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quantum teleportation 1 David Riethmiller 28 May 2007 The EPR Paradox • Einstein, Podolsky, Rosen – 1935 paper • Concluded quantum mechanics is not “complete.” 2 The EPR Paradox Spin zero Copenhagen Interpretation of QM: no state is attributable to a particle until that state is measured. 3 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.” 4 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” 5 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. 6 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. 7 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. 8 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. 9 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. 10 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. 11 Teleportation • Two spin-1/2 particles are prepared in an EPR singlet state: | (23) 1 (| | | | ) 2 3 2 2 3 • The pair is separated and distributed to Alice and Bob. 12 Teleportation • Writing the state of the initial particle as: | 1 a |1 b |1 • Note that initially Alice has a pure product state: 13 | 1 | () 23 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. 14 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. 15 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 16