Stern-Gerlach machine

Readers familiar with spin systems may recall that the study of spin yields a physical example of different bases for the same complex vector space. For instance, to study an electron, or any other particle of spin-1/2, one uses a basis of two kets. Which kets one chooses depends on the orientation of the Stern-Gerlach machine (real or imagined). One might use - -z> and — z) as a basis for one calculation and - -x> and — x) for another. No matter what... 

Figure 10.3. A schematic picture of a Stern-Gerlach machine and a beam of spin-1/2 particles.

At this point it is useful to conduct a thought experiment. Consider a system for which the only possible measurement is by a Stern-Gerlach machine oriented along the z-axis. In other words, assume that once the probability for coming out of the machine spin up is known, every physically predictable feature of the state is known. Then the pair (c+, c ) e would contain more information than is necessary. Only c+p and c P would have physical meaning, and because of the condition Ic+I - - c 1 = 1, even these two real numbers are dependent. Thus the phase space of this hypothetical system... 

For more about the physics of Stern-Gerlach machines, see the Feynman Lectures [FLS, III-5]. 

For example, consider a particle of spin 1/2. We can build a basis of the corresponding projective space by considering spin along the --axis. There are only two certain spin states, up and down. These are mutually exclusive if a particle is spin up, then it will not exit spin down from a --axis Stern-Gerlach machine, and vice versa. But is this set of states large enough Do either of these states have multiplicities In other words, is there some measurement that can distinguish between two pure spin-up particles, or between two pure spin-down particles The answer is no. As far as experiments have been done, any two z-spin-up (resp., spin-down) spin-1/2 particles are absolutely identical. So the list... 

As far as experiments have been done, the state of a spin-1/2 particle is completely determined by its probabilities of exiting x-, y- and c-spin up from Stern-Gerlach machines oriented along the coordinate axes. This fact is consistent with the mathematical model for a qubit, as the following proposition shows. 

The existence of spin-1/2 particles is evidence that the projective-space model is correct. For a description of the relevant experiments with Stern-Gerlach machines, see the Feynman Lectures [FLS, Vol. Ill, Chapter 6]. 

This bizarre prediction, known as the Einstein-Podolsky-Rosen paradox, has been verified many times in the laboratory. The most famous version involves two electrons manipulated into a mixed state with combined spin of 0, The electrons are separated in space before the spin of one (and only one) electron is measured, say, in a Stern-Gerlach machine. If that electron is found to be spin up, then by conservation of spin angular momentum, the other electron must be spin down, and vice versa. This holds true even if the ratio of the distance between the measurements to the time between the measurements is greater than the speed of light. See the discussion in Townsend [To, Sections 5,4 and 5,5] and the references therein. 

---