Quantum Communication from No-Cloning to the Quantum Repeater - PowerPoint PPT Presentation

Two-Mode Squeezed State = +   ˆ ˆ ˆ ' a a a   / 2 1   1 2 = −   ˆ ˆ ˆ ' a  a a  / 2 2  1 2  − + = + r r ( 0 ) ˆ ˆ ˆ ( 0 ) a e x i e p 1 1 1 + − = + r r ( 0 ) ( 0 ) ˆ ˆ ˆ a e x i e p 2 2 2

Two-Mode Squeezed State − +   = + r r ( 0 ) ( 0 ) ˆ ˆ ˆ ' x e x e x   / 2 1 1 2   − +   = r + r ( 0 ) ( 0 ) ˆ ˆ ˆ ' p e p e p   / 2 1 1 2   − +   = − r r ( 0 ) ( 0 ) ˆ ˆ ˆ ' x  e x e x  / 2 1 2 2   − +   r r = − ( 0 ) ( 0 ) ˆ ˆ ˆ ' p  e p e p  / 2 1 2 2  

Two-Mode Squeezed State − +   = + r r ( 0 ) ( 0 ) ˆ ˆ ˆ ' x e x e x   / 2 1 1 2   + −   − = − + r r ( 0 ) ( 0 ) ˆ ˆ ˆ ' x e x e x   / 2 1 2 2   − +   r r = + ( 0 ) ( 0 ) ˆ ˆ ˆ ' p  e p e p  / 2 1 1 2   − +   r r + = − ( 0 ) ( 0 ) ˆ ˆ ˆ ' p  e p e p  / 2 1 2 2  

Two-Mode Squeezed State ˆ ∝ + α + β + S 0 0 2 4 ... → 00 00 − 20 02 → 11 2

Two-Mode Squeezed State ˆ ∝ + α + β + S 0 0 2 4 ... → 00 00 − 20 02 Like (inverse) → 11 Hong-Ou-Mandel Effect! 2

Two-Mode Squeezed State 2 2 ∝ − − − + + +     2 2 r r ( , , , ) exp [ W x p x p e x x e p p     1 1 1 2 1 2 2 2     2 2 − + − − − −     2 2 r r ] e x x e p p     1 2 1 2    

Gaussian States 2 ∫ iyp − ρ + = 4 ˆ e ( , ) dy x y x y W x p π 2 p = − − − − 2 2 ( , ) exp [ 2 ( ) 2 ( ) ] W x p x x p p π 0 0 x 1 − = − ξ ξ 1 T ξ ( ) exp ( / 2 ) W V π ( 2 ) det V 2nd moment covariance matrix

Gaussian States 1 − = − ξ ξ 1 T ξ ( ) exp ( / 2 ) W V π ( 2 ) det V ( ) ξ = , , , x p x p 1 1 2 2 ( ) ξ ˆ = ˆ ˆ ˆ ˆ , , , x p x p 1 1 2 2     ρ ξ ˆ ξ ˆ + ξ ˆ ξ ˆ = / ˆ   Tr   V 2 ij   i j j i  

Gaussian States ξ ˆ ξ ˆ = Λ [ ] , ( / 2 ) i k l kl   0 1 N ⊕   Λ = = , J J   −   1 0 = 1 k + i Λ ≥ 0 V 4 R. Simon, PRL 84 , 2726 (2000)

Gaussian States   A C   = = ( 2 ) V N   T   C B + + ≤ + 1 det det 2 det 4 det A B C V 4 ( =  1 )

Gaussian States   A C   = = ( 2 ) V N   T   C B + − ≤ + 1 det det 2 det 4 det A B C V 4 ( = separability  1 )

Witnessing Entanglement [ ] ρ < Tr W 0 − + + < ˆ ˆ ˆ ˆ Var ( ) Var ( ) 1 x x p p 1 2 1 2 L. -M. Duan et al., PRL 84 , 2722 (2000)

Quantum Information

Classical versus Quantum Information ψ = α + β 0 1 0 or 1 Computational Basis States

Photonic Qubit α β + 0 1 Occupation-Number Qubit: α β + 10 01 Dual-Rail Qubit: e.g., polarization of a photon… α β ↔ + 

Quantum Information No Communication Heisenberg Faster Than Light Uncertainty Relation “No-Cloning” Theorem Entanglement

Quantum Information & Technology No Communication Heisenberg Faster Than Light Uncertainty Relation “No-Cloning” Theorem Entanglement Long-Distance Universal Quantum Communication Quantum Computer

