On the properties of spectral effect algebras

Jenčová, Anna; Plávala, Martin

doi:10.22331/q-2019-06-03-148

Cited by 11 publications

(8 citation statements)

References 17 publications

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“…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].…”

Section: Propositionmentioning

confidence: 99%

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The three types of normal sequential effect algebras

Westerbaan

Wetering

2020

Quantum

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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.

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“…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].…”

Section: Propositionmentioning

confidence: 99%

“…Effect algebras generalise and abstract the unit interval of effects in a C * -algebra. The study of effect algebras has become a flourishing field on its own [7,10,14,24,27,35,45] and covers a variety of topics [34,36,47,51]. An effect algebra that has an action of the real unit interval on it is called a convex effect algebra [25].…”

Section: Introductionmentioning

confidence: 99%

The three types of normal sequential effect algebras

Westerbaan

Wetering

2020

Quantum

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“…A universal computer is a 26 Conjugate coding (Wiesner, 1983) of position and velocity are mapped onto firing rates and co-firing rates of neural spike trains resulting in an uncertainty principle for neural coding (Grgurich and Blair, 2019). 24 There can be either only one or uncountably many contexts in a spectral effect algebra (Jencova and Plavala, 2018). A convex spectral effect algebra is either classical (the classical probability theory) or has infinite contexts (the non-classical probability theory described in this paper.)…”

Section: Cortical Operators Include Unitary Operatorsmentioning

confidence: 99%

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Jones¹

2020

Preprint

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A model of computation describes how units of information and an organizational architecture transform input to output. In the cerebral cortex the unit of information is a normalized collection of neuronal membrane potentials representing a vector of complex amplitudes. The organizational architecture of the cortex allows for programming operator matrices in dendritic arborizations on vectors of complex amplitudes. Three well-documented facts are sufficient to define the cortical model of computation: first, cortical dendritic inputs to compute membrane potentials are, in general, complex numbers (amplitudes) that may be configured as state vectors in complex Hilbert spaces; second, normalization is a canonical neural computation, specifically, for vectors of cortical membrane amplitudes representing orthogonal states; and, third, dendritic arborizations are programmable devices in a universal sense for mathematical and logical functions. These three facts, well studied and accepted for years, are all that is needed for a simple, well-defined, and well-known model of computation. A great deal of research is presently devoted to understanding particular neural algorithms, circuits, and programming. Understanding the algorithm that a particular circuit implements is not to be confused with identifying the much more basic underlying model of computation. The analogy for classical computing is that the classical model of computing stores and manipulates information fundamentally in the form of bits – essentially a finite set of possible states, typically, {0,1}. The primitive classical container of information is analogous to a two-state switch. Nevertheless, sophisticated computational circuitry and algorithms are developed on that simple classical realization. In contrast to the finite set of states found in the fundamental unit of classical information, the fundamental units of information in the cerebral cortex form a model of computation over the complex field in normalized sets of mutually inhibitory amplitudes. This means that the cortex represents information in a fundamental container of information (a vector of amplitudes in membrane potentials) that can encode discrete states over a continuum of values (no doubt down to some constant.) A well-known result from probability theory and standard neuroimaging analysis argue that these mutually inhibitory fundamental units of information may normalize complex amplitudes under the 2-norm. Sufficiently sophisticated computational means are available in the cortex to program unitary operators, state reduction and expansion operators and tensor products that operate on vectors of normalized membrane amplitudes. This model of computation is typically viewed as a probabilistic or predictive model of computation. Again, the model of computation is not a particular circuit or implementation of any one algorithm. The model of computation is the arena in which cortical circuits and algorithms are implemented. The fundamental arena then for cortical computations is a field of complex amplitudes normalized for probabilistic computation.

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“…The answer to this question is negative. In fact, if E has more than one context, then it has uncountably many [17].…”

Section: Convex Sequential Effect Algebrasmentioning

confidence: 99%

Contexts in Convex and Sequential Effect Algebras

Gudder

2019

Electron. Proc. Theor. Comput. Sci.

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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.

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On the properties of spectral effect algebras

Cited by 11 publications

References 17 publications

The three types of normal sequential effect algebras

The three types of normal sequential effect algebras

The Cerebral Cortex Realizes a Universal Probabilistic Model of Computation in Complex Hilbert Spaces

Contexts in Convex and Sequential Effect Algebras

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