# The three types of normal sequential effect algebras

Abraham Westerbaan1, Bas Westerbaan1,2, and John van de Wetering1

2University College London

### Abstract

A sequential effect algebra (SEA) is an effect algebra equipped with a $\textit{sequential product}$ operation modeled after the Lüders product $(a,b)\mapsto \sqrt{a}b\sqrt{a}$ on C$^*$-algebras. A SEA is called $\textit{normal}$ when it has all suprema of directed sets, and the sequential product interacts suitably with these suprema. The effects on a Hilbert space and the unit interval of a von Neumann or JBW algebra are examples of normal SEAs that are in addition $\textit{convex}$, i.e. possess a suitable action of the real unit interval on the algebra. Complete Boolean algebras form normal SEAs too, which are convex only when $0=1$.

We show that any normal SEA $E$ splits as a direct sum $E= E_b\oplus E_c \oplus E_{ac}$ of a complete Boolean algebra $E_b$, a convex normal SEA $E_c$, and a newly identified type of normal SEA $E_{ac}$ we dub $\textit{purely almost-convex}$.

Along the way we show, among other things, that a SEA which contains only idempotents must be a Boolean algebra; and we establish a spectral theorem using which we settle for the class of normal SEAs a problem of Gudder regarding the uniqueness of square roots. After establishing our main result, we propose a simple extra axiom for normal SEAs that excludes the seemingly pathological a-convex SEAs. We conclude the paper by a study of SEAs with an associative sequential product. We find that associativity forces normal SEAs satisfying our new axiom to be commutative, shedding light on the question of why the sequential product in quantum theory should be non-associative.

To better understand the properties and foundations of quantum theory it is useful to contrast it with hypothetical alternative physical theories and mathematical abstractions. By studying these alternatives it becomes clearer which parts of quantum theory are special to it, and which are present in any reasonable physical theory.

A framework that has been used extensively to study such alternatives is that of generalised probabilistic theories. These have built into their definition the classical concepts of probability theory, and hence they fundamentally rely on properties of real numbers. While this is certainly a useful framework, it precludes the study of physical theories that have a more exotic notion of probability.
To study systems that allow a broader notion of probability, a more general structure is needed.
This paper is about effect algebras, which generalise and abstract the set of measurement effects in a quantum system.
The study of effect algebras has become a flourishing field on its own and covers a variety of topics.
In this paper we look at effect algebras that come equipped with a 'sequential product'. This is an operation that corresponds to the process of sequential measurement where we first observe one effect and then another.

Examples of sequential effect algebras are Boolean algebras, which model classical deterministic systems, and certain convex subspaces of vector spaces, which model standard probabilistic models. A third type is given by what we dub 'almost-convex' sequential effect algebras, which resemble convex models but are somewhat weirder.
Our main result is showing that for a particularly nice class of sequential effect algebras that we call 'normal' these three types are the only possibilities.

This result is surprising because it gives a way to retrieve a dichotomy between deterministic models and probabilistic models from an abstract starting point. It might give some insight into why our physical reality needs a continuum of real numbers to describe our fundamental theories.

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### Cited by

[1] Martin Plávala, "One Measurement, Two Measurements: from Sequential Products to Convexity", Quantum Views 5, 49 (2021).

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