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Quantum teleportation for continuous variables

Quantum teleportation for continuous variables. Myungshik Kim Queen’s University, Belfast. Contents. Quantum teleportation for continuous variables Quantum channel embedded in environment Transfer of non-classical features Quantum channel decohered asymmetrically

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Quantum teleportation for continuous variables

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  1. Quantum teleportation for continuous variables Myungshik Kim Queen’s University, Belfast

  2. Contents • Quantum teleportation for continuous variables • Quantum channel embedded in environment • Transfer of non-classical features • Quantum channel decohered asymmetrically Braunstein & Kimble, PRL 80, 869 (1998); Furusawa & Kimble, Science 282, 706 (1998).

  3. Quantum teleportation for continuous variables (Braunstein, PRL 84, 3486 (2000)) D(): Displacement operator

  4. How to realise the joint measurement • Use two homodyne measurement setups = Beam Splitter

  5. How to realise the entangled quantum channel • Use a non-degenerate down converter • Two-mode squeezed state is generated Non-linear crystal pump ;

  6. Quantum channel embedded in environments • Two-mode squeezed state is entangled. • Entanglement grows as squeezing grows. • The von-Neumann entropy shows it. • The pure two-mode squeezed state becomes mixed when it interacts with the environment. • For a mixed continuous variables, a measure of entanglement is a problem to be settled. • For a Gaussian mixed state, we have the separability criterion.

  7. Separability criterionLee & Kim, PRA 62, 032305 (2001) • A two-mode Gaussian state is separable when it is possible to assign a positive well-defined P function to it after any local unitary operations. Quasiprobability functions Joint probability-like function in phase space Glauber P, Wigner W, Husimi Q functions Characteristic functions of P and W are related as

  8. Assuming two independent thermal environments, we solve the two-mode Fokker-Planck equation • We find that the two-mode squeezed state is separable when (R: Normalised interaction time) • For vacuum environment, the state is always entangled. Entangled-state Generator

  9. Transfer of non-classical features • Can we find any non-classical features in the teleported state? • What is a non-classical state? • State without a positive well-defined P function • After a little algebra, the Weyl characteristic function for the teleported state is found Wigner function for the original unknown state

  10. Characteristic function for Q Characteristic function for P • Using the relation between the characteristic functions, The Q function is always positive and well‑defined. • When a quantum channel is separable, no non-classical features implicit in the original state transferred by teleportation.

  11. Quantum channel decohered asymmetrically (Kim & Lee, PRA 64, 012309 (2001)) • How to perform a unitary displacement operation T: Transmittance of the beam splitter Transformed field Phase modulator High-transmittance Beam splitter Quantum channel generator

  12. The Wigner function for the transformed field • As T1 while holding not negligible, the exponential function becomes the following delta function and the Wigner function for the transformed field becomes

  13. = Perfect displacement operation • Experimental model of displacement operation • It is more appropriate to assume that each mode of the quantum channel decoheres under the different environment condition. Vacuum environment

  14. Quantum channel interacts with two different thermal environments • Separability of the quantum channel is determined by the possibility to Fourier transform the characteristic function na,Ra nb,Rb q.channel generator

  15. ; Where (i = a,b) & Ti= 1-Ri We see that ma 1, mb 1 so the characteristic function is integrable when • The noise factor is

  16. The channel is not separable. • The noise factor becomes • For Ta=Tb=1, the noise factor n = e-2s . • For Ta=1 & Tb= 0,

  17. Fidelity • The fidelity measures how close the teleported state is to the original state. • For any coherent original state, the average fidelity for teleportation is

  18. Why? • For a short interaction with the environment, the quantum channel is represented by the following Wigner function when squeezing is infinite. • The asymmetric channel has the EPR correlation between the scaled quadrature variables.

  19. Final Remarks • Computers in the future may weigh no more than 1.5 tones. • Popular mechanics, forecasting the relentless march of sciences, 1949. • I think there is a world market for maybe five computers. • Thomas Watson, Chairman of IBM, 1943.

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