Quantum Measurement and Steering
Stub. The measurement axiom (projective measurements): how a measurement's possible outcomes and their probabilities are described, and the state collapse that follows; and local measurements on part of an entangled system, including the striking phenomenon of quantum steering — measuring one half of an entangled pair appears to instantaneously steer the ensemble of states the other half could be in.
Corresponds to Lectures 9–10. The third axiom: a (projective) measurement is described by a set of orthogonal projectors summing to the identity, one per possible outcome; the probability of each outcome is given by the Born rule, and the post-measurement state collapses onto the corresponding projector’s image. Applied to one half of an entangled pair, a local measurement reveals a subtlety: although nothing physical is transmitted, the choice of which measurement to perform on one subsystem determines which ensemble of possible states the other subsystem’s reduced description decomposes into — the phenomenon of quantum steering, which looks paradoxical until it’s understood via the partial trace and reduced density matrices of the next note.
Outline (TODO — flesh out each)
- The measurement axiom — projective measurements, the Born rule
- State collapse
- Local measurements on a composite system
- Quantum steering