There are two notions we need to develop.
1) Quantum states are like vectors: _{}
a. Addition: _{}
i. _{}
ii. _{}
b. inner product: _{}
c. basis: there are a set of complete states that completely define any QS in that all QS can be written as a sum over these states with appropriate amplitudes.
_{}
d. orthogonal and normal = orthonormal: given an inner product vectors can be normalized and orthogonalized s.t.
_{}
To clarify the notion of basis we consider the the filter shown below. We imagine a measurable property of a system such as the spin of an electron or the polarization of a photon. We build an idealized system that can separate the two states and then recombine them. In addition we can choose to completely block one component of the state by inserting a blocking mechanism.
http://csma31.csm.jmu.edu/physics/giovanetti/particlePhysics/Stern.htm
1) If the state is unblocked any entering QS will exit with no change. _{} with no detectable change. This is almost perfectly attainable with beam optics. States of polarization can be split and recombined and any tests done on the beam before the split are reproduced after the open filter [on the beam after the split].
2) If a blocking part is inserted then the Filter picks out the specific amount of the unblocked state and absorbs all of the blocked state.
· _{}
You can imagine that you do a host of experiments with photons:
You make measurements with a filter device in its various states and count the number of photons before and after or measure intensity before and after. This step is important because the character of a quantum state is affected by the measurement. It is like observing the hit on a screen in the double slit experiment. You imagine that the final state is the interference pattern but when you measure the location you get one and only one value. If you view the counts over time you get an intensity distribution that matches the interference pattern. Thus quantum measurement is complicated.
Now we can rotate filters around the beam axis. The basis states that were defined by the original filter orientation are now changed to new basis states. Indeed, for polarized photons, we simply change the orientation of “up” and “down”. This is a simple coordinate transformation and the definition of polarization is arbitrary in this respec so there is nothing surprising about the interpretation.
Finally we can arrange filters of various orientations in succession and predict the outcome of an intensity measurement. The correct method is to decompose states into the appropriate basis states and then project when the state is blocked. The transmitted state will be unchanged if unblocked.
_{} are one basis
_{} are a different basis
but the state _{} is the same and all measurable aspects are repeated using identical basis representations.
Representation is an important vocabulary word. It implies that there is something that has an intrinsic character and that character is maintained in the way we express it. Both bases based on different orientations of our open filter carry the entire character of the QS.
For the photon which has spin 1 and two polarization states on can choose two open filter orientations that allow one to discuss the direction of the electric field along an x,y direction or 45^{ O} rotated system x’,y’. In addition we can choose states of circular polarization and we can imagine designing a filter which split the beam based on circular polarization. Therefore consider these three bases.
_{}
Now we can set up a filter that selects _{} followed by a filter that selects _{}. From this we can determine the intensity or count through the two filter system. This can lead to _{}, the probability for the state _{} to get through the second filter. On simple takes the ratio before and after filter 2.
The real interesting numbers are the amplitudes, _{}.
1) Consider the polarized photon problem and fill out the table below by finding how much of the beam will be transmitted for the following situations.
_{}
Prepared state 
Analyzed using 
Measured state 
Result 

Filter 1 
Filter 2 
final filter 



type 
blocking 


_{} 
xy 
none 
_{} 
100% 
_{} 
xy 
none 
_{} 
0% 
_{} 
xy 
Block x 
_{} 
0% 
_{} 
xy 
Block x 
_{} 
0% 
_{} 
XY 
none 
_{} 
0% 
_{} 
RL 
none 
_{} 
100% 
_{} 
RL 
Block R 
_{}č_{} _{} 
50% 
_{} 
RL 
Block R 
_{}č_{} _{} 
50% 
_{} 
XY 
Block Y 
_{}č_{} _{} 
_{}% 
_{} 
XY 
Block Y 
_{}č_{} _{} 
25% 
_____________________________________________w
Another excellent review by Abner Shimony appears in the New Physics. He breaks QM into a set of basic postulates summarized here. He also discusses the problem of entanglement and measurement. A brief summary is provided.
1 
“Associated with every physical system is a complex linear vector space V, such that each vector of unit length represents a state of the system. 
2 
“There is a one to one correspondence between the set of
eventualities (observables) concerning the system and the set of subspaces of
the vector space associate with the system, such that if e is an
eventuality (observable) and E Is the subspace that corresponds to it,
then e is true in a state S> if and only if any vector that
represents S belong to E; and is false in the state S if and only if
any vector that represents S belongs to E^{(orthogonal)} .” A states described by a vector that has
components in both subspaces represents a state with an unspecified value for
this observable e. 
3 
“If S> is a state and e is an eventuality (observable) corresponding to the subspace E, then the probability that e will turn out to be true if the initially the system is in state S> and an operation is performed to actualize (measure) it. _{} v is a unit vector representing S … and P_{E} is the projection.” 
4 
“If 1 and 2 are two physical systems, with which the vector spacesV_{1} and V_{2} are associated, then the composite system 1 + 2 consisting of 1 and 2 is associated with the tensor product V_{1} x V_{2}.”_{} 
5 
“If a system is in a nonreactive environment between 0 > t, then there is a linear operator U(t) such that U(t) v> represents the state of the system at time t if v> represents the state of the system at time 0. Furthermore, U(t) v^{2 }=  v^{2} for all v in the vector space.” 


The author (Shimony) is an expert on quantum measurement, EPR, and Schrodinger cat paradoxes. He carefully explores the basic structure of quantum theories.