotte strings of various colors attache, use by the ancient Peruvians for recoring events, keeping accounts, etc. * Peruvian Information Processing Unit for Communication an Computation

Quantum Computer. Jaewan Kim School of Computational Sciences Korea Institute for Advanced Study

Quantum Computer. Jaewan Kim School of Computational Sciences Korea Institute for Advanced Study Quantum Computer Jaewan Kim jaewan@kias.re.kr School of Computational Sciences Korea Institute for Advanced Study KIAS (Korea Institute for Advanced Study) Established in 1996 Located in Seoul, Korea Pure