Quantum Mechanics1 Schrodinger’s Cat

Quantum Mechanics2 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.

Quantum Mechanics3 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:

Quantum Mechanics4 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.

Quantum Mechanics5 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 different input numbers. Then I observe the state and get an answer.

Quantum Mechanics6 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.

Quantum Mechanics7 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.

Quantum Mechanics8 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 don’t 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!!

Quantum Mechanics9 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.

Quantum Mechanics10 To encrypt, send a photon to the Remote Site, entangled with one at Home. (If the Enemy detects the entangled photon, it can’t 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.