Effect algebra

Effect algebras are partial algebras which abstract the (partial) algebraic properties of events that can be observed in quantum mechanics. Structures equivalent to effect algebras were introduced by three different research groups in theoretical physics or mathematics in the late 1980s and early 1990s. Since then, their mathematical properties and physical as well as computational significance have been studied by researchers in theoretical physics, mathematics and computer science.
History.
In 1989, Giuntini and Greuling introduced structures for studying "unsharp properties", meaning those quantum events whose probability of occurring is strictly between zero and one (and is thus not an either-or event). In 1994, Chovanec and Kôpka introduced "D-posets" as posets with a partially defined "difference operation". In the same year, the paper by Bennet and Foulis "Effect algebras and unsharp quantum logics" was published. While it was this last paper that first used the term "effect algebra", it was shown that all three structures are equivalent. The proof of isomorphism of categories of D-posets and effect algebras is given for instance by Dvurecenskij and Pulmannova.
Motivation.
The operational approach to quantum mechanics takes the set of observable (experimental) outcomes as the constitutive notion of a physical system. That is, a physical system is seen as a collection of events which may occur and thus have a measurable effect on the reality. Such events are called "effects". This perspective already imposes some constrains on the mathematical structure describing the system: we need to be able to associate a probability to each effect.
In the Hilbert space formalism, effects correspond to positive semidefinite self-adjoint operators which lie below the identity operator in the following partial order: formula_1 if and only if formula_2 is positive semidefinite. The condition of being positive semidefinite guarantees that expectation values are non-negative, and being below the identity operator yields probabilities. Now we can define two operations on the Hilbert space effects: formula_3 and formula_4 "if formula_5", where formula_6 denotes the identity operator. Note that formula_7 is positive semidefinite and below formula_6 since formula_9 is, thus it is always defined. One can think of formula_7 as the negation of formula_9. While formula_4 is always positive semidefinite, it is not defined for all pairs: we have to restrict the domain of definition for those pairs of effects whose sum stays below the identity. Such pairs are called "orthogonal"; orthogonality reflects simultaneous measurability of observables.
Definition.
An "effect algebra" is a partial algebra consisting of a set formula_13, constants formula_14 and formula_15 in formula_13, a total unary operation formula_17, a binary relation formula_18, and a binary operation formula_19, such that the following hold for all formula_20:
The unary operation formula_36 is called "orthosupplementation" and formula_37 the "orthosupplement" of formula_38. The domain of definition formula_39 of formula_40 is called the "orthogonality relation" on formula_13, and formula_42 are called "orthogonal" if and only if formula_21. The operation formula_40 is referred to as the "orthogonal sum" or simply the "sum".
Properties.
The following can be shown for any elements formula_45 and formula_46 of an effect algebra, assuming formula_47:
Order properties.
Every effect algebra formula_13 is partially ordered as follows: formula_57 if and only if there is a formula_58 such that formula_59 and formula_60. This partial order satisfies:
Examples.
Orthoalgebras.
If the last axiom in the definition of an effect algebra is replaced by:
one obtains the definition of an "orthoalgebra". Since this axiom implies the last axiom for effect algebras (in the presence of the other axioms), every orthoalgebra is an effect algebra. Examples of orthoalgebras (and hence of effect algebras) include:
MV-algebras.
Any MV-algebra is an effect algebra (but not, in general, an orthoalgebra) with the unary operation as orthosupplementation and the binary operation restricted to orthogonal elements as the sum. In the context of MV-algebras, orthogonality of a pair of elements formula_45 is defined as formula_68. This coincides with orthogonality when an MV-algebra is viewed as an effect algebra.
An important example of an MV-algebra is the unit interval formula_69 with operations formula_70 and formula_71. Seen as an effect algebra, two elements of the unit interval are orthogonal if and only if formula_72 and then formula_73.
The set of effects of a unital C*-algebra.
Slightly generalising the motivating example of Hilbert space effects, take the set of effects on a unital C*-algebra formula_74, i.e. the elements formula_75 satisfying formula_76. The addition operation on formula_77 is defined when formula_72 and then formula_73. The orthosupplementation is given by formula_80.
Types of effect algebras.
There are various types of effect algebras that have been studied.
Morphisms.
A morphism from an effect algebra formula_13 to an effect algebra formula_85 is given by a function formula_86 such that formula_87 and for all formula_42
It then follows that morphisms preserve the orthosupplements.
Equipped with such morphisms, effect algebras form a category which has the following properties:
Positive operator-valued measures.
As an example of how effect algebras are used to expess concepts in quantum theory, the definition of a positive operator-valued measure may be cast in terms of effect algebra morphisms as follows. Let formula_92 be the algebra of effects of a Hilbert space formula_93, and let formula_94 be a "σ"-algebra. A "positive operator-valued measure" (POVM) is an effect algebra morphism formula_95 which preserves joins of countable chains. A POVM is a projection-valued measure precisely when its image is contained in the orthoalgebra of projections on the Hilbert space formula_93.