Schrodinger

1
Schrodingers Cat
2

• A particular quantum state, completely described
  by enough quantum numbers, is called a state
  function or wave function ?
• When we observe the state, we find out its
  quantum numbers, observables
• The state function of two different states ?1 ?2
  is equal to the linear sum of the two ?s
  ?1 ?2 and this is attested by many experimental
  facts, e.g. two electron states like He or the
  hydrogen molecule.
• The interpretation of this given by Borh and
  others seems very strange. Examples were given
  by Schrodinger and Einstein (who did not believe
  it) to show that it must be wrong. Schrodinger
  gave the example of his cat ironically, both
  would be paradoxes have important applications.

3

• The cat starts out waiting in a box. A single
  photon is shot through a two slit interference
  set up, and is detected on a screen,
  demonstrating its wave/quantum nature. Depending
  on where it hits, it does or does not release a
  giant stone block. We then observe two possible
  final states

4

• Schrodinger points out that we can wait as long
  as we like before observing the inside of the
  box, and that we cannot believe that there is
  really a superimposed state of cat alive and
  cat dead during all that time. (In practice,
  it is hard to keep a macroscopic object in a
  definite quantum state for a long time h is so
  small that very tiny disturbances move it to a
  different quantum state, which then evolves away
  to a very different state rather quickly). There
  followed fifty years of argument about how we
  should interpret this. More interesting is the
  idea of applying this physical phenomenon.

5

• The Quantum Bit or qubit -- we can use the
  cat alive/cat dead state to represent one bit
  cat alive 1, cat dead 0. If I have 128 cats,
  I can represent 128 qubits.
• I can make gadgets that can operate on quantum
  states. I will show some particularly nice ones
  that operate on atomic spins in a next lecture.
  I can use these to operate to add and multiply,
  for example. Then I can perform computations on
  that 128 qubit number and get results. If I
  operate on the 128 SUPERIMPOSED states, I am
  doing the computation for 2128different input
  numbers. Then I observe the state and get an
  answer.

6

• Note that the answer has to be expressed in just
  the 128 bits I can observe. That is a trillion
  computations. If I could handle 256 cats, I
  could do a trillion trillion computations in the
  same time.
• With cats, it is hard to build such a device, and
  when this was first suggested, it was thought
  that it might be too hard even with atomic spins.
  In the last two years, it has been done for
  several atomic spins, and error corrected codes
  have been demonstrated. No question that it
  works at some level. When will it be applied?
  We probably know to what, first code breaking.
  No present code would be secure. How to fight
  that?--fire with fire.

7

• Einstein hated quantum mechanics, even though his
  Nobel Prize was for the Quantum, not Relativity.
  He came up with a number of paradoxes he threw to
  Bohr, who batted most of them quickly. One that
  troubled lots of serious people was Quantum
  Teleportation as people call it now. Idea
• take superimposed state where the joint
  properties have some total property that is
  known, often the total angular momentum. (the two
  states are called entangled.)
• Separate into two parts very far apart.
• Then observe one part. Instantly the properties
  of the other part are determined.

8

• Example of entangled states photons with net
  polarization zero
• The entangled photons have to have opposite
  polarizations, but (unlike the classical
  picuture) they are opposite with respect to a
  measurement axis that can be chosen after the
  entangled photons are separated but you dont see
  effects until the comparison is made later
• Why does this not transfer information faster
  than the speed of light?
• Because no information is transferred!!

9

• It has been experimentally demonstrated by
  several groups in the last year that you can use
  a pair of entangled states as a carrier to carry
  the properties of a third state to a different
  place, without observing them. This is the basic
  function of teleportation (as in StarTrek).
• Encryptation based on teleportation can be shown
  to be secure in principle, as long as Quantum
  Mechanics is true.

10

• To encrypt, send a photon to the Remote Site,
  entangled with one at Home. (If the Enemy
  detects the entangled photon, it cant be sent
  again, entangled, because of the Uncertainty
  Principle.) Then send the message photon, and
  read it by teleportantion with the entangled
  photon
• Similarly, you can make quantum money that
  cannot be counterfeited. This was invented by a
  Columbia student, Stephen Wiesner many years ago,
  and received more notice recently with the rise
  of quantum communications and computing. It
  allows secure Quantum Encryptation without
  Teleportation.