Effect algebra

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).[1][2] In 1994, Chovanec and Kôpka introduced D-posets as posets with a partially defined difference operation.[3] In the same year, the paper by Bennet and Foulis Effect algebras and unsharp quantum logics was published.[4] While it was this last paper that first used the term effect algebra,[5] it was shown that all three structures are equivalent.[2] The proof of isomorphism of categories of D-posets and effect algebras is given for instance by Dvurecenskij and Pulmannova.[6]

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.[7] 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: ${\displaystyle A\leq B}$ if and only if ${\displaystyle B-A}$ is positive semidefinite.[5] 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: ${\displaystyle A':=I-A}$ and ${\displaystyle A+B}$ if ${\displaystyle A+B\leq I}$, where ${\displaystyle I}$ denotes the identity operator. Note that ${\displaystyle A'}$ is positive semidefinite and below ${\displaystyle I}$ since ${\displaystyle A}$ is, thus it is always defined. One can think of ${\displaystyle A'}$ as the negation of ${\displaystyle A}$. While ${\displaystyle A+B}$ 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.

An effect algebra is a partial algebra consisting of a set ${\displaystyle E}$, constants ${\displaystyle 0}$ and ${\displaystyle 1}$ in ${\displaystyle E}$, a total unary operation ${\displaystyle ':E\rightarrow E}$, a binary relation ${\displaystyle \bot \subseteq E\times E}$, and a binary operation ${\displaystyle \oplus$ :\bot \rightarrow E} , such that the following hold for all ${\displaystyle a,b,c\in E}$:

• commutativity: if ${\displaystyle a\perp b}$, then ${\displaystyle b\perp a}$ and ${\displaystyle a\oplus b=b\oplus a}$,
• associativity: if ${\displaystyle a\perp b}$ and ${\displaystyle (a\oplus b)\perp c}$, then ${\displaystyle b\perp c}$ and ${\displaystyle a\perp (b\oplus c)}$ as well as ${\displaystyle (a\oplus b)\oplus c=a\oplus (b\oplus c),}$
• orthosupplementation: ${\displaystyle a\perp a'}$ and ${\displaystyle a\oplus a'=1}$, and if ${\displaystyle a\perp b}$ such that ${\displaystyle a\oplus b=1}$, then ${\displaystyle b=a'}$,
• zero-one law: if ${\displaystyle a\perp 1}$, then ${\displaystyle a=0}$.[4]

The unary operation ${\displaystyle '}$ is called orthosupplementation and ${\displaystyle a'}$ the orthosupplement of ${\displaystyle a}$. The domain of definition ${\displaystyle \bot }$ of ${\displaystyle \oplus }$ is called the orthogonality relation on ${\displaystyle E}$, and ${\displaystyle a,b\in E}$ are called orthogonal if and only if ${\displaystyle a\perp b}$. The operation ${\displaystyle \oplus }$ is referred to as the orthogonal sum or simply the sum.[4]

The following can be shown for any elements ${\displaystyle a,b}$ and ${\displaystyle c}$ of an effect algebra, assuming ${\displaystyle a\perp b,c}$:

• ${\displaystyle a''=a}$,
• ${\displaystyle 0'=1}$,
• ${\displaystyle a\perp 0}$, and ${\displaystyle a\oplus 0=a}$,
• ${\displaystyle a\oplus b=0}$ implies ${\displaystyle a=b=0}$,
• ${\displaystyle a\oplus b=a\oplus c}$ implies ${\displaystyle b=c}$.[4]

There are various types of effect algebras that have been studied.

• Interval effect algebras that arise as an interval ${\displaystyle [0,u]_{G}}$ of some ordered Abelian group ${\displaystyle G}$.[4]
• Convex effect algebras have an action of the real unit interval ${\displaystyle [0,1]}$ on the algebra. A representation theorem of Gudder shows that these all arise as an interval effect algebra of a real ordered vector space.[12]
• Lattice effect algebras where the order structure forms a lattice.[13]
• Effect algebras satisfying the Riesz decomposition property:[14] an MV-algebra is precisely a lattice effect algebra with the Riesz decomposition property.[15]
• Sequential effect algebras have an additional sequential product operation that models the Lüders product on a C*-algebra.[16]
• Effect monoids are the monoids in the category of effect algebras. They are effect algebras that have an additional associative unital distributive multiplication operation.[17]

A morphism from an effect algebra ${\displaystyle E}$ to an effect algebra ${\displaystyle F}$ is given by a function ${\displaystyle f:E\rightarrow F}$ such that ${\displaystyle f(1)=1}$ and for all ${\displaystyle a,b\in E}$

${\displaystyle a\perp b}$ implies ${\displaystyle f(a)\perp f(b)}$ and ${\displaystyle f(a\oplus b)=f(a)\oplus f(b)}$.[4]

It then follows that morphisms preserve the orthosupplements.

Equipped with such morphisms, effect algebras form a category which has the following properties:

• the category of Boolean algebras is a full subcategory of the category of effect algebras,[18]
• every effect algebra is a colimit of finite Boolean algebras.[18]

• Effect algebra at the nLab