symmetric monoidal (∞,1)-category of spectra
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Caveat: There is an unrelated notion of “effect of a computation”; that is rather in proximity to the entry monad (in computer science).
In quantum mechanics a self-adjoint operator on the given Hilbert space such that its spectrum lies between 0 and 1 (hence a positive operator which is ) 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.
A partial commutative monoid? (PCM) consists of a set with a zero element and a partial binary operation satisfying the three requirements below. They involve the notation for: is defined; in that case are called orthogonal.
Commutativity: implies and .
Associativity: and implies and and .
Zero: and
(Foulis-Bennet 94 p.22)
In a PCM, we define: . This is a preorder on any PCM.
A PCM is preordered by .
Reflexivity is immediate from the Zero axiom, and transitivity follows easily from Associativity.
A generalized effect algebra is a PCM such that:
Cancellation Law: If , and then .
Positivity Law: If and then .
In a generalized effect algebra, we define: (which exists iff , and is unique by the Cancellation Law).
A generalized effect algebra is partially ordered by .
Suppose and . Let and . Then , and so by the Cancellation Law. Therefore, and so .
An effect algebra is a PCM with an orthocomplement. The latter is a unary operation satisfying:
Orthocomplement Law. is the unique element in with , where .
Zero-One Law. .
For such an effect algebra one defines: (Foulis-Bennet 94 p. 23)
A structure is an effect algebra iff it is a generalized effect algebra with a greatest element, in which case that greatest element is .
Let be an effect algebra. Then is a generalized effect algebra since:
Cancellation Law. If then , and so .
Positivity Law. If then , hence and by Associativity. Thus, by the Zero-One Law.
1 is the greatest elements since, for any , we have and so .
Conversely, let be a generalized effect algebra with greatest element 1. Define for all . Then:
Orthocomplement Law. is the unique element such that by definition.
Zero-One Law. If , then , so . Thus, , and so by the Cancellation Law.
If we consider and as functors between posets we have adjunctions
Hence these functors are a frobenius pair.
(Foulis-Bennet 94 p.25)
Let and be effect algebras. A morphism of effect algebras is a function such that:
f(1) = 1
If then and .
We write for the category of effect algebras and morphisms of effect algebras.
(1) effect algebra of predicates
(2) The real unit inteval with being addition of real numbers is an effect algebra since is a pcm with zero object and commutative, associative addition of real numbers and iff . The orthocomplement of is given by .
(3) Let denote the discrete-probability-distribution monad on which sends a set to the collection
of formal convex combinations of elements of and let denote the Kleisli category of which has as objects (just) sets and a morphism in is a function which can be interpreted as a Markov chain where the probability of the transition is the coefficient in the the convex sum . has as coproducts coproducts of . A predicate on is hence a function and means that is a convex combination of elements of the form such that we have with such that . Hence can be written as . In particular a predicate is (uniquely determined by) a function to the unit interval. In this view the orthocomplemet of is the function which is point-wise the orthocomplement of the unit interval in the second example.
(4) In the category of Hilbert spaces the coproduct coincides with the product and hence is a biproduct. In this case a predicate on a Hilbert space has the form of a pair of maps and is equivalent to where is point-wise addition. In particular and determine each other uniquely.
And now comes the eponymous feature: The category is a dagger category and the dagger morphism is the identity on objects and complex conjugation on morphisms. An endomorphism is called to be a positive endomorphism if there is a such that and a predicate on is called to be an effect (on ) if and are positive. Another name for effect is “unsharp predicate”; in this terminology a “sharp predicate” is a subset of the set of projections onto .
(5) In a C-algebra the elements between 0 and 1 form an effect algebra with as the complement of .
(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.
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.
A partial binary operation on a poset is called a difference operation (or simply difference) on iff:
(1) is defined,
(2) ,
(3) ,
(4) if implies that and .
A D-poset is a poset with a difference operation and greatest element .
Any effect algebra is automatically a D-poset under the difference , well-defined by the cancellation property of generalized effect algebras. Ultimately this determines an isomorphism of categories between D-posets and effect algebras.
Every Boolean algebra is an effect algebra, with whenever . Then the partial monoid of the effect algebra is a restriction of the join of the Boolean algebra. The orthocomplement is .
Starting from a powerset Boolean algebra, for example, we have when and are disjoint, and 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 of effect algebras. This means that every effect algebra is a canonical colimit of finite Boolean algebras.
Since the powerset functor is an equivalence of categories (Stone duality), this density property also means that we have a full and faithful functor . 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 . Hence
For example, starting from the unit interval, gives the -simplex.
On quantum effect operators:
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]
Teiko Heinosaari, Mário Ziman, Section 2 of: The Mathematical Language of Quantum Theory – From Uncertainty to Entanglement, Cambridge University Press (2011) [doi:10.1017/CBO9781139031103]
The notion of effect algebra is due to
Discussion of effect algebras in the context of categorical logic is in
Discussion in the context of quantum logic is in section 6 of
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 (hence in the classifying topos for objects) and density of Boolean algebras is in
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.
Last revised on December 1, 2023 at 08:27:10. See the history of this page for a list of all contributions to it.