Quantum Information & Technology No Communication Heisenberg Faster Than Light Uncertainty Relation “No-Cloning” Theorem Quantum Cryptography Quantum Teleportation Beats Beats Classical Cryptography Classical Teleportation Entanglement Long-Distance Universal Quantum Communication Quantum Computer

Entanglement: EPR, Schrödinger, and Bell “When two systems enter into temporary physical interaction due to known forces between them, and when after a time of mutual influence the systems separate again, then they can no longer be described in the same way as before. I would not call that one but rather the characteristic trait of quantum mechanics, the one that enforces its entire departure from classical lines of thought. By the interaction the two systems have become en entan angled ed .” Schrödinger (1935)

Entanglement: EPR, Schrödinger, and Bell Entanglement ∫ + − c = +   Ψ , / dx x x 01 10   2   Discrete Variables Continuous Variables E instein, P odolsky, and R osen (1935)... ... non-locality, non-realism, or hidden variables... QM incomplete

Entangled Photons over 144km Zeilinger

Quantum Information & Technology No Communication Heisenberg Faster Than Light Uncertainty Relation “No-Cloning” Theorem Quantum Cryptography Quantum Teleportation Beats Beats Classical Cryptography Classical Teleportation Entanglement Long-Distance Universal Quantum Communication Quantum Computer

Quantum Information & Technology No Communication Heisenberg Faster Than Light Uncertainty Relation “No-Cloning” Theorem Quantum Cryptography Quantum Teleportation Beats Beats Classical Cryptography Classical Teleportation Entanglement Long-Distance Universal Quantum Communication Quantum Computer … Photon Loss in Fiber Channel!

Cloning

No-Cloning ψ  → ψ ψ

No-Cloning ψ ψ ψ 0

No-Cloning there is no unitary such that… ψ ψ U ψ 0

No-Cloning there is no unitary such that… U

No-Cloning U

No-Cloning ψ = 0 0 U 0 0

No-Cloning ψ = 0 0 U 0 0 U

No-Cloning ψ = 0 0 U 0 0 ψ = 1 1 U 1 0

No-Cloning ψ = 0 0 U 0 0 ψ = 1 1 U 1 0 + U ≠

No-Cloning ψ = 0 0 U 0 0 ψ = 1 1 U 1 0 ψ = 0 + 1 + 0 0 1 1 U 0 ≠ ψ ψ

No-Cloning { ( ) ( ) } + + Tr 0 0 1 1 0 0 1 1 2 = + 0 0 1 1

No-Cloning { ( ) ( ) } + + Tr 0 0 1 1 0 0 1 1 2 = + 0 0 1 1 OR

No-Cloning { ( ) ( ) } + + Tr 0 0 1 1 0 0 1 1 2 = + 0 0 1 1 OR ≠ 0 + 1

No-Cloning there is no physical operation such that… ψ ψ ψ 0 U a

No-Cloning there is no physical operation such that… ψ ψ ψ 0 U a No-Cloning Theorem ( Wootters and Zurek, Dieks, 1982 ): perfect deterministic copying device for arbitrary quantum states does not exist

No Cloning No-Cloning Theorem ( Wootters and Zurek, Dieks, 1982 ): perfect deterministic copying device for arbitrary quantum states does not exist U ψ  →  ψ ψ 0 ' a a ψ  → ψ ψ 1 2 3 1 2 3 ψ ρ ψ = ψ ρ ψ ≠ out out Tr Tr 1 23 13 123 123

Quantum Information & Technology No Communication Heisenberg Faster Than Light Uncertainty Relation “No-Cloning” Theorem Quantum Cryptography Quantum Teleportation Beats Beats Classical Cryptography Classical Teleportation Entanglement Long-Distance Universal Quantum Communication Quantum Computer

Quantum Gates

Qubits σ σ σ ≡ ≡ ≡ , , (Pauli operators) X Y Z z x y ( ) ± in particular, ± = ± ± ± ≡ / 0 1 X with 2 = = − (computational 0 0 , 1 1 Z Z basis) and, − θ / 2 i Z ≡ Z e (rotation about Z axis, etc.) θ = + = − = = 0 , 1 , , H H H X H Z H Z H X (Hadamard)

Qubits ( ) nm = − 1 C n m n m controlled sign Z { } , , , H Z C Z universal set π π / 2 / 4 Z

Quantum Computation with Qubits ( ) nm = − 1 C n m n m controlled sign Z { } universal set , , , H Z C Z π π / 2 / 4 Z Clifford set

Dual-Rail Linear-Optics Quantum Gates ˆ k 00 a 2 modes … any single-qubit gate easy with linear elements !

