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arXiv:2004.12749 · Quantum

The three types of normal sequential effect algebras

Abraham Westerbaan, Bas Westerbaan, John van de Wetering

Published Dec 24, 2020• 4 citations

Abstract

A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C ∗ -algebras. A SEA is called 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 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=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb , a convex normal SEA Ec , and a newly identified type of normal SEA Eac we dub 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.

DOI: 10.22331/q-2020-12-24-378

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A sequential effect algebra (SEA) is an effect algebra equipped with a sequential product operation modeled after the Lüders product (a,b)↦aba on C ∗ -algebras. A SEA is called 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 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=Eb⊕Ec⊕Eac of a complete Boolean algebra Eb , a convex normal SEA Ec , and a newly identified type of normal SEA Eac we dub 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.

It has already been established that the properties required of an abstract sequential product as introduced by Gudder and Greechie are not enough to characterise the standard sequential product a◦b=aba on an operator algebra. We give three additional properties, each of which characterises the standard sequential product on either a von Neumann algebra or a Euclidean Jordan algebra. These properties are (1) invariance under the application of unital order isomorphisms, (2) symmetry of the sequential product with respect to a certain inner product, and (3) preservation of invertibility of the effects. To give these characterisations, we first have to study convex σ-sequential effect algebras. We show that these objects correspond to unit intervals of spectral order unit spaces with a homogeneous positive cone.

A convex sequential effect algebra (COSEA) is an algebraic system with three physically motivated operations, an orthogonal sum, a scalar product and a sequential product. The elements of a COSEA correspond to yes-no measurements and are called effects. In this work we stress the importance of contexts in a COSEA. A context is a finest sharp measurement and an effect will act differently according to the underlying context with which it is measured. Under a change of context, the possible values of an effect do not change but the way these values are obtained may be different. In this paper we discuss direct sums and the center of a COSEA. We also consider conditional probabilities and the spectra of effects. Finally, we characterize COSEA's that are isomorphic to COSEA's of positive operators on a complex Hilbert space. These result in the traditional quantum formalism. All of this work depends heavily on the concept of a context.

We study the sequential product the operation p∗q=pqp on the set of effects, [0, 1]𝒜, of a von Neumann algebra 𝒜 that represents sequential measurement of first p and then q. In their work [J. Math. Phys. 49(5), 052106 (2008)], Gudder and Latrémolière give a list of axioms based on physical grounds that completely determines the sequential product on a von Neumann algebra of type I, that is, a von Neumann algebra ℬ(ℋ) of all bounded operators on some Hilbert space ℋ. In this paper we give a list of axioms that completely determines the sequential product on all von Neumann algebras simultaneously (Theorem 4).

A quantum effect is an operator

A

on a complex Hilbert space

H

that satisfies

0\leq A\leq I

,

{\cal E} (H)

is the set of all quantum effects on

H

. In 2001, Professor Gudder and Nagy studied the sequential product

A\circ B=A^{1/2}BA^{1/2}

of

A, B\in {\cal E}(H)

. In 2005, Professor Gudder asked: Is

A\circ B=A^{1/2}BA^{1/2}

the only sequential product on

{\cal E} (H)

? Recently, Liu and Wu presented an example to show that the answer is negative. In this paper, firstly, we characterize some algebraic properties of the abstract sequential product on

{\cal E} (H)

; secondly, we present a general method for constructing sequential products on

{\cal E} (H)

; finally, we study some properties of the sequential products constructed by the method