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\section*{effect algebra}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra}

[[!include higher algebra - contents]]

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\begin{quote}%
Caveat: There is an unrelated notion of ``effect of a computation''; that is rather in proximity to the entry [[monad (in computer science)]].


\end{quote}
\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{morphisms}{Morphisms}\dotfill \pageref*{morphisms} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{relation_to_dposets}{Relation to D-Posets}\dotfill \pageref*{relation_to_dposets} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

In [[quantum mechanics]] a [[self-adjoint operator]] $A$ on the given [[Hilbert space]] such that its [[spectrum of an operator|spectrum]] lies between 0 and 1 (hence a [[positive operator]] which is $\leq 1$) is sometimes called an \emph{effect} or \emph{quantum effect} (see e.g. (\hyperlink{Ludwig}{Ludwig}, \hyperlink{Kraus}{Kraus})). These operators generalize [[projection operators]] and may be thought of as [[quantum observables]] with ``unsharp'' or ``fuzzy'' value.

The notion of \emph{effect algebra} (due to (\hyperlink{FoulisBennet}{Foulis-Bennet 94})) is an abstraction of the structure exhibited by the collection of such effect operators.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A [[partial monoid|partial commutative monoid]] (PCM) consists of a set $M$ with a zero element $0 \in  M$ and a partial binary operation $\vee : M \times M \to M$ satisfying the three requirements below. They involve the notation $x \perp y$ for: $x \vee y$ is defined; in that case $x, y$ are called orthogonal.

\begin{enumerate}%
\item Commutativity: $x\perp y$ implies $y\perp x$ and $x\vee y=y\vee x$.


\item Associativity: $y\perp z$ and $x\perp(y\vee z)$ implies $x\perp y$ and $(x\vee y)\perp z$ and $x\vee (y\vee z)=(x\vee y)\vee z$.


\item Zero: $0\perp x$ and $0\vee x=x$



\end{enumerate}
(\hyperlink{FoulisBennet}{Foulis-Bennet 94} p.22)

In a PCM, we define: $x \le y:\Leftrightarrow \exists_z. x \vee z = y$. This is a preorder on any PCM.

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
A PCM is preordered by $\le$.

\end{prop}
\begin{proof}
Reflexivity is immediate from the Zero axiom, and transitivity follows easily from Associativity.

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
A \emph{generalized effect algebra} is a PCM $(E, 0, \vee)$ such that:

\begin{enumerate}%
\item Cancellation Law: If $a \perp b$, $a \perp c$ and $a \vee b = a \vee c$ then $b = c$.


\item Positivity Law: If $a \perp b$ and $a \vee b = 0$ then $a = b = 0$.



\end{enumerate}
In a generalized effect algebra, we define: $y\ominus x=z:\Leftrightarrow y=x\vee z$ (which exists iff $x \le y$, and is unique by the Cancellation Law).

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
A generalized effect algebra is partially ordered by $\le$.

\end{prop}
\begin{proof}
Suppose $x \le y$ and $y \le x$. Let $x \vee a = y$ and $y \vee b = x$. Then $x \vee (a \vee b) = x = x \vee 0$, and so $a \vee b = 0$ by the Cancellation Law. Therefore, $a = b = 0$ and so $x = y$.

\end{proof}
\begin{defn}
\label{}\hypertarget{}{}
An \emph{effect algebra} is a PCM $(E,0,\vee)$ with an orthocomplement. The latter is a unary operation $(-)^\perp :E\to E$ satisfying:

\begin{enumerate}%
\item Orthocomplement Law. $x^\perp\in E$ is the unique element in $E$ with $x\vee x^\perp=1$, where $1=0^\perp$.


\item Zero-One Law. $x\perp 1\Rightarrow x=0$.



\end{enumerate}
For such an effect algebra one defines: $x\wedge y:=(x^\perp\vee y^\perp)^\perp$ (\hyperlink{FoulisBennet}{Foulis-Bennet 94} p. 23)

\end{defn}
\begin{prop}
\label{}\hypertarget{}{}
A structure $(E, 0, \vee)$ is an effect algebra iff it is a generalized effect algebra with a greatest element, in which case that greatest element is $1 = 0^\perp$.

\end{prop}
\begin{proof}
Let $(E, 0, \vee)$ be an effect algebra. Then $E$ is a generalized effect algebra since:

\begin{enumerate}%
\item Cancellation Law. If $a \vee b = a \vee c$ then $a \vee b \vee (a \vee b)^\perp = a \vee c \vee (a \vee b)^\perp = 1$, and so $b = c = (a \vee (a \vee b)^\perp)^\perp$.


\item Positivity Law. If $a \vee b = 0$ then $(a \vee b) \perp 1$, hence $a \perp 1$ and $b \perp 1$ by Associativity. Thus, $a = b = 0$ by the Zero-One Law.



