I am doing a term project on the BB84 Protocol and it makes use of the Heisenberg Uncertainty Principle. I think I understand the principle in theory. If we have two non-commuting observables, then we cannot simultaneously measure a state with these observables, since after the measurement the state collapses to one of the eigenstates of the observable, but non-commuting observables do not have common eigenstates. This is what I understood.
I also found that the Paulis spin matrices for x and z, which are used in the BB84 protocol do not commute. So they should (and no dot) have common eigenstates. I tried to find the Heisenberg inequality using the formula for the Heisenberg inequality where we use average values and each time for the 0 ket I find the inequality side 0. Shouldn't it give something other than 0?
Then I tried to justify the 0, thinking that maybe for one of the spin matrices the uncertainty is 0 since we are using the right operator, and that is why when we multiply the uncertainties we get 0?
Am I doing something wrong in my calculations, or have I completely misunderstood the subject.
Thanks for any kind of help.