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The collapse of a quantum state as a signal analytic sleight of hand

Peter Morgan

Published Jan 26, 2021• 0 citations
OpenAlex

Abstract

The collapse of a quantum state is a mathematical sleight of hand that allows the construction of a joint probability density even for operators that are noncommutative. We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. A collapse picture, in which a quantum state collapses after each measurement, can therefore be equivalent to a no-collapse picture that uses a different state and different, mutually commutative, Quantum Non-Demolition operators to construct the same joint probability density for consecutive measurements. The collapse of a state is a mathematical construction that can also be a useful tool when using the Koopman formalism for classical mechanics and signal analysis.

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The collapse of a quantum state as a joint probability construction*

Peter Morgan

Abstract The collapse of a quantum state can be understood as a mathematical way to construct a joint probability density even for operators that do not commute . We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. The familiar collapse picture , in which a quantum state collapses after each measurement as a way to construct a joint probability density for consecutive measurements, is equivalent to a no-collapse picture in which Lüders transformers applied to subsequent measurements construct a quantum-mechanics—free subsystem of quantum non-demolition operators, not as a dynamical process but as an alternative mathematical model for the same consecutive measurements. The no-collapse picture is particularly simpler when we apply signal analysis to millions or billions of consecutive measurements.

arXiv:2101.10931•Journal of Physics A Mathematical and Theoretical•May 12, 2022•5 citations

Title: Sequential Quantum State Collapse – Dynamic Probability Matrix Construction

Jincheng Zhang

Sequential Quantum State Collapse explores a novel approach to understanding and predicting quantum state collapse, moving beyond traditional guesswork towards a data-driven model. This paper introduces a method for constructing quantum state collapse probabilities by analyzing the dynamic evolution of a complex system and calculating the probabilities of collapse for different states. We propose a statistical analysis framework that leverages the system's dynamics to determine the probabilities of collapse for various outcomes, offering a fundamentally different perspective than current methods. The core mechanism involves quantifying the system's evolution to determine the likelihood of each state collapsing into a particular state. This paper details the methodology, provides an initial analysis of the model's performance, and discusses potential future research directions.

•Sep 6, 2026•0 citations

Beyond Kolmogorov: Towards a Non-Commutative Measure Theory of Quantum Probability Spaces

Revista, Zen, MATH, 10

This paper explores the limitations of Kolmogorov's measure-theoretic foundation of probability in the context of quantum mechanics and proposes an extension towards a non-commutative framework. We delve into the mathematical structures underlying quantum probability spaces, focusing on operator algebras and their associated states. We analyze the challenges in defining joint probabilities for non-commuting observables and present an overview of alternative approaches based on operator algebras, quantum logic, and non-commutative integration. Furthermore, we investigate the applications of non-commutative measure theory in quantum information theory, quantum field theory, and the foundations of quantum mechanics, highlighting the potential for a more comprehensive understanding of probabilistic phenomena in quantum systems.

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Quantum State Collapse and Quantum Simulation

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Quantum state collapse is a fundamental process in quantum mechanics, leading to the probabilistic emergence of a single, definite state. This paper explores the potential of leveraging this collapse process within a quantum simulation framework, focusing on the design of an algorithm that utilizes quantum state collapse to accurately model complex quantum systems. We propose a novel approach to simulation that directly incorporates quantum state collapse as a key operational mechanism, aiming to surpass the limitations of traditional methods in simulating intricate quantum systems. The paper details the core mechanism, outlines the simulation algorithm's design, and highlights the significant advantages offered by this integrated approach. We argue that this combined methodology holds the key to unlocking more precise and efficient simulations of complex quantum phenomena.

•Sep 6, 2026•0 citations

2. Basic quantum physics

Arnold Neumaier

Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)

De Gruyter eBooks•Oct 24, 2019•0 citations

Quantum Information Theory: Exploring the Influence of Nonlinear Collapse on Quantum State Evolution and Measurement

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Quantum information theory explores the fundamental principles governing the manipulation and utilization of quantum states. A crucial aspect of this field is the concept of non-linear collapse, a process that fundamentally alters the quantum state upon measurement. This paper investigates the impact of non-linear collapse on quantum state evolution and measurement results, specifically focusing on its implications for quantum algorithm design. We propose a novel quantum algorithm leveraging the principles of non-linear collapse to achieve enhanced quantum information processing efficiency. This work will detail the development of a quantum state evolution model incorporating non-linear collapse, highlighting its potential to accelerate quantum computation and offer a new approach to quantum information handling. This research represents a significant step towards refining quantum algorithms and pushing the boundaries of quantum information processing.

