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quantum teleportation

David Riethmiller

28 May 2007

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The EPR Paradox

• Einstein, Podolsky, Rosen – 1935 paper

• Concluded quantum mechanics is not “complete.”

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The EPR Paradox

Spin zero

Spacelike Separation

Copenhagen Interpretation of QM:

no state is attributable to a particle until that state is measured.

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The EPR Paradox

• 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.”

Spacelike Separation

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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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• Two spin-1/2 particles are prepared in an EPR singlet state:

• The pair is separated and distributed to Alice and Bob.

Teleportation

( )23 2 3 2 3

1| (| | | | )2

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Teleportation

• Writing the state of the initial particle as:

• Note that initially Alice has a pure product state:

1 1 1| | |a b

( )1 23| |

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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