Why two-state system = two-dimensional Hilbert space?

[V0ODO0CH1LD]

When we talk about a two-state quantum system being a two-dimensional complex Hilbert space are we implicitly considering the "existence of time"? Why is all this additional structure (of a two-dimensional complex Hilbert space) necessary if, even with a full quantum mechanical perspective, asking a yes-no question yields either yes or no as the answer? I know there are going to be times where the answer will be in a superposition of yes and no, but so far nothing has been said about time so that point is moot. Is it that the yes-no question is being combined with the question "what time is it?" and then that results in a two-dimensional complex Hilbert space? In this case, which mathematical structures represent the "two-state system" and time so we can take their tensor products and get the Hilbert space?

Also, are all the properties of a Hilbert space necessary for the two-state system with time? What is the quantum mechanical interpretation of the addition of two vectors?

 

[Nugatory]

A "two state system" is not a system that can be in only two states, it is a system with an infinite number of possible states, all of which can be written as a linear combination of two basis states. It's like standing in the middle of a field and being able to walk in any of an infinite number of directions - but they can all be written as a linear combination of North and East, or "towards the road" and "parallel to the road", or any of pair of linearly independent directions.

Don't be confused about superpositions being somehow different than unsuperimposed states; every pure state is a superposition if you choose the right basis vectors. For example, with spin 1/2 particles, spin-up is a superposition of spin-left and spin-right.

 

[bhobba]

Why is all this additional structure (of a two-dimensional complex Hilbert space) necessary if, even with a full quantum mechanical perspective, asking a yes-no question yields either yes or no as the answer?

Suppose we have a system in 2 states represented by the vectors [0,1] and [1,0] (that's the yes no you are talking about). These states are called pure. These can be randomly presented for observation and you get the vector [p1, p2] where p1 and p2 give the probabilities of observing the pure state. Such states are called mixed. Probability theory is basically the theory of such mixed states. Now consider the matrix A that say after 1 second transforms one pure state to another with rows [0, 1] and [1, 0]. But what happens when A is applied for half a second. Well that would be a matrix U^2 = A. You can work this out and low and behold U is complex. Apply it to a pure state and you get a complex vector. This is something new. Its not a mixed state - but you are forced to it if you want continuous transformations between pure states.

QM is basically the theory that makes sense of such weird complex pure states (which are required to have continuous transformations between pure states) - it does so by means of the so called Born rule.

What is the quantum mechanical interpretation of the addition of two vectors?

It is called a superposition of the two states. One of the principles of QM is the superposition principle which says given any two states |a> and |b> then a third state c1*|a> + c2*|b> where c1 and c2 are any complex numbers exists. Although valid as is if you superimpose a state with itself you get c1*|a> where c1 is any complex number. Such states are considered physically to be the same state as |a>. Because of this its usual to normalise the vectors although that still leaves an arbitrary phase - this is a symmetry known as a global gauge symmetry - which actually turns out to be important in EM:
http://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

The above is not itself fundamental but can be deduced from the Born rule which says given an observable O there exists a positive operator of unit trace p, called the state of the system, such the expected value of the observation associated with O is Trace(PO). Don't worry if that's gibberish - in that form its usually only presented in advanced texts - I mention it to bring home two points - first states are more general than elements of a complex vector space - then are in fact operators - and secondly there is a deeper reason behind my discussion above.

In fact, and this again is usually left to advanced treatments, the Born rule can be derived using what's called Gleason's theorem:
http://en.wikipedia.org/wiki/Gleason's_theorem

It requires the assumption of non-contextuality however - which is the real rock bottom essence of all this stuff:
http://scienceblogs.com/pontiff/2008/01/17/contextuality-of-quantum-theor/

The following interesting basis of QM may also interest you:
http://arxiv.org/pdf/quant-ph/0101012.pdf

Thanks
Bill