nLab
Gleason's theorem

Idea

The classical Gleason theorem says that a state of the C-star algebra $ℬ(\mathscr{H})$ of all bounded operators on a Hilbert space is uniquely described by the values it takes on the orthogonal projections $𝒫$, if the dimension of the Hilbert space $\mathscr{H}$ is not 2.

In other words: every quasi-state is already a state $(\mathrm{if}\dim (H)>2)$.

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).

Implications for Quantum Logic

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”.

Abstract

…

Definitions

Definition

Let $\rho :𝒫\to [0,1]$ such that for every finite family $\{P_1,...,P_n:P_i\in 𝒫\}$ 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 $𝒫$.

If the family is not finite, but countable, then $\rho$ is a sigma-finite measure.

The Theorem

Classical Gleason’s Theorem

Theorem

If $\dim (\mathscr{H})\ne 2$ then each finitely additive measure on $𝒫$ can be uniquely extended to a state on $ℬ(\mathscr{H})$. Conversly the restriction of every state to $𝒫$ is a finitley additive measure on $𝒫$.

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.

Extension to Certain Types of von Neumann Algebras

…

Gleason's Theorem for POVMs

In quantum information theory, one often considers positive operator-valued measures (POVM?s) instead of Hermitian operator?s 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 $\rho :ℰ\to [0,1]$, where $ℰ$ 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.

Examples

Counterexample For Dimension Two

See example 8.1 in the book by Parthasarathy (see references). Our Hilbert space is ${ℝ}^2$. 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

$$P=\mid u⟩⟨u\mid$$P = {|u \rangle} {\langle u|}

with a unit vector $u$, i.e. $u\in {ℝ}^2,\parallel u\parallel =1$. 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

$$f(cu)=f(u)$$f(c u) = f(u)

$$\sum _{i}f(u_i)=1$$\sum_i f(u_i) = 1

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

$$f(u)=⟨u\mid Tu⟩$$f(u) = {\langle u | T u \rangle}

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

$$f(u)=\{\begin{array}{l}g(\theta )\text{for}0\le \theta <\frac{\pi }{2}\\ 1-g(\theta -\frac{\pi }{2})\text{for}\frac{\pi }{2}\le \theta <\pi \\ f(-u)\text{as defined in the first two items, else}\end{array}$$f(u) = \begin{cases} g(\theta) \; \; \text{for} \; \; 0 \le \theta \lt \frac{\pi}{2} \\ 1 - g(\theta - \frac{\pi}{2}) \; \; \text{for} \; \; \frac{\pi}{2} \le \theta \lt \pi \\ f(-u) \; \; \text{as defined in the first two items, else} \end{cases}

This probability measure will in general not extend to a state.

Related theorem

• Kochen-Specker theorem

• Wigner theorem

References

Gleason’s original paper outlining the theorem is

• A.M. Gleason “Measures on the closed subspaces of a Hilbert space,” Journal of Mathematics and Mechanics, pdf.

A standard textbook exposition of the theorem and its meaning is

• Huzihiro Araki, Mathematical Theory of Quantum Fields ,

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

• Jan Hamhalter, Quantum measure theory (ZMATH entry)

The classical theorem is proved also in this monograph:

• K. R. Parthasarathy, An Introduction to Quantum Stochastic Calculus (ZMATH)

Gleason's Theorem for POVMs is proved here:

• Paul Busch, Quantum states and generalized observables: a simple proof of Gleason’s theorem; (1999) (arXiv)

The failure of Gleason’s theorem for classical states (on Poisson algebras) is discussed in

• Michael Entov, Leonid Polterovich, Symplectic quasi-states and semi-simplicity of quantum homology (arXiv).

See also

• Wikipedia on Gleason’s theorem