Gleason's theorem

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Gleason's theorem (named after Andrew M. Gleason) 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 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(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[edit]

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 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 non-zero 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 [1], has negative implications for the possibility of a P-adic quantum mechanics.

Application[edit]

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. \alpha_1, \alpha_2, \alpha_3, ..., \alpha_n . An "event", then, is a proposition x_i, which in natural language can be rendered "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 events is an atom.

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

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

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  \langle\,  ,\, \rangle 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.

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[edit]

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. (Wilce (2006), pg. 3) 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.

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