JK423 said:
I am quite confused.. You are right, Gleason's theorem applies to all those situation that Born's rule is relevant. I need to understand the differences between Gleason's and Zurek's work, so i cannot say anything else at the moment. Does Gleason's theorem assumes Born's rule for example?
Gleason only assumes that we're dealing with a Hilbert space that's at least 3-dimensional.
From a physicist's point of view, the reason we
want a theorem like Gleason's is that subspaces of the Hilbert space can be thought of as mathematical representations of yes-no experiments. For example, let a,b be real numbers such that b>a, and suppose that we design a device that measures an observable A and returns the value "yes", if the result of the A-measurement is in the interval [a,b] and "no" otherwise. The system's state vector after the measurement will be in the subspace spanned by eigenvectors of A with eigenvalues in [a,b]. There's always a subspace associated with each yes-no experiment, so we need to be able to assign them probabilities in a way that makes sense. For example, the probability associated with the trivial subspace {0} must be 0, and the probability associated with the entire Hilbert space must be 1. Further, if E and F are orthogonal subspaces, then the probability associated with the smallest subspace that contains E and F must be the sum of the probability associated with E and the probability associated with F.
These rules are similar to, but not identical to, the rules that define a probability measure. The main difference is that the domain of a probability measure, as defined in measure theory, is a σ-algebra, and the set of subspaces of a Hilbert space isn't. However, it is a
lattice, and σ-algebras can be thought of as a special kind of lattice. So what we need is a generalization of the term "probability measure" to lattices.
The generalization is straightforward. It involves a version of the rules I described above. With this definition in place, we can state the problem accurately: Find all probability measures on the lattice of subspaces of a Hilbert space. This is the problem Gleason solved. His answer goes like this:
For each probability measure P on the lattice L of subspaces, there's a unique state operator S such that ##P(M)=Tr(P_M S)## for all M in L, where ##P_M## is the projection operator associated with the subspace M. The map ##P\mapsto S## is a bijection from the set of probability measures onto the set of state operators.
In my opinion, it's far more natural to define a "state" as a probability measure than as a state operator, so to me, this theorem is the justification for why state operators can be thought of as states.
Now consider the special case of a pure state ##S=|\psi\rangle\langle\psi|## and let M be the eigenspace associated with some eigenvalue
a of a self-adjoint operator A with non-degenerate spectrum (i.e. a 1-dimensional eigenspace for each eigenvalue). Then the formula above reduces to
$$P\left(|a\rangle \langle a|\right) =\operatorname{Tr} |\psi\rangle \langle\psi|a\rangle\langle a| =\sum_{a'} \langle a'|\psi\rangle \langle|\psi|a\rangle \langle a|a'\rangle =\left|\langle a|\psi\rangle\right|^2.$$ This right-hand side is of course the probability that the Born rule assigns to the result
a when we're measuring A.
One final comment, when I say "subspace" in this post, I always mean a closed linear subspace. (A Hilbert space can have subsets that are inner product spaces but not Hilbert spaces, but we're not interested in those. By requiring that the subset isn't just a vector space, but also a closed set with respect to the norm topology, we ensure that the subset is also a Hilbert space).