\end{enumerate}
1 is the greatest elements since, for any $x$, we have $x \vee x^\perp = 1$ and so $x \leq 1$.

Conversely, let $(E, 0, \vee)$ be a generalized effect algebra with greatest element 1. Define $x^\perp = 1 \ominus x$ for all $x$. Then:

\begin{enumerate}%
\item Orthocomplement Law. $x^\perp$ is the unique element such that $x \vee x^\perp = 1$ by definition.


\item Zero-One Law. If $x \perp 1$, then $1 \leq x \vee 1$, so $x \vee 1 = 1$. Thus, $x \vee 1 = 0 \vee 1$, and so $x = 0$ by the Cancellation Law.



\end{enumerate}
\end{proof}
\begin{remark}
\label{}\hypertarget{}{}
If we consider $(-)\ominus y:up(y)\to down(y^\perp)$ and $(-)\vee y:down(y^\perp)\to up$ as functors between [[posets]] we have [[adjunctions]]

\begin{displaymath}
((-\wedge y)\dashv (-\ominus y)\dashv (-\wedge y)
\end{displaymath}
Hence these functors are a [[Frobenius functor|frobenius pair]].

(\hyperlink{FoulisBennet}{Foulis-Bennet 94} p.25)

\end{remark}
\hypertarget{morphisms}{}\subsection*{{Morphisms}}\label{morphisms}

\begin{defn}
\label{}\hypertarget{}{}
Let $E$ and $F$ be effect algebras. A \emph{morphism of effect algebras} $f : E \rightarrow F$ is a function such that:

\begin{enumerate}%
\item f(1) = 1


\item If $x \perp y$ then $f(x) \perp f(y)$ and $f(x \vee y) = f(x) \vee f(y)$.



\end{enumerate}
We write $\mathbf{EA}$ for the category of effect algebras and morphisms of effect algebras.

\end{defn}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

(1) [[effect algebra of predicates]]

(2) The real unit inteval $[0,1]$ with $\vee$ being addition of real numbers is an effect algebra since $[0,1]$ is a pcm with zero object $0$ and commutative, associative addition of real numbers and $x\perp y$ iff $x+y\le 1$. The orthocomplement of $x\in [0,1]$ is given by $\x^\perp=1-x$.

(3) Let $D$ denote the discrete-probability-distribution [[monad]] on $Set$ which sends a set $X$ to the collection

\begin{displaymath}
D(X):=\{r_1 x_1+\dots +r_n x_n|x_i\in X, r_i\in [0,1], \Sigma_i r_i=1\}
\end{displaymath}
of formal convex combinations of elements of $X$ and let $Kl(D)$ denote the [[Kleisli category]] of $D$ which has as objects (just) sets and a morphism $f:X\to Y$ in $Kl(D)$ is a function $f:X\to D(Y)$ which can be interpreted as a [[Markov chain]] where the [[probability]] of the transition $x\to x_i$ is the coefficient $r_i\in [0,1]$ in the the convex sum $f(x)=r_1 x_1+\dots+r_n x_n$. $Kl(D)$ has as coproducts coproducts of $Set$. A predicate on $X\in Kl(D)$is hence a function $p:X\to D(X+X)$ and $[id_X,id_X]\circ p =id_X$ means that $p(x)\in D(X+X)$ is a convex combination of elements of the form $k_1 x, k_2 x\in X+X$ such that we have $p(x)=\varphi(x)k_1 x +\psi(x)k_2 x$ with $\varphi(x),\psi(x)\in [0,1]$ such that $\varphi{x}+\psi(x)=1$. Hence $p(x)$ can be written as $p(x)=\varphi(x)k_1 x + (1-\varphi(x))k_2 x$. In particular a predicate is (uniquely determined by) a function $\varphi:X\to [0,1]$ to the [[unit interval]]. In this view the orthocomplemet of $\varphi(x)$ is the function $x\mapsto 1-\varphi(x)$ which is point-wise the orthocomplement of the unit interval in the second example.

(4) In the category $Hilb$ of [[Hilbert space|Hilbert spaces]] the coproduct coincides with the product and hence is a [[biproduct]]. In this case a predicate $p:X\to X\otimes X$ on a Hilbert space $X$ has the form $p=\lt p_1,p_2\gt$ of a pair of maps and $[id_X,id_X]\circ p$ is equivalent to $p_1 +p_2=id_X$ where $+$ is point-wise addition. In particular $p_1$ and $p_2$ determine each other uniquely.

And now comes the eponymous feature: The category $Hilb$ is a [[dagger category]] and the dagger morphism $(-)^\dagger:Hilb^{op}\to Hilb$ is the identity on objects and complex conjugationn on morphisms. An endomorphism $f:X\to X$ is called to be a \emph{positive endomorphism} if there is a $g$ such that $f=g^\dagger\circ g$ and a predicate on $X$ is called to be an \textbf{effect} (on $X$) if $p_1$ and $p_2$ are positive. Another name for effect is ``unsharp predicate''; in this terminology a ``sharp predicate'' is a subset of the set of projections onto $X$.

