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The classical Gleason theorem says that a state on the C*-algebra $\mathcal{B}(\mathcal{H})$ of all bounded operators on a Hilbert space is uniquely described by the values it takes on the orthogonal projections $\mathcal{P}$, if the dimension of the Hilbert space $\mathcal{H}$ is not 2.
In other words: every quasi-state is already a state if $dim(H) \gt 2$.
It is possible to extend the theorem to certain types of von Neumann algebras (e.g. obviously factors of type $I_2$ have to be excluded).
Roughly, Gleason’s theorem says that “a quantum state is completely determined by only knowing the answers to all of the possible yes/no questions”.
Let $\rho: \mathcal{P} \to [0, 1]$ such that for every finite family $\{ P_1, ..., P_n: P_i \in \mathcal{P} \}$ of pairwise orthogonal projections we have $\rho(\sum_{i=1}^n P_i) = \sum_{i=1}^n \rho(P_i)$, then $\rho$ is a finitely additive measure on $\mathcal{P}$.
If the family is not finite, but countable, then $\rho$ is a sigma-finite measure.
If $dim(\mathcal{H}) \neq 2$ then each finitely additive measure on $\mathcal{P}$ can be uniquely extended to a state on $\mathcal{B}(\mathcal{H})$. Conversly the restriction of every state to $\mathcal{P}$ is a finitley additive measure on $\mathcal{P}$.
The same holds for sigma-finite measures and normal states: Every sigma-finite measure can be extended to a normal state and every normal state restricts to a sigma-finite measure.
…
In quantum information theory, one often considers positive operator-valued measures (POVMs) instead of Hermitian operators as observables. While a Hermitian operator is given by a family of projection operators $P_i$ such that $\sum_i P_i = 1$, a POVM is given more generally by any family of positive-semidefinite operators $E_i$ such that $\sum_i E_i = 1$.
In the analog of Gleason’s Theorem for POVMs, therefore, we start with $\rho\colon \mathcal{E} \to [0,1]$, where $\mathcal{E}$ is the space of all positive-semidefinite operators. Then if $\sum_i \rho(E_i) = 1$ whenever $\rho(\sum_i E_i) = 1$, the theorem states that $\rho$ has a unique extension to a mixed quantum state.
As a theorem, Gleason's Theorem for POVMs is much weaker than the classical Gleason's Theorem, since we must begin with $\rho$ defined on a much larger space of operators. However, some content does remain, since we have not assumed any continuity properties of $\rho$. Also, Gleason's Theorem for POVMs has a much simpler proof, which works regardless of the dimension.
See example 8.1 in the book by Parthasarathy (see references). Our Hilbert space is $\mathbb{R}^2$. Projections $P$ on it are either identical zero, the identity, or projections on a one dimensional subspace, so that these $P$ can be written in the bra-ket notation? as
with a unit vector $u$, i.e. $u \in \mathbb{R}^2, {\|u\|} = 1$. In this finite dimensional case sigma-finite and finite are equivalent, and a finite probability measure is equivalent to a (complex valued) function such that
for every scalar $c$ of modulus one, every unit vector $u$ and every orthonormal basis $\{u_1, u_2\}$. If there is a state that extends such a measure and therefore restricts to such a measure on projections, there would be a linear operator $T$ such that
for all unit vectors $u$.
It turns out however that the conditions imposed on $f$ are not enough in two dimensions to enforce this kind of linearity of $f$. Heuristically, in three dimensions there are more rotations than in two, therefore the “rotational invariance” of (the conditions imposed on) $f$ is more restrictive in three dimensions than it is in two dimensions.
In two dimensions, choose a function $g$ on $[0, \frac{\pi}{2})$ such that $0 \le g(\theta) \le 1$ everywhere. There are no further restrictions, that is $g$ need not be continuous, for example. Now we can define a probability measure on the projections by
This probability measure will in general not extend to a state.
Other theorems about the foundations and interpretation of quantum mechanics include:
Gleason’s original paper outlining the theorem is
A standard textbook exposition of the theorem and its meaning is
where it appears as theorem 2.3 (without proof).
A monograph stating and proving both the classical theorem and extensions to von Neumann algebras is
The classical theorem is proved also in this monograph:
Gleason's Theorem for POVMs is proved here:
The failure of Gleason’s theorem for classical states (on Poisson algebras) is discussed in
See also