Quantum teleportation for continuous variables

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Quantum teleportation for continuous variables

• Myungshik Kim
• Queens University, Belfast

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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).

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• 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
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How to realise the joint measurement

• Use two homodyne measurement setups

Beam Splitter
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How to realise the entangled quantum channel

• Use a non-degenerate down converter
• Two-mode squeezed state is generated
• 
  

Non-linear crystal
pump

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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.

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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

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• 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
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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
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• 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.

Characteristic function for Q
Characteristic function for P
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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

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• 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

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

Perfect displacement operation
Vacuum environment
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• 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
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• 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

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• The channel is not separable.
• The noise factor becomes
• For TaTb1, the noise factor n? e-2s .
• For Ta1 Tb 0,

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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

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• 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.

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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.