Gleason's theorem
Encyclopedia
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:
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 subspace
s 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 number
s, complex number
s, or the quaternion
s. 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.
We let A represent an observable with finitely many potential outcomes: the eigenvalues of the Hermitian operator A, i.e.
. An "event", then, is a proposition 
, which in natural language can be rendered as "the outcome of measuring A on the system is 
". The events 
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:
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:
Here, we introduce Gleason's theorem itself:
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.
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-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
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. 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.
Andrew Gleason
Andrew Mattei Gleason was an American mathematician and the eponym of Gleason's theorem and the Greenwood–Gleason graph. After briefly attending Berkeley High School he graduated from Roosevelt High School in Yonkers, then Yale University in 1942, where he became a Putnam Fellow...
, is a mathematical result of particular importance for quantum logic
Quantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
. It proves that the Born rule
Born rule
The Born rule is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of quantum mechanics...
for the probability of obtaining specific results to a given measurement, follows naturally from the structure formed by the lattice
Lattice model (physics)
In physics, a lattice model is a physical model that is defined on a lattice, as opposed to the continuum of space or spacetime. Lattice models originally occurred in the context of condensed matter physics, where the atoms of a crystal automatically form a lattice. Currently, lattice models are...
of events in a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
or complex
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
. The essence of the theorem is that:
- For a Hilbert spaceHilbert spaceThe mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
of dimension 3 or greater, the only possible measureMeasure (mathematics)In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
of the probability of the state associated with a particular linear subspaceLinear subspaceThe concept of a linear subspace is important in linear algebra and related fields of mathematics.A linear subspace is usually called simply a subspace when the context serves to distinguish it from other kinds of subspaces....
a of the Hilbert space will have the form Tr(μ(a) W), the traceTrace (linear algebra)In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of the operator product of the projection operator μ(a) and the density matrixDensity matrixIn quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
W for the system.
Context
Quantum logicQuantum logic
In quantum mechanics, quantum logic is a set of rules for reasoning about propositions which takes the principles of quantum theory into account...
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
Measurement in quantum mechanics
The framework of quantum mechanics requires a careful definition of measurement. The issue of measurement lies at the heart of the problem of the interpretation of quantum mechanics, for which there is currently no consensus....
. More formally, a quantum logic is a set of events that is closed
Closure (mathematics)
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
under a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
disjunction of countably many mutually exclusive events. The representation theorem
Representation theorem
In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to a concrete structure.For example,*in algebra,...
in quantum logic shows that these logics form a lattice
Lattice (group)
In mathematics, especially in geometry and group theory, a lattice in Rn is a discrete subgroup of Rn which spans the real vector space Rn. Every lattice in Rn can be generated from a basis for the vector space by forming all linear combinations with integer coefficients...
which is isomorphic
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
to the lattice of subspace
Subspace
-In mathematics:* Euclidean subspace, in linear algebra, a set of vectors in n-dimensional Euclidean space that is closed under addition and scalar multiplication...
s of a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
with a scalar product.
It remains an open problem in quantum logic to prove that the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K over which the vector space is defined, is either the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s, or the quaternion
Quaternion
In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space...
s. 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.
. An "event", then, is a proposition , which in natural language can be rendered as "the outcome of measuring A on the system is ". The events 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:
-
and 
for all 
-
if 
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:
Here, we introduce Gleason's theorem itself:
- Given a state P on a space of dimension
, there is an Hermitian, non-negative operator W on H, whose trace is unity, such that 
for all atoms 
, where < · , · > is the inner product, and 
is a unit vector along 
. In particular, if some 
satisfies 
, then 
for all 
This is, of course, the Born rule
Born rule
The Born rule is a law of quantum mechanics which gives the probability that a measurement on a quantum system will yield a given result. It is named after its originator, the physicist Max Born. The Born rule is one of the key principles of quantum mechanics...
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 stochasticStochastic
Stochastic refers to systems whose behaviour is intrinsically non-deterministic. A stochastic process is one whose behavior is non-deterministic, in that a system's subsequent state is determined both by the process's predictable actions and by a random element. However, according to M. Kac and E...
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
Relational quantum mechanics
Relational quantum mechanics is an interpretation of quantum mechanics which treats the state of a quantum system as being observer-dependent, that is, the state is the relation between the observer and the system. This interpretation was first delineated by Carlo Rovelli in a 1994 preprint, and...
make use of Gleason's theorem as an essential step in deriving the quantum formalism from information-theoretic
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
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
is continuous on the unit sphereUnit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point...
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. But, a hidden variables
Hidden variables
Hidden variables may refer to:* Hidden variable theories, in physics a class of theories trying to explain away the statistical nature of quantum mechanics* Latent variables, in statistics, variables that are inferred from other observed variables...
theory which is deterministic
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...
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
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
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.