Download Quantum Teleportation

quantum teleportation
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David Riethmiller
28 May 2007
The EPR Paradox
• Einstein, Podolsky, Rosen – 1935 paper
• Concluded quantum mechanics is not “complete.”
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The EPR Paradox
Spin zero
Copenhagen Interpretation of QM:
no state is attributable to a particle until that state is measured.
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Spacelike Separation
The EPR Paradox
Spacelike Separation
• Measurement on one particle collapses wave functions of both
• Appear to have superluminal propagation of information
• If we can’t account for “hidden variables” which allow this propagation,
QM must not be “complete.”
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Non-Locality and Bell’s Inequalities
• Local Interactions
– Particle interacts only with adjacent particles
• Non-Local Interactions
– Particle allowed to interact with non-adjacent particles
– “Action at a distance”
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Non-Locality and Bell’s Inequalities
• J.S. Bell, 1964
– Calculated series of inequalities based on probability of measuring
entangled (correlated) photons in certain states
– If observations obeyed these inequalities, only LOCAL interactions
allowed
– If observations violated inequalities, NON-LOCAL interactions allowed.
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Non-Locality and Bell’s Inequalities
• Experiments showed violation of Bell’s Inequalites.
• Then non-locality is a necessary condition to arrive at the
statistical predictions of quantum mechanics.
• Gives rise to principle mechanism behind quantum
teleportation.
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Meet Alice and Bob
• Let’s say Alice has some arbitrary quantum particle in
state |f> that she doesn’t know, but she wants to send
this information to Bob.
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Meet Alice and Bob
• Alice has 2 classical options:
– 1) She can try to physically transport this info to Bob.
– 2) She can measure the state in her possession and communicate the
measurement to Bob, who prepares an identical state.
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Problems
• 1) She can try to physically transport this info to Bob.
– Not a good idea. Quantum states are fragile and
unstable under small perturbations. It will never reach
Bob without being perturbed out of its original state.
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Problems
• 2) She can measure the state in her possession and
communicate the measurement to Bob, who prepares an
identical state.
– Quantum measurement is unreliable unless Alice knows beforehand
that her state belongs to an orthonormal set.
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Teleportation
• Two spin-1/2 particles are prepared in an EPR singlet
state:
|  (23)  
1 (|  |  |  |  )
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2 2 3
• The pair is separated and distributed to Alice and Bob.
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Teleportation
• Writing the state of the initial particle as:
| 1   a |1   b |1 
• Note that initially Alice has a pure product state:
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| 1  |  
()
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Teleportation
• Alice’s measurement on her own correlated system collapses
the wave functions of BOTH EPR particles, since they are
entangled.
• All Alice has to do is communicate the (classical) results of her
measurement to Bob.
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Teleportation
• Bob’s EPR particle wave function has been collapsed – Alice just needs to
tell him HOW it should collapse, according to her measurement:
• Bob only needs to know which of the unitary transformations to apply in
order to reconstruct |f>, and the teleportation is complete.
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Conclusions
• Non-locality necessary condition to for statistical
predictions of QM
• QM Complete?
– Complete enough to predict states of EPR pairs
• Predictions principle mechanism behind
quantum teleportation
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