# nLab effect algebra

Contents

### Context

#### Higher algebra

higher algebra

universal algebra

## Surveys, textbooks and lecture notes

Caveat: There is an unrelated notion of “effect of a computation”; that is rather in proximity to the entry monad (in computer science).

# Contents

## Idea

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

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

## Definition

###### Definition

A 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.

1. Commutativity: $x\perp y$ implies $y\perp x$ and $x\vee y=y\vee x$.

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

3. Zero: $0\perp x$ and $0\vee x=x$

(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.

###### Proposition

A PCM is preordered by $\le$.

###### Proof

Reflexivity is immediate from the Zero axiom, and transitivity follows easily from Associativity.

###### Definition

A generalized effect algebra is a PCM $(E, 0, \vee)$ such that:

1. Cancellation Law: If $a \perp b$, $a \perp c$ and $a \vee b = a \vee c$ then $b = c$.

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

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).

###### Proposition

A generalized effect algebra is partially ordered by $\le$.

###### 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 $x = y$.

###### Definition

An effect algebra is a PCM $(E,0,\vee)$ with an orthocomplement. The latter is a unary operation $(-)^\perp :E\to E$ satisfying:

1. Orthocomplement Law. $x^\perp\in E$ is the unique element in $E$ with $x\vee x^\perp=1$, where $1=0^\perp$.

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

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

###### Proposition

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 $1 = 0^\perp$.

###### Proof

Let $(E, 0, \vee)$ be an effect algebra. Then $E$ is a generalized effect algebra since:

1. 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 $b = c = (a \vee (a \vee b)^\perp)^\perp$.

2. 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.

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:

1. Orthocomplement Law. $x^\perp$ is the unique element such that $x \vee x^\perp = 1$ by definition.

2. 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.

###### Remark

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

$((-\wedge y)\dashv (-\ominus y)\dashv (-\wedge y)$

Hence these functors are a frobenius pair.

(Foulis-Bennet 94 p.25)

## Morphisms

###### Definition

Let $E$ and $F$ be effect algebras. A morphism of effect algebras $f : E \rightarrow F$ is a function such that:

1. f(1) = 1

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

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

## Examples

(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

$D(X):=\{r_1 x_1+\dots +r_n x_n|x_i\in X, r_i\in [0,1], \Sigma_i r_i=1\}$

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 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 conjugation on morphisms. An endomorphism $f:X\to X$ is called to be a positive endomorphism if there is a $g$ such that $f=g^\dagger\circ g$ and a predicate on $X$ is called to be an 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. De Groote defines a 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.

## Relation to D-Posets

A parallel concept in the literature is that of D-poset (sometimes called D-lattice), originally introduced for the same purpose of studying fuzzy or quantum logics. These first appeared in Kôpka 92 Chovanec-Kôpka 95.

###### Definition

A partial binary operation $\ominus$ on a poset $(P, \leq)$ is called a difference operation (or simply 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 D-poset is a poset $(P, \leq, \ominus, 1)$ with a difference operation and greatest element $1 \in P$.

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.

## Relation to Boolean algebras

Every Boolean algebra is an effect algebra, with $a\bot b$ whenever $a \wedge b=0$. Then the partial monoid of the effect algebra is a restriction of the join of the Boolean algebra. The orthocomplement is $a^\bot=\neg a$.

Starting from a powerset Boolean algebra, for example, we have $a\bot b$ when $a$ and $b$ are disjoint, and $a\vee b$ is the disjoint union.

A function is a Boolean algebra homomorphism if and only if it is a morphism of effect algebras.

In fact the finite Boolean algebras form a dense subcategory of the category $\mathbf{EA}$ of effect algebras. This means that every effect algebra is a canonical colimit of finite Boolean algebras.

Since the powerset functor $\mathbf{FinSet}\to \mathbf{FinBool}^{\mathrm{op}}$ is an equivalence of categories (Stone duality), this density property also means that we have a full and faithful functor $T:\mathbf{EA}\to [\mathbf{FinSet},\mathbf{Set}]$. This functor can be given explicitly as finding the tests of an effect algebra. A test is a sequence of orthogonal elements that sum to $1$. Hence

$T(E) (n) = \{ (a_1,\dots,a_n)\ |\ a_1\vee\dots \vee a_n=1\}$

For example, starting from the unit interval, $T([0,1])(n)$ gives the $(n-1)$-simplex.

Quantum effect operators are discussed for instance in

• G. Ludwig, Foundations of Quantum Mechanics I Springer Verlag, New York, (1983)

• Karl Kraus, States, Effects, and Operations – Fundamental Notions of Quantum Theory, Lecture Notes in Physics 190 Springer (1983) [doi:10.1007/3-540-12732-1]

The notion of effect algebra is due to

• D. J. Foulis, M. K. Bennet, Effect algebras and unsharp quantum logics, Found. Phys. 24 (1994), 1 331–1 352.

Discussion of effect algebras in the context of categorical logic is in

Discussion in the context of quantum logic is in section 6 of

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

A survey of the use of effect algebras in quantum mechanics is in

• Teiko Heinosaari, Mario Ziman, Guide to Mathematical Concepts of Quantum Theory (arXiv:0810.3536)

also appeared as:

The Mathematical Language of Quantum Theory, Cambridge University Press (2011)

• Hans de Groote, On a canonical lattice structure on the effect algebra of a von Neumann algebra arXiv:0410018

Discussion in relation to presheaves on FinSet${}^{op}$ (hence in the classifying topos for objects) and density of Boolean algebras is in

• Sam Staton, Sander Uijlen, Effect algebras, presheaves, non-locality and contextuality (pdf)

D-posets were first introduced in:

• F. Kôpka,_D-posets of fuzzy sets_, Tatra Mountains Mathematical Publications, vol. 1, no. 1, pp. 83-87, 1992.

• F. Chovanec and F. Kôpka, D-lattices, International Journal of Theoretical Physics, vol. 34, no. 8, pp. 1297–1302, 1995.