(5) In a C$^\ast$-algebra the elements between 0 and 1 form an effect algebra with $(1-a)$ as the complement of $a$.

(6) As a special case, we obtain the effect algebra of a [[von Neumann algebra]]. In general, this is not a lattice. \hyperlink{DeGroote}{De Groote} defines a \emph{spectral order} on self-adjoint operators which makes the collection of effects a [[boundedly complete lattice]]. However, this is not the canonical order on an effect algebra, as defined above.

\hypertarget{relation_to_dposets}{}\subsection*{{Relation to D-Posets}}\label{relation_to_dposets}

A parallel concept in the literature is that of \emph{D-poset} (sometimes called \emph{D-lattice}), originally introduced for the same purpose of studying fuzzy or quantum logics. These first appeared in \hyperlink{ChovKop}{ChovKop}.

\begin{defn}
\label{}\hypertarget{}{}
A partial binary operation $\ominus$ on a poset $(P, \leq)$ is called a \emph{difference operation} (or simply \emph{difference}) on $P$ iff:

(1) $a \leq b \leftrightarrow b \ominus a$ is defined,

(2) $b \ominus a \leq b$,

(3) $b \ominus (b \ominus a) = a$,

(4) if $a \leq b \leq c$ implies that $c \ominus b \leq c \ominus a$ and $(c \ominus a) \ominus (c \ominus  b) = b \ominus a$.

A \emph{D-poset} is a poset $(P, \leq, \ominus, 1)$ with a difference operation and greatest element $1 \in P$.

\end{defn}
Any effect algebra is automatically a D-poset under the difference $c := b \ominus a \iff a \oplus c = b$, well-defined by the cancellation property of generalized effect algebras. Ultimately this determines an isomorphism of categories between D-posets and effect algebras.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[density matrix]]


\item [[separation algebra]]



\end{itemize}
\hypertarget{reference}{}\subsection*{{Reference}}\label{reference}

Quantum effect operators are discussed for instance in

\begin{itemize}%
\item G. Ludwig, \emph{Foundations of Quantum Mechanics I} Springer Verlag, New York, (1983)

\end{itemize}
\begin{itemize}%
\item K. Kraus, \emph{States, Effects, and Operations} Springer Verlag, Berlin, (1983)

\end{itemize}
The notion of effect algebra is due to

\begin{itemize}%
\item D. J. Foulis, M. K. Bennet, \emph{Effect algebras and unsharp quantum logics}, Found. Phys. 24 (1994), 1 331--1 352.

\end{itemize}
Discussion of effect algebras in the context of [[categorical logic]] is in

\begin{itemize}%
\item [[Bart Jacobs]], \emph{New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic}, (2012) (\href{http://arxiv.org/abs/1205.3940}{arXiv:1205.3940})

\end{itemize}
Discussion in the context of [[quantum logic]] is in section 6 of

\begin{itemize}%
\item Gianpiero Cattaneo, Maria Luisa Dalla Chiara, Roberto Giuntini and Francesco Paoli, \emph{Quantum Logic and Nonclassical Logics}, p. 127 in Kurt Engesser, Dov M. Gabbay, Daniel Lehmann (eds.) \emph{Handbook of Quantum Logic and Quantum Structures: Quantum Logic}, 2009 North Holland

\end{itemize}
A survey of the use of effect algebras in [[quantum mechanics]] is in

\begin{itemize}%
\item Teiko Heinosaari, Mario Ziman, \emph{Guide to Mathematical Concepts of Quantum Theory} (\href{http://arxiv.org/abs/0810.3536}{arXiv:0810.3536})

also appeared as:

\emph{The Mathematical Language of Quantum Theory}, Cambridge University Press (2011)


\item Hans de Groote, \emph{On a canonical lattice structure on the effect algebra of a von Neumann algebra} \href{http://arxiv.org/abs/math-ph/0410018}{arXiv:0410018}



\end{itemize}
Discussion in relation to [[presheaves]] on [[FinSet]]${}^{op}$ (hence in the [[classifying topos]] for objects) is in

\begin{itemize}%
\item Sam Staton, Sander Uijlen, \emph{Effect algebras, presheaves, non-locality and contextuality} (\href{http://www.cs.ru.nl/S.Uijlen/ICALP-camera-ready.pdf}{pdf})

\end{itemize}
D-posets were first introduced in:

\begin{itemize}%
\item F. Chovanec and F. Kopka, \emph{D-lattices}, International Journal of Theoretical Physics, vol. 34, no. 8, pp. 1297–1302, 1995.

\end{itemize}
[[!redirects effect algebras]]

[[!redirects effect]] [[!redirects effects]]

[[!redirects quantum effect]] [[!redirects quantum effects]] [[!redirects effect algebra]]



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