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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).
Gleason’s theorem is also valid for real and quaternionic Hilbert spaces as proved by Varadarajan in 1968. A gap of that proof has been fixed in 2018 by V.Moretti and M.Oppio.
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
Gleason's Theorem proved for real, complex and quaternionic Hilbert spaces using the notion of real trace.
See also