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'''Gleason's theorem''' is a mathematical result which is of particular importance for the field of [[quantum logic]]. It proves that the [[Born rule]] for the probability of obtaining specific results to a given measurement, follows naturally from the structure formed by the [[Lattice (order)|lattice]] of events in a [[Real number|real]] or [[Complex number|complex]] [[Hilbert space]]. The essence of the theorem is that:

:''For a [[Hilbert space]] of dimension 3 or greater, the only possible [[measure (mathematics)|measure]] of the probability of the state associated with a particular [[Linear subspace]] ''a'' of the Hilbert Space will have the form '''Tr'''(P(''a'') W), where '''Tr''' is a [[trace class]] operator of the matrix product of the projection operator P(a) and the density matrix for the system W.''

== Context ==
[[Quantum logic]] treats quantum events (or measurement outcomes) as [[propositional logic|logical propositions]], and studies the relationships and structures formed by these events, with specific emphasis on [[Measurement in quantum mechanics|quantum measurement]]. More formally, a quantum logic is a set of events that is [[Closure|closed]] under a [[Countable set|countable]] [[disjunction]] of countably many [[Mutual exclusivity|mutually exclusive]] events. The ''[[representation theorem]]'' in quantum logic shows that these logics form a [[lattice]]{{dn|date=October 2013}} which is [[Isomorphism|isomorphic]] to the lattice of [[subspace]]s of a [[vector space]] with a [[scalar product]].

It remains an open problem in quantum logic to prove that the [[field (mathematics)|field]] ''K'' over which the vector space is defined, is either the [[real number]]s, [[complex number]]s, or the [[quaternion]]s. This has negative implications for the possibility of a [[P-adic quantum mechanics]]. This is a necessary result for Gleason's theorem to be applicable, since in all these cases we know that the definition of the [[inner product]] of a [[Zero vector|non-zero]] [[Euclidean vector|vector]] with itself will satisfy the requirements to make the vector space in question a Hilbert space. Soler's Result, the restriction of the field to just these three fields [http://golem.ph.utexas.edu/category/2010/12/solers_theorem.html], has negative implications for the possibility of a [[P-adic quantum mechanics]].

== Application ==
The representation theorem allows us to treat quantum events as a lattice ''L = L(H)'' of subspaces of a real or complex Hilbert space. Gleason's theorem allows us to "attach" these events to probabilities. This section draws extensively from the analysis presented in Pitowsky (2005).

We let ''A'' represent an observable with finitely many potential outcomes: the eigenvalues of the [[Hermitian operator]] ''A'', i.e. <math>\alpha_1, \alpha_2, \alpha_3, ..., \alpha_n </math>. An "event", then, is a proposition <math>x_i</math>, which in natural language can be rendered "the outcome of measuring ''A'' on the system is <math>\alpha_i</math>". The events <math>x_i</math> generate a sublattice of the Hilbert space which is a finite [[Boolean algebra]], and if ''n'' is the dimension of the Hilbert space, then each events is an [[atom]].

A ''state'', or ''probability function'', is a [[real function]] ''P'' on the atoms in ''L'', with the following properties:
# <math>P(0) = 0</math>, and <math>P(y) \ge 0</math> for all <math>y \in L</math>
# <math>\sum_{j=1}^n P(x_j) = 1</math>, if <math>x_1, x_2, x_3, ..., x_n</math> are orthogonal atoms

This means for every lattice element ''y'', the probability of obtaining ''y'' as a measurement outcome is fixed, since it may be expressed as the union of a set of orthogonal atoms: <math>P(y) = \sum_{j=1}^r P(x_j) = 1</math>

Here, we introduce Gleason's theorem itself:

:''Given a state P on a space of dimension <math>\ge 3 </math>, there is an Hermitian, non-negative operator W on H, whose trace is unity, such that <math>P(x) = \langle \mathbf{x}, W \mathbf{x} \rangle </math> for all atoms <math>x \in L</math>, where < , > is the inner product, and <math>\mathbf{x}</math> is a unit vector along <math>x</math>. In particular, if some <math>x_0</math> satisfies <math>P(x_0) = 1</math>, then <math> P(x) = \left| \langle\mathbf{x_0}, \mathbf{x} \rangle \right|^2</math> for all <math>x \in L</math>.

This is, of course, the [[Born rule]] for probability in quantum mechanics. The theorem presumes that the underlying set of numbers that the functions are defined over are [[real numbers]] or [[complex numbers]]. A constructive proof exists.

== Implications ==
Gleason's theorem highlights a number of fundamental issues in quantum measurement theory. The fact that the logical structure of quantum events dictates the probability measure of the formalism is taken by some to demonstrate an inherent [[stochastic]]ity in the very fabric of the world. To some researchers, such as Pitowski, the result is convincing enough to conclude that quantum mechanics represents a new theory of probability. Alternatively, such approaches as [[relational quantum mechanics]] make use of Gleason's theorem as an essential step in deriving the quantum formalism from [[Information theory|information-theoretic]] postulates.

The theorem is often taken to rule out the possibility of hidden variables in quantum mechanics. This is because the theorem implies that there can be no [[Bivalence|bivalent]] probability measures, i.e. probability measures having only the values 1 and 0. Because the mapping <math>u \rightarrow \langle Wu, u \rangle </math> is continuous on the [[unit sphere]] of the Hilbert space for any density operator ''W''. Since this unit sphere is [[Connected (topology)|connected]], no continuous function on it can take only the value of 0 and 1. (Wilce (2006), pg. 3) But, a [[hidden variables]] theory which is [[determinism|deterministic]] implies that the probability of a given outcome is ''always'' either 0 or 1: either the electron's spin is up, or it isn't (which accords with [[Classical mechanics|classical]] intuitions). Gleason's theorem therefore seems to hint that quantum theory represents a deep and fundamental departure from the classical way of looking at the world, and that this departure is ''logical'', not ''interpretational'', in nature.

== See also ==
* [[Quantum logic]]
* [[Born's rule]]
* [[Measurement in quantum mechanics]]
* [[Hilbert space]]

== References ==
* {{cite journal
| author = Gleason, A. M.
| title = Measures on the closed subspaces of a Hilbert space
| journal = [[Journal of Mathematics and Mechanics]]
| volume = 6
| year = 1957
| pages = 885–893
| url = http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050
| id = {{MathSciNet | id = 0096113}}
| doi=10.1512/iumj.1957.6.56050}}
* {{cite journal
| author = Pitowsky, I.
| year = 2005
| title = Quantum mechanics as a theory of probability
| id =
| arxiv=quant-ph/0510095|bibcode = 2005quant.ph.10095P
| pages = 10095 }}
* Wilce, A. (2006). "Quantum Logic and Probability Theory". In ''[http://plato.stanford.edu/archives/spr2006/entries/qt-quantlog/ The Stanford Encyclopedia of Philosophy]'' (Spring 2006 Edition), Edward N. Zalta (ed.).
* {{cite book
| author = Dvurecenskij, Anatolij
| title = Gleason's Theorem and Its Applications
| year = 1992
| series = Mathematics and its Applications , Vol. 60
| pages = 348
| isbn = 978-0-7923-1990-0
| publisher = Kluwer Acad. Publ.
| location = Dordrecht
}}

{{DEFAULTSORT:Gleasons theorem}}
[[Category:Hilbert space]]
[[Category:Quantum measurement]]
[[Category:Probability theorems]]

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