Citation Context

...ion: a map between von Neumann algebras is called normal if it preserves joins of countable increasing chains. This is equivalent to preservation of countable sums of orthogonal projections, see e.g. =-=[19]-=- for details. The category of unital von Neumann algebras with normal ∗-homomorphisms will be denoted vN. Sometimes it is more appropriate to use a weaker notion of morphism. The category with von Neu...

Citation Context

...urable subset M ⊆ X . Here Pos(H) denotes the set of positive operators on H . The derivative, if it exists, is unique up to equality almost everywhere. Conditions for existence are discussed in e.g. =-=[5]-=-. Here we will only briefly state the result that we need for the remainder of this paper. The POVM A is called µ-continuous if A(M) = 0 whenever µ(M) = 0. It has bounded variation if sup n ∑ i=1 ||ϕ(...

Citation Context

...alence between the morphisms of the comma categories, the above correspondences are natural. Many POVMs occuring in physics are covariant with respect to a symmetry group or groupoid, as discussed in =-=[20, 22]-=-. For future research, it would be interesting to see how our results can be extended to the covariant setting using convolution algebras. Another possible direction would be to study the sequential c...

Citation Context

...ribe both the probabilistic and the logical aspects of quantum mechanics. An overview of the theory of effect algebras is given in [6]. The principal example of an effect algebra is the unit interval =-=[0,1]-=-. Addition serves as a partially defined binary operation, and the orthocomplement is given by x⊥ = 1− x. Another important example comes from quantum logic. An effect on a Hilbert space H is an opera...

Citation Context

...• If x⊕1 is defined, then x = 0. Effect algebras constitute a category EA, in which the morphisms are functions preserving ⊕, (−)⊥, and 0. Effect algebras originated in the study of quantum logics in =-=[7]-=-, and can be used to describe both the probabilistic and the logical aspects of quantum mechanics. An overview of the theory of effect algebras is given in [6]. The principal example of an effect alge...

Citation Context

...from X to the unit interval. In other words, Meas(X , [0,1]) is the free σ -effect module generated by the σ -algebra ΣX . Giry initiated the categorical approach to measure and integration theory in =-=[8]-=- by defining the Giry monad G on the category Meas as G (X) = σEA(ΣX , [0,1]). Thus the elements of G (X) are probability measures. A measurable map p : X → [0,1] can be integrated along a probability...

Citation Context

...phic. 136 Categorical characterizations of operator-valued measures 2. The σ -effect modules Alg(G )(DM (H), [0,1]) and E f (H) are isomorphic. Proof. 1. This is a reformulation of Busch’s theorem in =-=[3]-=-. 2. In [18] this result is proven for affine maps DM (H)→ [0,1] instead of G -algebra maps, so the statement follows because every G -algebra map is in particular affine. Since [0,1] ∼= G (2), measur...

Citation Context

...], in which Birkhoff and von Neumann propose to use the orthomodular lattice of projections on a Hilbert space. However, this approach has been criticized for its lack of generality, see for instance =-=[22]-=- for an overview of experiments that do not fit in the Birkhoff-von Neumann scheme. The operational approach to quantum physics generalizes the approach based on projective measurements. In this appro...

Citation Context

...lts. The logic of effects is useful in describing the semantics of quantum programs via weakest preconditions, as argued in [4]. A more general treatment of the logical aspects of effects is given in =-=[15]-=-. Both references use a duality between effects and convex sets to relate syntax and semantics of the logic. More generally, measurements with an arbitrary space of results can be modeled as maps from...

Citation Context

... : [0,1]×X → X , such that • r · (s · x) = (rs) · x. • If r+ s≤ 1, then (r+ s) · x = r · x⊕ s · x. • If x⊕ y is defined, then r · (x⊕ y) = r · x⊕ r · y. • 1 · x = x. Effect modules were introduced in =-=[13]-=- under the name ‘convex effect algebras’, and generalized in [18] to modules over arbitrary effect algebras with a monoid structure, rather than just over the interval [0,1]. Morphisms of effect modul...

