Gleason's theorem

Gleason's theorem, named after Andrew Gleason, is a mathematical result of particular importance for 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 of events in a real or complex Hilbert space. The essence of the theorem is that:

For a Hilbert space of dimension 3 or greater, the only possible measure of the probability of the state associated with a particular linear subspace a of the Hilbert space will have the form Tr(μ(a) W), the trace of the operator product of the projection operator μ(a) and the density matrix W for the system.

1 Context
2 Application
3 Implications
4 See also
5 Notes
6 References

Context

Quantum logic treats quantum events (or measurement outcomes) as logical propositions, and studies the relationships and structures formed by these events, with specific emphasis on quantum measurement. More formally, a quantum logic is a set of events that is closed under a countable disjunction of countably many mutually exclusive events. The representation theorem in quantum logic shows that these logics form a lattice which is isomorphic to the lattice of subspaces of a vector space with a scalar product.

It remains an open problem in quantum logic to prove that the field K over which the vector space is defined, is either the real numbers, complex numbers, or the quaternions. This is a necessary result for Gleason's theorem to be applicable, since a Hilbert space is by definition defined over one of these fields.

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 assign probabilities to these events. This section draws extensively on 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. $\alpha_1, \alpha_2, \alpha_3, ..., \alpha_n$ . An "event", then, is a proposition $x_i$ , which in natural language can be rendered as "the outcome of measuring A on the system is $\alpha_i$ ". The events $x_i$ 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 event is an atom.

A state, or probability function, is a real function P on the atoms in L, with the following properties:

$P(0) = 0,\,$ and $P(y) \ge 0$ for all $y \in L;$
$\sum_{j=1}^n P(x_j) = 1$ if $x_1, x_2, x_3, \dots , x_n\,$

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:

$P(y) = \sum_{j=1}^r P(x_j).$

Here, we introduce Gleason's theorem itself:

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

This is, of course, the Born rule for probability in quantum mechanics. The probability rule of quantum mechanics is therefore dictated by the event structure generated by propositions governing measurement.

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 stochasticity 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-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 bivalent probability measures, i.e. probability measures having only the values 1 and 0. Because the mapping $u \rightarrow \langle Wu, u \rangle$ is continuous on the unit sphere of the Hilbert space for any density operator W. Since this unit sphere is connected, no continuous function on it can take only the value of 0 and 1. ^[2] But, a hidden variables theory which is 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 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.

Notes

^ Pitowsky (2005), pg. 14
^ Wirce (2006), pg. 3

References

Gleason, A. M. (1957). "Measures on the closed subspaces of a Hilbert space". Journal of Mathematics and Mechanics 6: 885–893. doi:10.1512/iumj.1957.6.56050. MR 0096113. http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050.
Pitowsky, I. (2005). Quantum mechanics as a theory of probability. pp. 10095. arXiv:quant-ph/0510095. Bibcode 2005quant.ph.10095P.
Wilce, A. (2006). "Quantum Logic and Probability Theory". In The Stanford Encyclopedia of Philosophy (Spring 2006 Edition), Edward N. Zalta (ed.).
Dvurecenskij, Anatolij (1992). Gleason's Theorem and Its Applications. Mathematics and its Applications , Vol. 60. Dordrecht: Kluwer Acad. Publ.. pp. 348. ISBN 978-0-7923-1990-0.