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)) 1.Unknown state to teleport 2. Share entangled pair 3. Joint measurement 4. Send classical message 5. Unitary transformation 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 criterion Lee & 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. 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. Charac teristi c functio n for P Characteristic function for Q

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

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. 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 Perfect displacement operation =

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 q.channel generator n a,R a n b,R b

15. Where (i = a,b) & T i = 1-R i We see that m a  1, m b  1 so the characteristic function is integrable when The noise factor is ;

16. The channel is not separable. The noise factor becomes 1)For T a =T b =1, the noise factor n  = e -2s. 2)For T a =1 & T b = 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.