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Quantum Mathematical Integrated Information Theory: From the Classical Version

Siddharth Sharma

Published Jan 12, 2021• 0 citations
Publisher

Abstract

This paper is an attempt to give a Quantum theory of Mathematical integrated information theory which is mathematical version of integrated information theory by Masafumi Oizumi, Larissa Albantakis, Giulio Tononi. Using the definitions of Classical Mathematical Integrated Information Theory. And considering that the Quantum theory is given by the functor which maps from a category whose objects is topology to a linear category whose objects are Hilbert spaces indexed with the objects from previous category. Also we will be using the definition of conditional density matrix to define repertoire.

DOI
10.31219/osf.io/smxd9

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This paper is an attempt to give a Quantum theory of Mathematical integrated information theory which is mathematical version of integrated information theory by Masafumi Oizumi, Larissa Albantakis, Giulio Tononi. Using the definitions of Classical Mathematical Integrated Information Theory. And considering that the Quantum theory is given by the functor which maps from a category whose objects is topology to a linear category whose objects are Hilbert spaces indexed with the objects from previous category. Also we will be using the definition of conditional density matrix to define repertoire.
This paper is an attempt to give a Quantum theory of Mathematical integrated information theory which is mathematical version of integrated information theory by Masafumi Oizumi, Larissa Albantakis, Giulio Tononi [2]. Using the definitions of Classical Mathematical Integrated Information Theory [1]. And considering that the Quantum theory is given by the functor which maps from a category whose objects is topology to a linear category whose objects are Hilbert spaces indexed with the objects from previous category. Also, we will be using the definition of conditional density matrix to define repertoire. We will be also discussing the relationship between consciousness and entanglement.
The concept of quantum information is introduced as both normed superposition of two orthogonal subspaces of the separable complex Hilbert space and invariance of Hamilton and Lagrange representation of any mechanical system. The base is the isomorphism of the standard introduction and the representation of a qubit to a 3D unit ball, in which two points are chosen.The separable complex Hilbert space is considered as the free variable of quantum information and any point in it (a wave function describing a state of a quantum system) as its value as the bound variable.A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives. Then, that Hilbert space is considered as a generalization of Peano arithmetic where any unit is substituted by a qubit and thus the set of natural number is mappable within any qubit as the complex internal structure of the unit or a different state of it. Thus, any mathematical structure being reducible to set theory is representable as a set of wave functions and a subspace of the separable complex Hilbert space, and it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another. Thus, category theory is isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as above.Given any value of quantum information, i.e. a point in the separable complex Hilbert space, it always admits two equally acceptable interpretations: the one is physical, the other is mathematical. The former is a wave function as the exhausted description of a certain state of a certain quantum system. The latter chooses a certain mathematical structure among a certain category. Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information. This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa. Further, that definition is equivalent to both standard definition as the normed superposition and invariance of Hamilton and Lagrange interpretation of mechanical motion introduced in the beginning of the paper.Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: Lagrange representation and each counterpart of the pair of Hamilton representation. The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as both (privileged) reference frame and the symmetries 𝑈𝑈(1), 𝑆𝑆𝑆 (2), and 𝑆𝑆𝑆 (3) of the Standard model.
In this paper I am going to give a mathematical theory of Integrated Information Theory, using entropy as measure of information and hence, as the information distance function. Also, we will consider a set, whose open subsets are mechanisms and topology is the system, we use these two modification in the structure of Integrated Information Theory [2] and Quantum Integrated Information theory [3], to define Entropic Integrated Information Theory, we will also justify our claims to use why entropy should be use as a measure of cause/effect information and as information distance function using [1]. We will also see the relationship of entanglement with concept and conceptual information. This paper is an attempt to binds consciousness, quantum information, entanglement and quantum mechanics together.
The Global Hilbert Space Boundary Screening Theory (GHS-BST) defines physical reality as the projection of informational structures through boundary screening. This paper provides the rigorous mathematical foundations for the theory. We employ category theory to reformulate the global Hilbert space \mathcal{H}{\text{global}} and the boundary screening mechanism, and topos theory as the mathematical framework for information ontology. We construct the information category \mathcal{I}, whose objects are all possible informational structures and whose morphisms are transformations between informational structures. We prove that the five axioms of GHS-BST are consistent within this mathematical framework, and demonstrate that Hilbert space is a realization of the information category under specific representational conditions. Furthermore, we introduce the von Neumann algebraic criterion for indecomposability: \mathcal{H}{\text{global}} is indecomposable if and only if its associated operator algebra \mathcal{A}_{\text{global}} has trivial center (i.e., is a Type III factor). This provides a rigorous mathematical definition of "informational primacy." Keywords: Category Theory; Topos Theory; Information Ontology; von Neumann Algebra; Type III Factor; GHS-BST; Mathematical Foundations