An effect-theoretic reconstruction of quantum theory

Abstract

An often used model for quantum theory is to associate to every physical system a C∗-algebra. From a physical point of view it is unclear why operator algebras would form a good description of nature. In this paper, we find a set of physically meaningful assumptions such that any physical theory satisfying these assumptions must embed into the category of finite-dimensional C∗-algebras. These assumptions were originally introduced in the setting of effectus theory, a categorical logical framework generalizing classical and quantum logic. As these assumptions have a physical interpretation, this motivates the usage of operator algebras as a model for quantum theory.In contrast to other reconstructions of quantum theory, we do not start with the framework of generalized probabilistic theories and instead use effect theories where no convex structure and no tensor product needs to be present. The lack of this structure in effectus theory has led to a different notion of pure maps. A map in an effectus is pure when it is a composition of a compression and a filter. These maps satisfy particular universal properties and respectively correspond to `forgetting' and `measuring' the validity of an effect.We define a pure effect theory (PET) to be an effect theory where the pure maps form a dagger-category and filters and compressions are adjoint. We show that any convex finite-dimensional PET must embed into the category of Euclidean Jordan algebras. Moreover, if the PET also has monoidal structure, then we show that it must embed into either the category of real or complex C∗-algebras, which completes our reconstruction.

The three types of normal sequential effect algebras

“…What is also known is that when the normal sequential product satisfies for all idempotents p and q the identity (p • q) 2 = p •(q • p) and the implication ω(q) = 1 =⇒ ω(q • p) = ω(p) for any state ω, then the space must be a JB-algebra [55]. There are many elegant characterisations of Jordan algebras [3,4,42] that in turn can be used to characterise quantum theory [5,54]. If the existence of a sequential product also characterises Jordan algebras this would give another pathway to understanding the fundamental structures of quantum mechanics.…”

“…Convex effect algebras have been well-studied, see e.g. [19,32,36,53,54]. In the literature on effectus theory, convex effect algebras are also called effect modules [8,9].…”

Spacetime symmetries and the qubit Bloch ball: a physical derivation of finite dimensional quantum theory and the number of spatial dimensions

“…We use the definition (38) of Λ prest (p), for all p ∈ Π prest . We complete the proof by showing (35). From the definition (32) of P little 0,prest (a, x, Λ, p), we have…”

“…for all O = 1 0 0 Õ with Õ ∈ SO(n), for all a, x ∈ R 1+n and for all p ∈ Π prest . Thus, the transformations P little 0,prest (a, x, Λ, p) generate the whole little group SO(n), for all a, x ∈ R 1+n , p ∈ Π prest and Λ ∈ L. Therefore, it follows from (30), (33), (34) and (35), and from the fact that Λ, Λ ′ ∈ L are arbitrary that R st prest is a representation of the little group SO(n), as claimed.…”

Reconstructing quantum theory from diagrammatic postulates

“…More recently, a third wave of reconstruction attempts took off [14,15,116], which can be seen as resurrecting the mathematical spirit of the first wave, while still embracing the principled underpinning that guides the second wave. Recently, there have been various other reconstructions of quantum theory from a variety of perspectives, for example [24,66,73,74,92,109,113].…”