Citation Context

...of G (X) are probability measures. A measurable map p : X → [0,1] can be integrated along a probability measure ϕ ∈ G (X) to obtain ∫ pdϕ ∈ [0,1], sometimes written as ∫ p(x)dx if ϕ is understood. In =-=[16]-=- it is shown that there is a dual adjunction between Eilenberg-Moore algebras for the Giry monad and σ -effect modules: Alg(G ) Hom(−,[0,1]) --⊥ σEModop Hom(−,[0,1]) mm (1) This gives a foundation for...

Citation Context

...effect is a predicate, or equivalently, a measurement with two possible results. The logic of effects is useful in describing the semantics of quantum programs via weakest preconditions, as argued in =-=[4]-=-. A more general treatment of the logical aspects of effects is given in [15]. Both references use a duality between effects and convex sets to relate syntax and semantics of the logic. More generally...

Citation Context

...and B : HomKl(Y,2)→ E f (K) is given by a diagram HomKl(X ,2) A HomKl(Y,2) (−)◦ f oo B E f (H) E f (K) E f (g) oo (3) in σEMod, for a measurable map f : X → G (Y ) and an isometry g : H → K. In =-=[14]-=- it is shown that there is a correspondence between POVMs and G -algebra homomorphisms DM (H)→ G (X), called statistical maps. From a categorical perspective, this can be phrased as an equivalence bet...

Citation Context

...inated in the study of quantum logics in [7], and can be used to describe both the probabilistic and the logical aspects of quantum mechanics. An overview of the theory of effect algebras is given in =-=[6]-=-. The principal example of an effect algebra is the unit interval [0,1]. Addition serves as a partially defined binary operation, and the orthocomplement is given by x⊥ = 1− x. Another important examp...

Citation Context

...s the free effect module generated by X . The situation is more subtle for σ -effect algebras, because the tensor product of two σ -effect algebras need not always exist. This problem is discussed in =-=[9]-=-. Effect algebras also occur in measure theory. A measurable space consists of a set X together with a σ -algebra of subsets of X , denoted ΣX . Measurable spaces constitute a category Meas, in which ...

Citation Context

...operties are modeled by effects A and B, then the composite test corresponds to the effect √ AB √ A, which is called the sequential product of A and B. The properties of this operation are studied in =-=[12, 11, 10]-=-. We will now define an extension of this operation to POVMs, which can be used if we want to measure two POVMs sequentially. We start by measuring a POVM A : ΣX → E f (H). The outcome of this measure...

Citation Context

...the syntax via duality. In our quantum example, the semantics is given by density matrices, since density matrices and effects are related via the duality between convex sets and effect algebras, see =-=[17]-=- for details. In the remainder of this paper, we will try to establish a similar picture for POVMs. This section considers a generalization of the duality for effects to POVMs. The duality for POVMs w...

Citation Context

..., then (r+ s) · x = r · x⊕ s · x. • If x⊕ y is defined, then r · (x⊕ y) = r · x⊕ r · y. • 1 · x = x. Effect modules were introduced in [13] under the name ‘convex effect algebras’, and generalized in =-=[18]-=- to modules over arbitrary effect algebras with a monoid structure, rather than just over the interval [0,1]. Morphisms of effect modules are morphisms of effect algebras that additionally preserve th...

Citation Context

...operties are modeled by effects A and B, then the composite test corresponds to the effect √ AB √ A, which is called the sequential product of A and B. The properties of this operation are studied in =-=[12, 11, 10]-=-. We will now define an extension of this operation to POVMs, which can be used if we want to measure two POVMs sequentially. We start by measuring a POVM A : ΣX → E f (H). The outcome of this measure...

Citation Context

...operties are modeled by effects A and B, then the composite test corresponds to the effect √ AB √ A, which is called the sequential product of A and B. The properties of this operation are studied in =-=[12, 11, 10]-=-. We will now define an extension of this operation to POVMs, which can be used if we want to measure two POVMs sequentially. We start by measuring a POVM A : ΣX → E f (H). The outcome of this measure...

Citation Context

...ctions f : X → C such that the integral ∫X | f |dµ is finite. In this case, the trace map is integration ∫ X(−)dµ . The structure of a predual can be captured abstractly by base norm spaces, see e.g. =-=[1, 21]-=-. Let V be an ordered vector space, and τ : V → C a positive linear functional. A convex subset C of V is called linearly bounded if C∩L is bounded for every line L through the origin. Let K = τ−1(1) ...