•Sep 7, 2026•0 citations
The collapse of a quantum state is a mathematical sleight of hand that allows the construction of a joint probability density even for operators that are noncommutative. We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. A collapse picture, in which a quantum state collapses after each measurement, can therefore be equivalent to a no-collapse picture that uses a different state and different, mutually commutative, Quantum Non-Demolition operators to construct the same joint probability density for consecutive measurements. The collapse of a state is a mathematical construction that can also be a useful tool when using the Koopman formalism for classical mechanics and signal analysis.
Abstract The collapse of a quantum state can be understood as a mathematical way to construct a joint probability density even for operators that do not commute . We can formalize that construction as a non-commutative, non-associative collapse product that is nonlinear in its left operand as a model for joint measurements at time-like separation, in part inspired by the sequential product for positive semi-definite operators. The familiar collapse picture , in which a quantum state collapses after each measurement as a way to construct a joint probability density for consecutive measurements, is equivalent to a no-collapse picture in which Lüders transformers applied to subsequent measurements construct a quantum-mechanics—free subsystem of quantum non-demolition operators, not as a dynamical process but as an alternative mathematical model for the same consecutive measurements. The no-collapse picture is particularly simpler when we apply signal analysis to millions or billions of consecutive measurements.
Sequential Quantum State Collapse explores a novel approach to understanding and predicting quantum state collapse, moving beyond traditional guesswork towards a data-driven model. This paper introduces a method for constructing quantum state collapse probabilities by analyzing the dynamic evolution of a complex system and calculating the probabilities of collapse for different states. We propose a statistical analysis framework that leverages the system's dynamics to determine the probabilities of collapse for various outcomes, offering a fundamentally different perspective than current methods. The core mechanism involves quantifying the system's evolution to determine the likelihood of each state collapsing into a particular state. This paper details the methodology, provides an initial analysis of the model's performance, and discusses potential future research directions.
This paper explores the limitations of Kolmogorov's measure-theoretic foundation of probability in the context of quantum mechanics and proposes an extension towards a non-commutative framework. We delve into the mathematical structures underlying quantum probability spaces, focusing on operator algebras and their associated states. We analyze the challenges in defining joint probabilities for non-commuting observables and present an overview of alternative approaches based on operator algebras, quantum logic, and non-commutative integration. Furthermore, we investigate the applications of non-commutative measure theory in quantum information theory, quantum field theory, and the foundations of quantum mechanics, highlighting the potential for a more comprehensive understanding of probabilistic phenomena in quantum systems.
Quantum state collapse is a fundamental process in quantum mechanics, leading to the probabilistic emergence of a single, definite state. This paper explores the potential of leveraging this collapse process within a quantum simulation framework, focusing on the design of an algorithm that utilizes quantum state collapse to accurately model complex quantum systems. We propose a novel approach to simulation that directly incorporates quantum state collapse as a key operational mechanism, aiming to surpass the limitations of traditional methods in simulating intricate quantum systems. The paper details the core mechanism, outlines the simulation algorithm's design, and highlights the significant advantages offered by this integrated approach. We argue that this combined methodology holds the key to unlocking more precise and efficient simulations of complex quantum phenomena.
Quantum physics consists of a formal core that is universally agreed upon (basically being a piece of mathematics with suggestive names for concepts matching related concepts of experimental culture) and an interpretational halo (trying to make this relation more precise) that remains highly disputed even after more than 90 years of modern quantum physics. The latter is the subject of the interpretation of quantum mechanics (see Part II), where many interpretations coexist and compete for the attention of those interested in what quantum physics really means. In this chapter, an axiomatic introduction to the undisputed formal core of quantum physics is given. The axioms described present nonrelativistic quantum statistical mechanics in the Schrödinger picture. The relativistic case is outside the scope of these axioms, as it must be treated by quantum field theory; see Chapter 7. As in any axiomatic setting (necessary for a formal discipline), there are a number of different but equivalent sets of axioms or postulates that can be used to define formal quantum physics. Since they are equivalent, their choice is a matter of convenience. The choice presented here is a formulation featuring three aspects: It emphasises the similarity of quantum mechanics and classical mechanics, gives most direct access to statistical mechanics, and is free from allusions to measurement. The reason for the first is that this similarity serves as a guide to meaning. From the axiomatic setting for quantum mechanics presented below, classical mechanics (in the Koopman setting discussed in Section 7.10) is obtained by simply restricting the operators to be diagonal, so that all operations happen pointwise on the diagonal elements. Thus, multiplication is commutative, and one can identify operators and functions. In particular, the density operator degenerates into a probability density. Therefore, the quantum case appears as a generalization of the classical case allowing for noncommutativity, so that both q-observables and the density operator are (usually infinite-dimensional) operators. The reason for the second is that statistical mechanics is the main tool for applications of quantum physics to the macroscopic systems we are familiar with. The reason for the third is that realistic measurements constitute a complex process involving macroscopic detectors, hence should be explained by quantum statistical mechanics rather than be part of the axiomatic foundations themselves. (This is in marked contrast to other foundations, and distinguishes the present system of axioms.)
Quantum information theory explores the fundamental principles governing the manipulation and utilization of quantum states. A crucial aspect of this field is the concept of non-linear collapse, a process that fundamentally alters the quantum state upon measurement. This paper investigates the impact of non-linear collapse on quantum state evolution and measurement results, specifically focusing on its implications for quantum algorithm design. We propose a novel quantum algorithm leveraging the principles of non-linear collapse to achieve enhanced quantum information processing efficiency. This work will detail the development of a quantum state evolution model incorporating non-linear collapse, highlighting its potential to accelerate quantum computation and offer a new approach to quantum information handling. This research represents a significant step towards refining quantum algorithms and pushing the boundaries of quantum information processing.