Multiple-Rail Linear-Optics Quantum Gates Adami, Cerf (1998) ˆ k 00 . ... 0 a d modes

Multiple-Rail Linear-Optics Quantum Gates Adami, Cerf (1998) ˆ k 00 . ... 0 a d modes = N 2 d … bad scaling of optical elements… N qubit space needs control of modes

Nonlinear Quantum Gates ˆ ˆ π ˆ = ˆ ˆ i a a b b Cross Kerr … U e ˆ ˆ ˆ ˆ  →  →  →  → U U U U - , , , 01 10 10 11 11 00 00 01 controlled sign gate C Z

Nonlinear Quantum Gates ˆ ˆ π ˆ = ˆ ˆ i a a b b Cross Kerr … U e ˆ ˆ ˆ ˆ  →  →  →  → U U U U - , , , 01 10 10 11 11 00 00 01 controlled sign gate C Z …hard to obtain for single-photon states

Nonlinear Quantum Gates π − ˆ = ˆ ˆ ˆ ˆ ( / 2 ) ( 1 ) i a a a a U e

Nonlinear Quantum Gates → → → 00 00 00 00 π − ˆ = ˆ ˆ ˆ ˆ ( / 2 ) ( 1 ) i a a a a U e

Nonlinear Quantum Gates + + 10 01 10 01 → → → 10 10 2 2 π − ˆ = ˆ ˆ ˆ ˆ ( / 2 ) ( 1 ) i a a a a U e

Nonlinear Quantum Gates − − 10 01 10 01 → → → 01 01 2 2 π − ˆ = ˆ ˆ ˆ ˆ ( / 2 ) ( 1 ) i a a a a U e

Nonlinear Quantum Gates − − 20 02 02 20 → → → − 11 11 2 2 π − ˆ = ˆ ˆ ˆ ˆ ( / 2 ) ( 1 ) i a a a a U e

Nonlinear Quantum Gates

Continuous Variables − ˆ ˆ 2 2 i s p i t x ≡ ≡ (WH operators) ( ) , ( ) X s e Z t e − 2 = i s p = + p p ( ) p , ( ) p X s Z t t e = 2 i t x = + ( ) , ( ) Z t x x X s x x s e (position/computational basis)

Continuous Variables − ˆ ˆ 2 2 i s p i t x ≡ ≡ (WH operators) ( ) , ( ) X s e Z t e − = 2 = + i s p p p ( ) p , ( ) p X s Z t t e = 2 = + i t x ( ) , ( ) Z t x x X s x x s e (position/computational basis) ˆ ( x ) i f D ≡ e

Continuous Variables − ˆ ˆ 2 2 i s p i t x ≡ ≡ (WH operators) ( ) , ( ) X s e Z t e − = 2 = + i s p p p ( ) p , ( ) p X s Z t t e = 2 = + i t x ( ) , ( ) Z t x x X s x x s e (position/computational basis) ˆ ( x ) i f D ≡ e = = = − = ˆ ˆ ˆ ˆ p x , etc. , , F x F x F p F p F x

Continuous Variables (controlled Z gate) ⊗ ˆ ˆ 2 i x x = C e (QND gate) Z

Continuous Variables (controlled Z gate) ⊗ ˆ ˆ 2 i x x = C e (QND gate) Z   κ 2 λ 3 ˆ ˆ i x i x   κ λ ∈ R , ( ), , , F Z t e C e ; , , t   Z “ cubic phase gate ” (universal set)

Quantum Channels/Operations

Quantum Channels ρ ˆ input U ρ ˆ ancilla ε = ρ → ρ ρ ⊗ ρ ( ) Tr [ ] ˆ ˆ ˆ ˆ ( ) U U anc in in in anc

Gaussian Formalism: Channels ρ ˆ input U ρ ˆ ancilla Gaussian

Gaussian Formalism: Channels ρ ˆ input U ρ ˆ ancilla If input state has N modes, the most general Gaussian CPTP map has at most 2 N ancilla modes F. Caruso, J. Eisert, V. Giovannetti, and A. S. Holevo, New J. of Phys. 10 , 083030 (2008)